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dissertation entitled

MANY-ELECTRON TUNNELING IN A MAGNETIC FIELD

presented by

TATYANA O. SHARPEE

has been accepted towards fulﬁllment
of the requirements for

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MANY-ELECTRON TUNNELING IN A MAGNETIC FIELD
By

Tatyana Sharpee

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2001

ABSTRACT
MANY-ELECTRON TUNNELING IN A MAGNETIC FIELD
By

Tatyana Sharpee

The dissertation is devoted to the study of tunneling decay in a magnetic ﬁeld. In
its standard form, the semiclassical solution of the decay problem relies on tunneling
trajectories in real space with purely imaginary momenta and time. Because of the
broken time-reversal symmetry, trajectories of such form do not exist in a magnetic
ﬁeld. A semiclassical solution is presented which is valid for an arbitrary magnetic
ﬁeld and a three-dimensional potential, so that there is no need to treat magnetic
ﬁeld or some part of a potential as a perturbation. The decay rate and the outcoming
wave packet have been found from the analysis of the set of Hamiltonian trajectories
and its singularities in the complex phase space. A path-integral formulation for the
tunneling decay problem in a magnetic ﬁeld is provided as well.

The developed semiclassical solution to the problem of tunneling decay in a mag-
netic ﬁeld was used to analyze tunneling from a strongly correlated system of in-
teracting electrons. We show that the electron—electron interaction in a low density
two—dimensional system can affect the rate of out-of the layer tunneling exponentially.

The strongest and most interesting effects arise in the presence of a magnetic ﬁeld

parallel to the layer. The tunneling rate becomes exponentially larger than that in the
single-electron approximation. The physical mechanism is a dynamical Mossbauer-
type recoil, in which the in-plane Hall momentum of the tunneling electron is partly
transferred to the electron system as a whole. The interrelation between the char-
acteristic rate of momentum exchange between electrons (plasma frequency) and the
imaginary time of motion under the barrier determines what portion of the total mo-
mentum is transferred in a recoil-free way. The remaining part of the Hall momentum
of the tunneling electron corresponds to excitation of phonons. Explicit results are
obtained assuming that the electrons form a 2D Wigner crystal.

We show that, at higher temperatures, there is a possibility that the magnetic ﬁeld
parallel to the layer will increase, rather than suppress the out-of—plane tunneling
rate. The B-enhanced tunneling allows to control the tunneling rate over several
orders of magnitude just by changing magnetic ﬁeld and temperature, without altering
parameters of the tunneling barrier. Magnetic ﬁeld can also induce switching between
tunneling from different intra-well states, as well as switching from escape via over—
barrier activation to tunneling escape, and vice versa.

The results, obtained with no adjustable parameters, were compared with data on
tunneling from a correlated electron system on liquid helium surface. They are in both
qualitative and quantitative agreement with the experimental data in a broad range
of magnetic ﬁelds and temperatures. Therefore, tunneling experiments in a magnetic
ﬁeld can be used as a relatively simple and direct way to reveal electron correlations
in other 2D systems, such as those in semiconductor heterostructures. The range of

parameters for the tunneling experiments in heterostructures was provided.

T0 Brian

iv

ACKNOWLEDGMENTS

I write with the deepest gratitude to my teacher, Professor Mark Dykman, whose
extreme patience and attention to my problems cannot be underestimated. I can only
hope to be worthy of his efforts.

I greatly appreciate interesting conversations and much valuable advice given to
me by Prof. Philip Platzman and Prof. Leonid Pryadko. It was a pleasure to be in
one working group with Frank Kuehnel, Vadim Smelyanskiy, and Alex Zhukov. I
am indebted to Professors R. Brock, P. Duxbury, S. D. Mahanti, M. Thorpe, and
V. Zelevinsky for teaching interesting classes that helped me understand more of the
physics around. I am very thankful for the time and effort of Professors N. Birge,
P. Danielewicz, and W. Tung spent in serving on my guidance committee.

My life during the years spent in working on this dissertation was enlighten by
attentive kindness of many people, all of whom I thank. Of course, all of this would

not come to be without the support of my family.

TABLE OF CONTENTS

List of Figures viii
1 Introduction 1
2 Semiclassical solution of a 1D tunneling problem 12
3 Multidimensional tunneling decay in a magnetic ﬁeld 20
3.1 Tunneling exponent .............................. 23
3.2 The action manifold ............................. 26
3.2.1 Branching lines - caustics ......................... 26
3.2.2 Local analysis near caustics ........................ 28
3.2.3 Projection onto real space ......................... 36
3.2.4 Exactly solvable non-symmetric model .................. 41
3.3 The path-integral formulation in a magnetic ﬁeld ............. 45
3.4 Summary ................................... 49
4 Tunneling transverse to a magnetic ﬁeld from correlated 2D electron
systems. - 51
4.1 The model: tunneling from a harmonic Wigner crystal .......... 54
4.2 A many-body WKB approximation ..................... 58
4.2.1 General formulation ............................ 58
4.2.2 The initial conditions ............................ 61
4.2.3 A three-segment optimal trajectory .................... 63
4.3 The tunneling exponent ........................... 67
4.3.1 Zero temperature limit ........................... 68
4.3.2 High temperatures and small phonon frequencies ............ 72
4.4 Effect of in-plane conﬁnement on the tunneling rate ............ 74
4.4.1 The Einstein approximation for a Wigner crystal ............ 74
4.4.2 Triangular barrier ............................. 76
4.4.3 Square barrier ................................ 81
4.5 Summary ................................... 84
5 Magnetic-ﬁeld—enhanced tunneling 87
5.1 General properties of the transition ..................... 89
5.1.1 The temperature of crossover for small magnetic ﬁelds ......... 89
5.1.2 Upper temperature limit for enhancement for small magnetic ﬁelds . . 90
5.1.3 Field-induced switching between tunneling from the ground state, ex-
cited states, and over-barrier activation ............. 92

vi

5.2 Tunneling enhancement for the Einstein model of a Wigner crystal . . . 93
5.2.1 Smooth potentials: ﬁeld-induced tunneling enhancement and switching

from activation to tunneling ................... 93
5.2.2 Square barrier: ﬁeld-induced crossover to thermal activation ...... 97
5.3 Summary ................................... 100
6 Comparison with experimental data on tunneling from helium sur-
face 102
6.1 The tunneling potential for electrons on helium .............. 103
6.2 Exponent of the tunneling rate ....................... 105
6.3 The prefactor ................................. 112
7 Conclusions 115
APPENDICES 120

A The instanton method in a magnetic ﬁeld: analysis of the prefactor 121
B Many-electron inﬂuence functional Ree[z] at zero temperature 128
C Square barrier: calculation of the tunneling exponent. 131

Bibliography 135

vii

1.1

1.2

1.3

3.1

LIST or FIGURES

Tunneling in a two-dimensional potential U (:13, z) transverse to a magnetic
ﬁeld B pointing in the y direction. Initially the particle is localized in a
metastable state behind the barrier, with energy E. In contrast to the
case B = O, a particle emerges from the barrier with a ﬁnite velocity,
and therefore the exit point is located away from the line U (r) = E. .

The geometry of tunneling from a correlated 2DES transverse to a mag-
netic ﬁeld; electrons vibrate in the plane with frequencies of the order
of the plasma frequency wp. .......................

Magnetic ﬁeld induced lowering of the tunneling barrier by thermal in-
plane motion (schematically; the lowering is superimposed on the mag-
netic barrier for T = 0). The effective electric ﬁeld 8 is determined by
the T-dependent optimal in-plane velocity, 8 = vopt x B / c .......

(a) Complex t plane for integrating the Hamiltonian equations (3.3) in the
escape problem. The line Im t = const corresponds to the classical tra-
jectory of outgoing electron. (b) The classical trajectory on the (:r, 2)
plane. The solid lines in (a) and (b) show the range of Re t where
the amplitude of the propagating wave exceeds the amplitude of the
decaying underbarrier wave function, and the corresponding “visible”
part of the trajectory. The escape occurs at the point where classical
trajectory intersects the anti-Stokes line (thin solid line in (b)). Note
that particle emerges from under the potential barrier with ﬁnite veloc-
ity and for :I: 75 0. Thus the initial symmetry x -—> —a: is broken by the
magnetic ﬁeld. The data refer to the potential (3.17) with (.0070 = 1.2
and worn = 1.2, time in (a) is in the units of To = 2mL/7 ........

3.2 The complex tunneling trajectories (solid lines) and the complex caustic

surface (dashed line) are shown in the (Re z, Im 2:) plane. The tunnel-
ing trajectory (2) reaches the clasical trajectory of an escaped particle
for Im a: = 0. At this point (Shown by empty circle) the momentum is
real along the trajectory (2). Even though other trajectories (1) and
(3) cross the line of Im :1: = 0, it can be seen from the ﬁgure that
Im pz 76 0 for Im 51: = 0. Therefore these trajectories do not touch the
classical trajectory. Note that the point z = 26 where the caustic goes
through real space is not the point where the tunneling trajectory (2)
becomes real. ...............................

viii

24

3.3

3.4

3.5

3.6

3.7

3.8

Sectors of complex z’-plane with different asymptotic behavior of the wave
function around a 1D turning point with a ﬁrst order zero. The anti-
Stokes (solid) lines correspond to values of 9, where both asymptotes
have equal amplitudes, 6 = 0, 27r/3, 47r/3 .................

Complex 2’ plane perpendicular to the caustic surface at z’ = 0. The anti-
Stokes lines (solid) are located at arg z’ = 0, 27r/ 3, 47r/ 3. They divide
regions where the solution is exponentially large and small. If there are
two solutions on the anti-Stokes line, the one that was exponentially
smaller in one region becomes exponentially larger after crossing the
line. The subscripts s and d denote whether the solution is exponen-
tially small or large, respectively, in a particular region. The dashed
lines show Stokes lines, where the difference between the exponentially
small and large solution is maximal. The wavy line shows the branch
cut ......................................

T wo branches of the action on the symmetry axis a: = 0 as a function of
the tunneling coordinate 2 before the branching point, for the same
parameter values as in Fig. 3.1. ImS is shown before the branching
point zC in the real space. The vicinity of the cusp 2C is zoomed in the
inset to show that the upper branch is nonmonotonic. Its extremum
at zm lies on the classical trajectory of the escaping particle shown in
Fig. 3.1(b). However, the particle emerge from the barrier for z > zm
and :1: ¢ 0 ..................................

Cross sections of function ImS for constant .2 near the branching point zc:
(a) zm < z < 26; (b) 2 = 20; (c) z > 26. The parameter values are the
same as in Figs. 31,35. The solid line shows the branches of Im S
that determine the exponent of |I,/)|. The minima of the branch 2 lie on
the classical trajectory shown in Fig. 3.1(b). ..............

The classical trajectory of the escaping particle for potential (3.20) with
a = 0.5 and same values of parameters wcro and won, as in Figs. 3.1,
3.5, 3.6. The cross marks the branching point for function ImS - the
point where caustic goes through the real space. The anti-Stokes line
starts from the branching point. The escape takes place at the point
where classical trajectory intersects the anti-Stokes line. Note that the
trajectory lies away from the line of zero velocity U (:r, z) = E. . . . .

Cross-section of function Im S for constant 2 near a branching point
(are, zc): (a) zm < z < zc; (b) .2 = zc; (c) z > zc. The tunneling poten-
tial is (3.20) with the symmetry-breaking parameter a = 0.5, and other
parameters are the same as in Figs. 3.1, 3.5, 3.6, so that won) 2 1.2
and ono = 1.2. Although the symmetry :1: —> —x is no longer present,
all of the main features of the solution remain the same. The minima
of the branch 2 lie on the classical trajectory shown in Fig. 3.7.

ix

34

35

38

40

43

44

4.1 The optimal trajectories of the tunneling electron Z(T) and of one of the
vibrational modes 1911(7) for ,8 > 27', (a) and ,8 < 27, (b). The numer-
ical data refer to the Einstein model of the Wigner crystal, pH is the
p—component of the vibrational momentum in the Hall direction 2 x B.
The arrows Show the direction of motion along the optimal trajectory
when [3 < 27,. The tunneling potential is of the form (4.38), with di-
mensionless cyclotron frequency wcro = 2.0, where To = 2mL/7 is the
imaginary transit time for B = 0. The phonon frequency is wpro = 1.0. 66
4.2 The dependence of the tunneling rate at zero temperature on mag-
netic ﬁeld, W = W(B)/W(0). The curves 1 to 4 refer to wpro =
0, 0.2, 0.4, 0.6. Magnetic ﬁeld eliminates Single-electron tunneling for
won, 2 1 (cf. curve 1). Inset: tunneling exponent vs in-plane fre—
quency cup for (1)273 = 1.0, 2.0, 3.0 (curves a,b,c). ............ 78
4.3 The tunneling exponent in the ground state for a triangular potential bar-
rier (4.38) as a function of the phonon frequency (up in the Einstein
model of the Wigner crystal for wcro = 2. The time To = mL/y is the
duration of tunneling for B = 0 and T = 0. The curves 1 to 3 refer to
reciprocal temperatures B/TO = 7, 5, 3. The dashed line is the result
of the direct variational method, with one variational parameter 7}.
The relative importance of many-electron effects is demonstrated in
the inset. Here, the difference between the full many-electron tunnel-
ing exponent and that obtained in the single-electron approximation
is plotted as a function of inverse temperature. The many—electron
tunneling exponent was calculated for wpro = 3.0. ........... 80
4.4 Exponent of the tunneling rate —R from a 2D WC in a semiconductor
heterostructure as a function of scaled electron density (2 = x/2u‘rro
(To = mL/y). Electron correlations increase the tunneling exponent
both for B = 0 (dashed line) and in the presence of magnetic ﬁled
(solid line refers to were = 1.0). With increasing 9 the tunneling rate
in the magnetic ﬁeld approaches the zero-ﬁeld line. Inset (a): relative
tunneling rate W = W(B)/W(0) vs magnetic ﬁeld for ([270 =' 0.5.
Inset (b): electron potential with (bold line) and without (thin line)
the reduction of the tunneling barrier due to the effect of electron
correlations. ................................ 83

5.1 The dependence of the tunneling exponent R(B) 5 129(8) on the magnetic
ﬁeld (4.39) for wpro = 1 / 3 near the crossover temperature BC x 1.6770
(5.4). The curves 1 to 3 correspond to (13’ — 133/70 2 0.2,0, —0.3 . . . 94

5.2 Magnetic ﬁeld induced switching from activation (a) and from tunneling
from the excited state (b) to tunneling from the ground state, for
wpro = 1 / 3. In (a), there is only one intrawell state in the potential well
U (z), and the transition to activation for B = 0 occurs for B/To = 4/3.
The curves 1, 2 correspond to (B — BC)/TO = —0.35, —0.4. In (b), the
position E2 of the excited level (n = 2) is chosen at 0.272/2m below
the barrier top (E1 = 0). The temperature is chosen at (B — ﬂc)/TO =
—0.16, so that for B = 0 the system tunnels from the excited state.
The observable (smaller) tunneling exponents for a given B are shown
with bold lines, whereas dashed lines Show the bigger exponents, which
correspond to smaller tunneling rates ................... 95
5.3 The logarithm of the escape rate R(B) compared to its B = 0 value
R(O) = 230(E9) E 27L, which is determined by tunneling through the
square barrier (5.5). Curves 1-4 correspond to (B — 53/70 = 3,4,5,
with To :2 mL/ry, and BC 2 for chosen (up 2 1/270. AS wcro increases,
there occurs a transition from tunneling to thermal activation. . . . . 99

6.1 The relative rate of electron tunneling from helium surface W(B)/W(O)

as a function of the magnetic ﬁeld B for the electron density n =

0.8 x 108cm“2 and the calculated pulling ﬁeld E; = 24.7 V/cm (solid

curve). Solid lines show how the theory compares to the experimental

data [34]. The errorbars show the uncertainty in the theoretical values

due to the uncertainty in the parameters of the experiment. ..... 108
6.2 The rate of electron tunneling from helium surface W(B) as a function of

the magnetic ﬁeld B for the electron density n = 0.8 x 108cm‘2 and

the calculated pulling ﬁeld EL 2 24.7 V/cm (solid curve). Solid line

is the theoretical calculation for T = 0. The experimental data are

taken from [34] for T = 0.04 K. For such low temperature, predictions

of ﬁnite and zero temperature theory are very close to each other, as

can be noted by comparing with Fig. 6.1, where the ﬁnite-temperature

curve is given. The error bars show the uncertainty in the theoretical

values due to the uncertainty in the parameters of the experiment.

The dashed curve is the calculation [34] for T = 0.04 K without inter-

electron momentum exchange. ...................... 109
6.3 The rate of electron tunneling from the helium surface W(O) for B = 0 as a

function of the electron density. The clots Show the experimental data

[34]. The pulling ﬁeld 8i for n —> 0 is calculated for the parameters

used in the experiment to be 26.7V / cm. ................ 114

xi

Chapter 1

Introduction

Tunneling is a basic quantum phenomenon. As soon as a particle is described by
a wave function, there is a ﬁnite probability to ﬁnd it in regions of Space that are
classically inaccessible to it, that is where the potential energy U (r) is bigger than
the total energy E of the particle. Moreover, if this classically forbidden region has a
ﬁnite width, then there is a ﬁnite probability to classically observe the particle behind
the barrier.

Tunneling lies at the core of many physical and chemical phenomena ranging
from alpha-particle decay in nuclear physics [1, 2], ﬁeld ionization of neutral atoms
[3], to tunnel splitting of molecular spectra [4] and scanning tunneling microscopy in
condensed matter physics. In particular, Fowler and Oppenheimer [5] Showed that
tunneling could explain cold emission of electrons from a metal, a phenomenon that
remained unexplained since Lilienfeld [6] discovered it in 1922.

Over the course of time much progress has been made in solving various tunneling

problems. The semiclassical approximation turned out to be particularly useful in

solving tunneling problems, because the general solution can be obtained for an arbi-
trary potential, as long as it is sufficiently smooth [7]. The semiclassical approxima-
tion was originally used by Wentzel, Kramers, Brillouin [8] to solve a one—dimensional
Schriidinger equation, and since then the method is called the WKB approxima-
tion. In the classically allowed region, the motion of a particle is semiclassical if its
de Broglie wavelength /\D = h/ p is much smaller than the characteristic length of the
potential. A semiclassical approximation can also be applied to describe the wave
function in a classical forbidden region. Here, the appropriate condition is that the
decay length is much smaller than the characteristic length of the potential [the decay
length can be thought of as an imaginary part of a de Broglie wave length, which
becomes imaginary in the region where there are no classically propagating solutions].
In other words, the tunneling barrier should be much wider than the particle’s decay
length. Therefore, the tunneling rate obtained in the semiclassical approximation
will be exponentially small. A semiclassical solution of a one-dimensional tunneling
problem is discussed in Chapter 2.

Numerous physical applications, chemical reactions being one of them, stimulated
the extension of the semiclassical method to ﬁnite temperatures and beyond the one-
dimensional approximation [9]-[15]. It is also very important to understand how
tunneling occurs in cases where the particle motion is coupled to the bath [16] and
where the underlying classical dynamics used to construct the semiclassical solution
becomes chaotic [17]-[21].

A magnetic ﬁeld can have a strong effect on the tunneling rate of charged parti-

cles. The exponential increase of resistance in semiconductors with increasing mag-

netic ﬁeld has been known for years [22]. The conductance mechanism there is that
of electron hopping between sites localized on defects. The magnetic ﬁeld B leads
to an exponential suppression of tails of the wave functions in the direction perpen-
dicular to B, thus leading to an exponentially smaller overlap, and ultimately, to
the exponentially smaller conductance value. Recently, this effect was used to probe
two-dimensional electron systems (2DESS) in semiconductor heterostructures [23]-[33]
and on a helium surface [34]. In the case of electron tunneling from a 2DES, as we
shall Show the rate of tunneling transverse to the magnetic ﬁeld is very sensitive not
only to the value of magnetic ﬁeld, but also to electron correlations and temperature.

However, despite its interest and generality, even the problem of single-particle
decay in a magnetic ﬁeld lacked a semiclassical solution. Existing results, although
highly non-trivial, are limited to the cases where the potential has a special form
[16, 35, 36, 37], e.g. parabolic [35], or a part of the potential or the magnetic ﬁeld are
in some sense weak [38]—[45]. Chapter 3 of this dissertation will contain a discussion
of a semiclassical solution to the problem of single—particle decay in a magnetic ﬁeld
[46, 47]. The proposed method applies to a three-dimensional potential of a general
form and arbitrary magnetic ﬁelds. We assume, however, that the intrawell wave
function is known and use to obtain the initial conditions which parametrize the set
of tunneling trajectories. One of the unexpected results is that the tunneling particle
has a ﬁnite momentum and velocity when it escapes from the barrier (see Fig. 1.1).
This is in contrast to what happens for B = 0 where the particle comes out of the
barrier with zero momentum on the line E = U (r). The magnetic ﬁeld breaks the

time-reversal symmetry of the classical equations of motion, which in turn leads to

complex momentum of the tunneling particle under the barrier. The imaginary part
goes to zero at the escape point, but the real part stays ﬁnite. To calculate the
tunneling probability it is therefore insuﬁicient to just ﬁnd the probability for the
particle to reach the classically allowed region E = U (r) The probability to reach
the escape point can be exponentially smaller.

The physical situation to which the proposed method will be applied is tunneling
transverse to the ﬁeld from low density 2DESS. In the low density limit Coulomb
interaction dominates the exchange interaction, and as a result, electrons in the layer
form a strongly correlated liquid, or, for yet lower densities, a 2D Wigner crystal
(WC) cf. Fig 1.2. Strong electron correlations Show up dramatically in many unusual
transport properties [48]-[53]. The out-of—plane tunneling can also be very sensitive
to electron correlations [54]. In double layer heterostructures, for example, a giant
increase of the interlayer tunneling was recently observed and was related to interlayer
electron correlations in the quantum Hall regime [55]. In the case of 2DESs on a
liquid helium surface, it was known experimentally since 1993 [34] that the tunneling
rate at low temperatures is exponentially larger than predicted by the single-electron
approximation. This fact, however, was unexplained until the present work [56, 57].

A magnetic ﬁeld parallel to a 2DES usually exponentially suppresses the out-of-
plane tunneling. This can be understood from the following arguments. Consider
an isolated electron, which is separated from the continuum states by a 1D potential
barrier U (2), see Fig. 1.2, and is free to move in the plane. When the electron moves a
distance 25 away from the layer, it acquires the in-plane Hall velocity V” = (e/c)B x z.

The corresponding kinetic energy mvf, / 2 E 771(1)sz / 2 is subtracted from the energy

4

 

 

 

 

X

Figure 1.1: Tunneling in a two-dimensional potential U (m, z) transverse to a magnetic
ﬁeld B pointing in the y direction. Initially the particle is localized in a metastable
state behind the barrier, with energy E. In contrast to the case B = 0, a particle
emerges from the barrier with a ﬁnite velocity, and therefore the exit point is located
away from the line U (r) = E.

of the out-of—plane tunneling motion (a)C = |eB|/mc is the cyclotron frequency), or
equivalently, there emerges a “magnetic barrier” mwfz2/2. This leads to a sharp
decrease of the decay rate.

For an electron that is conﬁned in-plane, however, the in-plane force from the
conﬁnement can partly compensate the Lorentz force, thus reducing the suppression
of the tunneling rate caused by B. In this way, the conﬁning potential absorbs part of
the in-plane Hall momentum of the tunneling electron. In a strongly correlated 2DES

the in-plane conﬁnement originates from Coulomb interactions with other electrons.

The idea that the momentum transfer may lead to strong increase of tunneling was
ﬁrst discussed in [38]-[40] in the context of scattering by defects. A conﬁning potential
from a defect has a limited range (in particular, in the tunneling direction), and could
be considered by perturbation theory. In our problem, the conﬁning potential is
formed by Coulomb interaction with other electrons. It remains strong for distances
~ 71—1/2 (12 is the electron density), which, for a strongly correlated electron system,
are larger than the tunneling distance L. Therefore a perturbation theory may not
be used, and an exact analysis is required.

If one adOpts the Einstein model in which the in-plane electron motion is a
harmonic vibration about an equilibrium position, with one frequency w, then the
problem is effectively reduced to a single-particle problem with in-plane potential
Tru.a2(:zr2 + y2)/2 [directions it and '1] are in-plane, 2 is the out-of—plane direction].
Characteristic frequencies (.2 are of the order ofthe plasma frequency cap, which is

1/2. As we will see below,

related to the electron density n by top = (27r62n3/2/m)
modeling electron-electron interaction by such an effective single-particle potential
results in an adequate explanation of experimental results.

The electron system accommodates the in—plane Hall momentum of the tunneling
electron in a way similar to how it happens in the Mossbauer effect. In the latter effect
the atom of a crystal emits the gamma quantum without recoil, if the momentum
of the quantum is distributed to all atoms in the crystal. In the case of tunneling
from a 2DES, however, the dynamics of the interelectron momentum exchange is
very substantial [56]. The characteristic momentum exchange rate is also given by

the zone-boundary plasma frequency w,,. In the limit where wp exceeds the reciprocal

6

 

 

 

' Electrons

 

Figure 1.2: The geometry of tunneling from a correlated 2DES transverse to a mag-
netic ﬁeld; electrons vibrate in the plane with frequencies of the order of the plasma
frequency cup.

duration of under-barrier motion in imaginary time T; 1 and we, other electrons in the
WC adiabatically follow the momentum of the tunneling electron. As a result, the
Hall velocity is the same for all electrons, and ’0” oc l/N —) 0 (N is the number of
electrons). The effect of the magnetic ﬁeld on tunneling is then fully compensated.
For wp'r, ~ 1 the compensation is only partial, yet very substantial. One can say
that tunneling is accompanied by creation of phonons of the WC, and the associated

energy adds to the magnetic barrier]. However, the barrier turns out to be smaller

 

1The problem of tunneling between the lattice sites of WCs at the edges of a. quantum Hall
system was discussed by M.B. Hastings and LS. Levitov, Phys. Rev. Lett. 77, 4422 (1996). This
problem is qualitatively different from that investigated in the present work, as are the results. In
particular, opposite to the present. case, the tunneling probability was determined by coupling to
the low-frequency long-wavelength WC modes, it oscillated with B, and went to zero for T —> 0.

than for a free electron, and the tunneling rate is then exponentially larger. Still, for
T = 0 it is much smaller than for B : 0.

The ﬁeld B parallel to a 2DES couples the out-of—plane tunneling motion of an
electron to its in-plane motion. As discussed, for T = 0 this results in the energy
transfer from the out-of—plane direction to the in-plane one, and ultimately, in the sup-
pression of the tunneling rate by B. For T > 0, however, the transfer of energy may go
in the Opposite direction: thermal in-plane energy is converted into the out-of—plane
motion. One can say that the in-plane motion with a velocity v changes the tunnel-
ing barrier by adding an effective out-of—plane electric ﬁeld c‘lv x B, as illustrated
schematically in Fig. 1.3. For an appropriate direction of v the ﬁeld pulls an electron
from the layer, and only these velocity directions contribute to the thermal-averaged
tunneling rate. This result is rather unexpected, because it opens a possibility for
an exponential increase of the tunneling rate with B [57]. The effect is analyzed in
detail in Chapter 5, including speciﬁc calculations for some model potentials.

The crossover from suppression to enhancement of tunneling by the ﬁeld occurs
for a crossover temperature Tc. This temperature can be estimated by noticing that,
for B = 0, the tunneling rate from the ground state If}, o< exp[—2SO] exponentially
depends on the energy E9 of the intrawell electron motion transverse to the layer
[So is the mechanical action for under-barrier motion; in what follows we use units
where h 2 k3 = 1]. The derivative T0 = BSD/OEg gives the imaginary duration of the
under-barrier motion. The magnetic field effectively transfers the in plane electron
energy Eplane into the out-of-plane energy E], at least in part. The probability to
have an energy Eplane is oc exp(—E,,la,,e/T). Therefore the overall probability, which

8

 

 

 

\ \5U= -e8 2

Z

Figure 1.3: Magnetic ﬁeld induced lowering of the tunneling barrier by thermal in-
plane motion (schematically; the lowering is superimposed on the magnetic barrier
for T = 0). The effective electric ﬁeld 8 is determined by the T-dependent optimal
in—plane velocity, 5 = vopt x B / c.

is determined by the product of the two exponentials, depends on the interrelation
between T and T0, and one may expect that TC ~ 70-1.

The time To also often determines the temperature Ta for which there occurs
a crossover from tunneling decay to decay via activated over-barrier transitions for
B = 0 [58, 59]. Therefore Ta and TC are of the same order of magnitude. The
interrelation between these temperatures is determined by the parameters of the
system, and various interesting situations may occur depending on these parameters.
For example, the logarithm of the escape rate may increase with B even for T > Ta,
because in a certain B-range, the rate of tunneling from the ground state exceeds the
activation rate, even though it is smaller than the activation rate for B = 0. Similarly,
with increasing B there may occur switching from tunneling from the excited intrawell

states (see Fig. 1.3) to tunneling from the ground state. Different switching processes

are considered in Chapter 5.

For T < TC, on the other hand, the tunneling rate decreases with increasing B.
For large enough B the tunneling rate becomes smaller than the rate of activated
escape, which then determines the overall escape rate and generally weakly depends
on B.

Although the thermal B—induced tunneling enhancement is generic, as we show
it arises only in systems where intrawell motion transverse to the layer is not semi-
classical. This is typical for 2DESS, where the conﬁning potential U (z) is usually
non-parabolic near the minimum, and even non-analytic 2. In contrast, the enhance-
ment does not arise if the tunneling rate can be found using the instanton (bounce)
technique [9], which is traditionally applied to describe tunneling for B = 0 [37].
The approach given in Chapter 4 allows one to calculate the tunneling rate from a
potential well that is strongly non-parabolic both at ﬁnite and zero temperature. In
addition to strong quantization of the intrawell motion, the instanton method has to
be modiﬁed, because the n'iagnetic ﬁeld breaks time-reversal symmetry, and therefore,
except for the case where the Hamiltonian of the system has a special form [16], there
are no escape trajectories in real space and imaginary time, and the system comes
out from the barrier with a ﬁnite velocity [46].

Explicit results on the effect of electron correlations on tunneling will be obtained
assuming that the electrons form a Wigner crystal. Because of strong correlations,

overlapping of the wave functions of individual electrons is small, and electrons can

 

2For example, in the case of a 2DES in heterostructures the potential has a step, and in the case
of electrons on helium, the part of the potential which is due to image forces has a singularity on
the surface.

10

be “identiﬁed”. The problem is then reduced to the tunneling of an electron coupled
to in—plane vibrations of the Wigner crystal. As discussed below, the results provide
a good approximation also for a correlated electron liquid.

Comparison with experimental data on tunneling from a liquid helium surface
is carried out in Chapter 6. The results, obtained with no adjustable parameters,
provide full qualitative and quantitative explanation of the experimental data [34] in
a broad rage of magnetic field and temperature. We show that the corresponding
experiments on tunneling in heterostructures will be very sensitive to both electron
correlations and in-plane electron dynamics. The range of parameters for such exper-
iments is provided.

Part of the material presented in this dissertation has been published [46, 56, 77],

submitted for publication [57], or is being prepared for publication [47].

11

Chapter 2

Semiclassical solution of a 1D

tunneling problem

It is natural to suppose that the motion of a particle with a small de Broglie wave
length A(:z:) = fi/p(a:), i.e. much smaller than the characteristic length of a potential,
should be similar to the motion described by classical equations of motion. We start

with a Scrodinger equation
___ + U(IIW’ = 31/) (2.1)

In the limit where Plank’s constant Ii ——> 0, one should recover the classical equations
of motion. However, it would be inappropriate to set it = 0 directly in the Eq. 2.1.
Instead, similarly to how the limit of geometric optics is obtained through the eikonal

equation, we look for the wave function in the form:

'z/)(:i:) : exp[iS(:r)/h], (2.2)

12

and substitute (2.2) back into the Schrodinger equation. This gives us an equation

for the function S (SE)

1 d 2 r 2
_(3) +U($)_.;_Lfi_§=E (2.3)

2m dz2
Now, we can use our assumption of small 6, and look for S as a power series in terms

of h:
n a 2
S = So + ‘Z-Sl + “z,— SQ. (2.4)

In the lowest order, Eq. (2.3) becomes just the Hamilton-Jacobi equation of classical

mechanics:

1 as, 2 _

2m

Therefore, it is the classical action SO that determines the exponent of the wave

function (2.2). According to (2.5), for a 1D case we have:

 

30 = i /p(:1:)d:r, p = \/2m[E — U(x)]. (2.6)

Here, p(:r) = dSo/dzr is the classical momentum of a particle. One can also calculate
the next order in expansion (2.4) to ﬁnd that S 1 : —% 1n p(:i:). The semiclassical wave

function consists then of two waves:

Cl 3 _ CQ 7,
1M3) :2 p(.’L‘) exp [E /pd:1,] + ___—12(1) exp {—fi fpdx] , (2.7)

where the ﬁrst term describes the wave propagating in the positive x-direction, while

 

the second term corresponds to a wave propagating in the negative x-direction. Most
of the essential physics is already present in (2.7). Taking higher order terms in h
leads to changes of the ﬁrst and higher orders of h in the prefactors.

l3

The turning points where p(:z:) = 0 have to be treated with care in the semiclassical
approximation. The prefactor diverges at a turning point, signaling the fact that
the approximation becomes invalid. Indeed, the semiclassical approximation can be
justiﬁed only if the zeroth order term in (2.3) is much larger than the one that we

neglect:

dZS/dzr2
(dS/dzr)2

_1
—27r

(1A

7
L dlL'

<<1

 

 

 

 

Since the de Broglie wave length /\ 2 13‘1, the approximation certainly does not work
where p(:1:) = 0, i.e. at a turning point.

So far we have found the semiclassical wave function in a classically allowed region
where E > U (3:). In a similar fashion to (2.2)-(2.6) the semiclassical approximation
can be used to ﬁnd the wave function in a classically forbidden region where E < U (3:)
One of the ﬁrst distinctions to be seen, is that momentum p(:1:) from (2.6) is no longer
real, but is purely imaginary. Because of the imaginary momentum, action So also

becomes imaginary:

 

so = iz‘ / Iptaldx. |p(w)l = «mu/(2:) — E]. (2.8)

with the next order correction being SI 2 —%ln[p(1‘)[. Therefore, deep under the
barrier (far from the turning points p(:r) = O), the semiclassical wave function consists

of an exponentially decaying and exponentially growing waves:

we): lifoCXp [-,1,/Iz2ldx]+ mil—>12 [Helm]. (2.9)

The semiclassical solutions (2.7) and (2.9) together describe the wave function

 

 

everywhere except near the turning points. What one would like to have is a relation

14

between amplitudes (31,2 of propagating waves in the classically allowed region and
amplitudes C],2 of evanescent waves in the classically forbidden region. There are two
possibilities to achieve this. The ﬁrst one is to solve the Schrtidinger equation near a
turning point using a linear function to approximate the potential. The solution can
then be extended to regions far enough from from the turning point, where both the
WKB and the linear approximation of the potential are valid.

Another method, which we will use below, is to analytically continue the semi-
classical solutions (2.7) and (2.9) into the complex :1: plane [7]. Then it becomes
possible to match the solutions continued from classically forbidden and allowed re-
gions without going near the turning point where the WKB approximation breaks
down.

Let us consider a speciﬁc example in which the two classically allowed regions are
separated by a tunneling barrier for 331 < a: < 2:2 [mm are the corresponding turning
points]. The solution we would like to ﬁnd is that of a tunneling problem: the particle
wave packet is incident on the barrier from the left, part of it is reﬂected back, and an
exponentially small part is emitted from the barrier on the other side. The boundary
condition for this problem is the absence of a semiclassical wave incident on the barrier

from the right. The wave function has the form:

{/l—ﬁ exp [if f; dasp(:1:)] + -% exp [—,—: f1: dzp(:r)] :1: < 2:1

 

 

1()(33) = [2m(U(x)C_E)]_,,, exp [—% [:1 (1:1:\/2n2,[U(1:) — E]] 2:1 < :1: < :52 (2-10)
% exp [% f1: (1:1:p(:r)] 1132 < 1?

where for :1: < 51:1, the ﬁrst term describes the wave incident on the barrier from the
left, and the second describes the wave reflected back.

15

To ﬁnd the relation between coefﬁcients A, B, and C, let us track what happens
to the exponentially decaying solution as we go around the turning point $1 in the
upper half plane. The phase of the difference [U(:1;) — E] (X (a: —— ml) is incremented

by 7r, so that the decaying solution transforms to the reflected wave

 

(j -*in/4 . x
eﬁ exp [—% d:rp(:r)] (2.11)

As we go around the turning point in the lower half plane, the phase of U (2:) — E is

incremented by —7r, and the decaying solution goes over to the incident wave

iﬂ/4 ' x
C317 exp [%/ d1:p(:1:)] (2.12)

By comparing expressions (2.11) and (2.12) with the original one (2.10), we ﬁnd that

 

B = —iA, C = exp[—i7r/4]A. (2.13)

To ﬁnd the coefﬁcient D, and the tunneling probability with it, let us rewrite the

decaying under barrier solution as:

 

 

 

zPdecatyingC’I?) = [2m(U($)CI_ E)]-1/4 exp [*% [1: da:\/2m[U(:2:) " El] ,
C' = Cexp [—% [:2 dzt\/2m[U($) —— E]] (2.14)

The function z/idecayingtr) describes the wave with an amplitude that exponentially
increases into the barrier (with increasing [1: —:r2|). The tunneling current is described
by the emitted wave in (2.10) for :1: > $2. Let us analytically continue this wave into
the upper half plane. As we go around the turning point 1:2, the difference U (:13) — E

acquires an extra phase of 7r, so that the emitted wave goes over into ’t/Jdecaying($) with

C’ = De-W“. (2.15)

16

We could also continue the emitted wave from (2.10) in the lower half plane, but
this would result in a semiclassical solution that exponentially decreases in amplitude
with increasing x2 — :r. We neglect this wave, because it is exponentially smaller than
I/Jdecaying(x) everywhere in the tunneling barrier region.

By comparing (2.13), (2.15), we ﬁnd the amplitude of the emitted wave D:

 

g- = exp[—h‘IImS($2,:I:1)] = exp [— [$2 h—1\/2m[U(a:) — E]da:] . (2.15)

Therefore, we have constructed a semiclassical solution for all value of :r by matching
different semiclassical solutions across the turning points. An important feature of
the ﬁnal wave function is that the semiclassical wave valid on opposite sides of a
turning point differ only by a phase factor, and have the same amplitudes. The decay
of a wave function across the tunneling barrier is described by the imaginary part of
a classical action S, as given by (2.16). The tunneling probability is given by |D/A|2.
Note that it does not depend on phase factors accumulated as a result of matching
the semiclassical solution across the turning points (although in this case, the total
phase is 0).

In preparation for the upcoming discussion of tunneling problems where the mo-
tion is not restricted to 1D, it is useful to write the action So (2.6) in the classically

allowed region as an integral along a classical trajectory r(t):
So = :t/p(r) ~dr, i' = ﬁ, 15 = —VU. (2.17)

Such a generalization becomes less straightforward in the classically forbidden region,
where there is no real classical trajectory because momentum is imaginary under
the barrier. Therefore we need to ﬁnd some other trajectory (possibly in a different

17

potential) that would result in the action equal to the right hand side of (2.16).
In order to achieve this, one could employ the following trick. Consider the usual
Hamiltonian trajectory with dr/dt = P/m and (IP /dt = —VU (r) and change t —>

—iT. With P being imaginary, these equations can be solved! They have the form

1 1
‘1’ z 2., LB — +VU(r), [P = 2p] (2-18)

_(1—7: m (17' _

which describes classical equations of motion in time 7' in the potential —U (r) The

 

absolute value of momentum is given by \/ 2m[U (r) — E]. The corresponding classical

action SE evaluated along the trajectory

550‘) = /p(r) -dr (2.19)

is related to that evaluated in real time by SE = 25'. Therefore, an appropriate

generalization to the 1D answer (2.16) would be

D
_ : exp[—h—ISE(rla 1.2)],

with the tunneling probability given by

2

T =[-12 = exp [_Qh“lSE(r1,r2)] a

A

 

In summary, the 1D tunneling problem has a general solution in the semiclassical
approximation. It uses the semiclassical solutions in regions that are separated by
turning points, where the approximation breaks down. However, it is possible to relate
the amplitudes of these semiclassical solutions by analytically continuing them into
the complex plane, where they should match one another. The tunneling probability
is given by the ratio of amplitudes of the wave incident on the barrier and the wave

18

emitted from the barrier on the other side. The method could also be extended to

tunneling problems where the motion is not restricted to one dimension [15, 62].

19

Chapter 3

Multidimensional tunneling decay

in a magnetic ﬁeld

This chapter is devoted to the discussion of the problem of a single-particle tunneling
decay in a magnetic ﬁeld. The semiclassical solution applies to a smooth three-
dimensional potential of a general form and for arbitrary magnetic ﬁelds that also
have to be smooth (i.e., the characteristic lengths of the potential and the magnetic
ﬁeld should much larger than the particle’s de Broglie wave length, if we discuss a
classically allowed region, or the decrement of the semiclassical wave function, if we
discuss a classically forbidden region. The method developed here will be used sub-
sequently to analyze tunneling from a strongly correlated 2DES. In the semiclassical

approximation we look for the wave function in the form:

‘(/)(I‘) = D(r) exp[iS(r)] (h = 1). (3.1)

20

Here, S (r) is the classical action. It satisﬁes the classical Hamilton-Jacobi equation

with the Hamiltonian:
H = (p + eA(r))2/2m + U(r), (3.2)

where A(r) is the vector potential. The action S can be found from the corresponding

Hamiltonian equations
5" = p - r, i- = OH/Bp, p = —aH/ar. (3.3)

The difﬁculty of solving the tunneling problem in a magnetic ﬁeld lies in the
absence of a time-reversal symmetry of the Hamiltonian equations (3.3). As discussed
in the previous Chapter, this symmetry is essential for the standard approach to the
problem of tunneling decay [11]-[13], [15] where after changing to imaginary time
and momentum we have obtained the real Hamiltonian equations of motion. The
equations then take the form of equations of classical motion in an inverted potential
—U (r), with energy —E 2 —U(r). Such a procedure is appropriate in the absence of
a magnetic ﬁeld when the Hamiltonian (3.2) remains real. When the corresponding
action S (r) is purely imaginary, it describes a decay of wave function in the classically
forbidden region (E < U (r)) The solution in this region then has to be matched to
a wave function in the classically accessible region (E > U (r)) that is found from the
classical Hamiltonian equations of motion with real time. The tunneling exponent
is given by twice ImS at the end point of motion in imaginary time that lies on
the boundary of the classically forbidden region E = U (r). The tunneling particle
“emerges” from under the barrier at this line with p -: 0.

The presence of a magnetic ﬁeld breaks the time-reversal symmetry of trajectories

21

(3.3). For B 7E O, we can no longer change If —> —7§t, p —> ip, r ——) r, U(r) —> —U(r)
and keep the Hamiltonian real. As a result it is impossible to match trajectories
(and, therefore, wave functions) in the classically allowed and forbidden regions that
have respectively real and complex E. It becomes clear that to solve the tunneling
problem in the presence of a magnetic ﬁeld, it is necessary to look for trajectories of
a different kind.

Let me note in passing that there are some important cases where certain speciﬁc
symmetry of a potential U (r) allows us to use the standard method after some mod-
iﬁcations. A particular case is when a canonical transformation of variables can be
found that would restore the time-reversal symmetry of the Hamiltonian. This, for
example, applies to tunneling of a particle coupled, for ﬁnite B, to a bath of harmonic
oscillators [16], where the potential U (r) is parabolic in the coordinates of the bath
oscillators. We will encounter such coupling of the form [2 x B]pkj in Chapter 4 in
the discussion of tunneling from a Wigner crystal. Even though the time reversal
symmetry may be broken by a magnetic ﬁeld, it can be restored by a canonical trans-
formation from coordinates ukj and momenta pk] to the new canonical coordinates
and momenta ij = pkj and ij 2 —uk,-. In the new representation, the tunneling
trajectories can go with real coordinates ij and imaginary momenta ij.

In Sec. 3.1 we concentrate on the exponent of the tunneling rate in the presence of
a magnetic ﬁeld. The technique can be applied both in the case where the potential is
parabolic near the intrawell minimum and in the case where the potential is singular,
which is of relevance to tunneling from a 2DES. In a magnetic ﬁeld the tunneling
trajectories start to intersect, giving rise to a branching form of the action S (r) and

22

corresponding singularities of the set of Hamiltonian trajectories. For the problem
of tunneling decay, the most important among these singularities are caustics, or 2
envelopes of trajectories. Section 3.2 deals with matching of different semiclassical
solutions across caustics. We discuss generic features of the manifold of Im S near the
branching point in complex space and its projection onto real space. In Sec. 3.3 we
discuss the tunneling decay problem for B yé 0 in the path-integral formulation. The
corresponding “bounce” technique for tunneling out of a potential minimum is closely
related to the instanton method for turmeling between two minima of a potential. The

method applies in cases where the intrawell potential is parabolic near its minimum.

3.1 Tunneling exponent

The main idea is to consider Hamiltonian trajectories (3.3) that evolve in complex,
rather than imaginary, time t in complex phase space (p,r). Motion along the tra-
jectories occurs with real energy E equal to that of the metastable state. This cor-
responds to the analytical continuation of the WKB wave function (3.1) to complex
variables. The action S will now have both real and imaginary parts, so that the wave
function will be oscillating and decaying in space, which is natural in the presence of
a magnetic ﬁeld.

The tunneling trajectories (3.3) start in the vicinity of the localized metastable
state at t = 0. The initial conditions can be obtained from the usually known form of
the wave function near the potential well. This can be done both in the case where

the potential is parabolic near its metastable minimum, so that z/2(r) is semiclassical

23

 

 

 

 

 

 

 

(a) E Re t
l I l |
-0.2 0 0.2
-1 _
Classically forbidden region
-2 - -0.2 0 X/L 0.2

Figure 3.1: (a) Complex t plane for integrating the Hamiltonian equations (3.3) in
the escape problem. The line Im t = const corresponds to the classical trajectory
of outgoing electron. (b) The classical trajectory on the (a3,z) plane. The solid
lines in (a) and (b) show the range of Re t where the amplitude of the propagating
wave exceeds the amplitude of the decaying underbarrier wave function, and the
corresponding “visible” part of the trajectory. The escape occurs at the point where
classical trajectory intersects the anti—Stokes line (thin solid line in (b)). Note that
particle emerges from under the potential barrier with ﬁnite velocity and for a: 75 0.
Thus the initial symmetry 2: —> —:z: is broken by the magnetic ﬁeld. The data refer
to the potential (3.17) with wore : 1.2 and wcro = 1.2, time in (a) is in the units of
To = 2mL/cy.

inside the well [9]-[14], and where the potential is nonanalytic, which is of interest
for 2D electron systems. In either case the trajectories (3.3) can be parameterized
by two complex parameters 1313(0), e.g. the in-plane coordinates for a given out-
of—plane coordinate, for tunneling from a 2DES. The in-plane momenta are then
1213(0) = 85(0)/8:r1,2.

With initial conditions at hand, the equations of motion (3.3) can be solved to ﬁnd
the action S(t,:r1,2(0)), together with p(t,:r1,2(0))) and r(t,:1:1,2(0)). The tunneling

rate is determined by 21m S at the point where the particle emerges from the barrier

24

as a semiclassical wave packet that propagates in real time along a real classical
trajectory rc1(t). This trajectory is yet another classical real-time solution of the
Hamiltonian equations, the ﬁrst one corresponding to the particle trapped in the
well. To ﬁnd the tunneling exponent we have to ﬁnd such paths r(t),p(t) (3.3) that
would start in the vicinity of the well with some complex 2713(0), then go in complex
time and space to reach the classical trajectory r,,(t). In other words, one has to ﬁnd

such 3313(0) that, for some t, both r(t) and p(t) become real,

Im r(t) = Im p(t) 2 0. (3.4)

This is a set of equations for complex 3512(0) and Im t. The number of equations
is equal to the number of variables, taking into account that H is real. The Re t
remains undetermined: a change in Re t in (3.4) results just in a shift of the particle
along the classical trajectory r,,,(t), see Fig. 3.1. Such a shift does not change Im S.
We note that 3313(0) are real for B = 0.

The tunneling exponent R is given by the value of Im S at any point on the

trajectory rd,
72 = 2 Im S(rd), (3.5)

For a physically meaningful solution, Im S should have a parabolic minimum at rd as
a function of the coordinates transverse to the trajectory. Respectively, the outgoing
beam will be Gaussian near the maximum.

From (3.4), the tunneling exponent can be obtained by solving the equations of
motion (3.3) in imaginary time, with complex r. However, that solution does not
give the wave function for real r between the well and the classical trajectory rd.

25

Neither does it tell us where the particle shows up on the classical trajectory.

To obtain a complete solution of the tunneling problem, we have to take into
account that S is a multivalued function of r even though it is a single-valued function
oft and 1313(0). This means that several trajectories (3.3) with different t and $1,2(0)
can go through one and the same point r. The wave function 1,0(r) is determined by
one of the branches of the action S (r). How to match different branches to construct

the wave function is discussed in the next section.

3.2 The action manifold

3.2.1 Branching lines - caustics

In multidimensional systems, branching generally occurs on caustics, or envelopes of
trajectories [60, 62]. Their role in a multidimensional tunneling problem is analogous
to that of turning points zt in a 1D tunneling decay [see Chapter 2 for a discussion].
The common feature is the divergence of the prefactor D(r) in the WKB wave function
(3.1). In the case of a 1D tunneling, D o< p‘l/z, leading to a branching action
3 — St oc (z — 2,)3/2 near the turning point z]. The caustic is the line where the
transformation between coordinates 3:1, .732 , .3 on the trajectory and parameters t,

1:1(0), $2(0) loses its uniqueness:

0(1L'1, 1:2, 2:)

8(:1:1(0),;1;2(())7 t) ' (3.6)

J(r) = 0, J(r) =

 

As we will now Show, the prefactor in the WKB wave function is D = const x J ‘1”.

Indeed, the next-to leading order term in the expansion of action in powers of h,

26

S = 50 + z'hSl satisﬁes the equation:

2vV51 = —divv, (3.7)

where v = r" is the velocity along a trajectory, and the Coulomb gauge was used for
the vector potential. The left-hand side of (3.7) is 2S 1. In the right-hand side, let us

express the derivatives B/Br in terms of those with respect to t, 3313(0):

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

27

 

 

 

 

 

 

at or or
1
j. _ a 017-) a:
(m .1 0mm) 0131(0) 6mm
(9 0.732 02
0.152(0) 0.132(0) 0.112(0)
at or, 0!
l
#2- _. an 0 0::
(”2 J 01:,(0) 0171(0) 0mm)
8:“ 8 Oz
01:2(0) (9132(0) 8172(0)
in (9—2 9—
oz 0t at
a 1 ,. ,
,— - 0L 0r; a (3.8)
(1’2 ,] 011(0) 01,-, (0) (mm)
(911 0:17;; ('3
032(0) (Tr-2(0) (hr-2(0)
Therefore,
82x; M Q: d~_r1. _202x Q
8% 0t 6t 6t 0% 6t
. 1 , .. 1 . .
leV= _ 8:171 ()IIT-g 6:: +_ drl 83:2 6::
J arm-1(0) emu) ohm) J 01:1(0) 6t6x1(0) 611(0)
8217] (9.17;) (92 3.1"] (92122 62
awe-2(0) 01:2(0) 013(0) air-2(0) meme) 623(0)
8t at 6‘3:
1 . .. .2
+—. (’42; 01:2 0 z (39)
J 02:1(0) 811(0) (9661:1(0)
0:111 ()1‘2 (322
6.122(0) 011:2(0) afar-2(0)

and we ﬁnd that S1 = —.j/2J, so that the next correction is
S] OC J—l/Q,

and indeed diverges on the caustic. Therefore the WKB approximation does not
apply close to it (cf. [60, 61]). Similarly to the 1D case, the different branches of

action can be matched by going around the caustic line in complex space.

3.2.2 Local analysis near caustics

In our problem, in contrast to the usual case, the trajectories r(t) will be complex,
as will also be the caustics. The caustic of interest is the one where the WKB wave
function corresponding to the tail of the intrawell state is connected with the WKB
wave function that describes the escaped particle [cf Fig 3.2]. Both of the WKB
solutions are analytically continued to complex r, and merge on the caustic in the
complex place.

Local analysis near the caustic will be similar to that in the 1D case [60, 61]. It
is convenient to change to the variables 23’, y’, and 2’ which are locally parallel and
perpendicular to the caustic surface, respectively. We set 2’ = 0 on the caustic. Let

us write the wave function near the caustic surface as:
l ' I
W(I‘c + r’) : e'pcr ¢>(z'; re). (3.10)

Here, pc is the momentum along a trajectory going through the point rc of the
caustic surface. For B 75 0, the component of momentum perpendicular to the caustic

surface is ﬁnite, but the velocity pc + eA(rC) should be tangent to the surface. As a

28

 

I
U),—
I
I
l

0.4
Im x/L

I
l

 

l
p—A
1

-0.4

0.8 1 Z' 1.2
Re z/L

 

 

 

Figure 3.2: The complex tunneling trajectories (solid lines) and the complex caustic
surface (dashed line) are shown in the (Re 2, Im 515) plane. The tunneling trajectory
(2) reaches the clasical trajectory of an escaped particle for Im :17 = 0. At this point
(shown by empty circle) the momentum is real along the trajectory (2). Even though
other trajectories (1) and (3) cross the line of Im :1: = 0, it can be seen from the ﬁgure
that Im 19,, 72 0 for Im :1: = 0. Therefore these trajectories do not touch the classical
trajectory. Note that the point z 2 2C where the caustic goes through real space is
not the point where the tunneling trajectory (2) becomes real.

result, in the Schrodinger equation for (23(z’; rc) motions along the caustic surface and
perpendicular to it separate. In the perpendicular direction, the equation for (15(2' ; r.)

is identical to that near a 1D turning point in the absence of a magnetic ﬁeld:

f1,"Z ([2

2m (iz’2

+ U'(rc)z' 95(3'; rc) = 0. (3.11)

 

Therefore, we will be able to match semiclassical solutions of the wave function 1,0
across the caustic line, if we can ﬁnd the proper asymptotes of the function ¢(z’, ; rc).
As a solution of Eq. (3.11), gb(z’) may be given in terms of a linear combination of Airy

functions. Even though the function (j)(z’) is single-valued, its asymptotic behavior is

29

determined by one of the two branching functions:

 

101,2 2 [271‘1U’(rc)z']‘1/4 exp (3.12)

I” 311.

 

.2 27nU’(rc) Z13/2]

Combining Eqs. (3.10) and (3.12), we ﬁnd that the action for small |z’| behaves

S(zr’, y', 2') z S(:L", y', 0) + (112' + (122'3/2 (3.13)

The linear term reﬂects the fact that. momentum perpendicular to the caustic remains
ﬁnite, and, in general, complex (see Fig. 3.5). Therefore, the classical trajectory does
not necessarily go through the caustic surface, contrary to the case without magnetic
ﬁeld.

Another way to understand the branching form (3.13) of the action S is to consider
Hamiltonian trajectories near the caustic surface. Because of v2, = 0, z’ is quadratic
in the increments 6.1312(0), 6t. Therefore 6:r1,2(0), 6t are nonanalytic in 2’, as is also
the action S. Taking into account. cubic terms in 6:1:1,2(0), 6t we obtain (3.13), where
the coefﬁcients c112 E a1,2(:1:’, y’) can be expressed in terms of the derivatives of S, 1‘
over $13 (0),t on the caustic.

Having obtained the branching form for the action S (3.13), we now need to ﬁnd
out which of the branches describes the proﬁle of the wave function |¢| in the global
variables. The boundary conditions usually specify the asymptotic behavior at a
particular value of 0 = arg z’, the angle in the complex z'-plane. Our goal is then to
ﬁnd the asymptotic behavior throughout the complex z’-plane. When, and how, it

can be done in the general case is discussed next.

30

Asymptotic analysis near a 1D turning point: the Stokes phenomenon.

The matching of asymptotes across a 1D turning point was extensively studied in the
complex plane [7, 60, 61]. The difference with the analysis done in the context of 1D
tunneling without magnetic ﬁeld [7] is that now we need to know not only asymp-
totes for 6 = 0, :trr but for all values of 6 [in a 1D tunneling problem, the classically
allowed region may, for example, correspond to 6 2 0, and classically forbidden re-
gion to 6 = :l:7r]. The asymptotic behavior at large p = [z’ |[2771U'(rc)/f12]1/3 >> 1 [a
combination [1‘12/2171U’(rc)]1/3 is appropriate to scale 2’ , since it has the correct dimen-
sion of length] of the function qb is given by a linear combination A(6)w1 + B (6)2122.
The functions 1111;; are the ﬁrst terms in the asymptotic expansion in h‘l. In our

case, the inherent error associated with an asymptotic expansion is of the order of

 

h/\/2mU’(rc)]z’|3. For some 6, the function 1121 is exponentially larger (dominant
wave) than 1112 (subdominant wave). At other values of 6 the asymptote 1112 be-
comes dominant, and 1121 becomes subdominant. The subdominant asymptote may,
and should, be omitted, when it is smaller than the inherent error of an asymptotic
expansion.

The branch cut from 2’ = 0 may be inserted arbitrarily. When going across
the branch out in the positive direction asymptotes interchange w] —) —z'w2 and
1112 —-) —iw1, with the property of dominance and subdominance preserved in the
process. G.G. Stokes [63] was the ﬁrst to notice that in order for the function (15(2’)
to be single-valued, the coefﬁcients .4 and B should change with 6. Indeed, suppose

the asymptotic behavior for 6 = 0 was described by .4101 + BIUQ. Then incrementing

31

6 by 271‘ leads to —z'B'w1 — 1.41122, which can be satisfied only if both A and B are zero.

The coefﬁcient A can change without causing a change in 6(2’) only if the appro-
priate term is less than the error of the asymptotic expansion. In other words, the
coefﬁcients A (or B) can change only when (i) B (.4) is non-zero, (ii) the asymp—
tote wl (1122) is subdominant. The change is ccmcentrated in a narrow region around
the so called Stokes line where the difference in anmlitudes between the subdomi-
nant and dominant waves is maximal. In our problem the Stokes lines correspond
to Re 2’3/220: 6 = ia/3,7r. The width of the region where coefficients change is
66 ~ [hlz’ |3/2 /W]1/2 ~ [fl/2 [60]. For our purposes, this change is equiva—
lent to a jump [65], which is usually described in terms of a Stokes constant T. Upon
crossing a Stokes line where 11:1 is subdominant the coefﬁcient changes as A —+ A+BT.

One can ﬁnd all three Stokes constants from the requirement that the function
¢(z’) should be single-valued. Indeed, let the asymptotic behavior at 6 = 0 be given
by Awl + 81122. The asymptotes 1111 and 1112 interchange the property of dominance
and subdominance upon crossing lines where their amplitudes are equal. In our
case, these so-called anti-Stokes lines are found from the condition Im 2’3/2 2 0
( = 0, i27r/ 3). The branch cut, together with three Stokes and three anti-Stokes

lines, divides complex plane into seven sectors, as shown in Fig. 3.3. The asymptotic

32

behavior in each sector is:

I : Awld + B11225; 2 : —7l.411)2d —1§Btols; 3: —1I.411)1d — i(B + AT1)w15;
4: —z'Aw23 — 1(B + AT1)1111,1; 5: —'1'[A + T2(B + AT1)]w23 — z'(B + AT1)wld;
6: —Z[A + T2(B + 14T1)]'LU2(1—Z(B + AT1)IL713;

72 —Z[A + T2(B + AT1)]‘?L’2(1 — ([B + 4T1 "i“ AT; 'I' T2T3(B + AT1)]’IU13 (3.14)
The single-valuedness for the function 6(3’) is achieved if
A = —z‘[B + AT1+ AT3 + T2T3(B + AT1)]; B = —z‘[A + T2(B + AT,)], (3.15)

which gives equal Stokes constants on all three of the Stokes lines: T1 = T2 = T3 = i.
This calculation also shows that if we know the asymptotic behavior on one of the anti-
Stokes lines, we can ﬁnd it at all values of 6. In contrast, if the asymptotic behavior is
speciﬁed on a Stokes line, then we have no knowledge about the coefﬁcient in front of
the subdominant wave, and therefore, are not able to uniquely specify the asymptotic

behavior through out the complex plane.

Boundary conditions.

In the problem of tunneling decay the boundary condition that we need to satisfy is the
absence of the wave incident on the barrier from large positive 2. After transformation
(3.10) to local variables 12’, y’, z’, the incident and outgoing waves in global variables
become familiar 1D waves described by exp[i1f0‘~'l dz’p(z’)/h] with arg z’ = = 0.
Indeed, the coordinate-dependent amplitude is eliminated by a factor exp[ipcr’], which

has complex pc. The direction of incident and outgoing classical waves is therefore

33

 

 

Figure 3.3: Sectors of complex z’-plane with different asymptotic behavior of the
wave function around a 1D turning point with a first order zero. The anti-Stokes

(solid) lines correspond to values of 6, where both asymptotes have equal amplitudes,
6 = 0, 27r/3,47r/3.

projected onto one of the anti-Stokes lines, and we can always choose it to be the one
with 6 = 0. As a result, the absence of the incident on the barrier wave in global
variables eliminates the asymptote '11)] for 6 : 0.

To ﬁnd the asymptotic behavior across the caustic line, we now can set coefficients
A = 0 and B = 1 and use the previously calculated asymptotes (3.14) in complex
space around a 1D turning point. The result is shown in Fig. 3.4. In the semiclassical
approximation, we take into account only the exponentially larger, dominant wave.
Therefore different waves will determine the proﬁle of the wave function on different
sides of an anti-Stokes line, where two waves have equal amplitude. It is interesting
to note that because we have speciﬁed that there is only one solution 1112 at the

anti-Stokes line for 6 = 0, there is also only one solution on the anti-Stokes line

34

 

the line of
switching

 

Figure 3.4: Complex 2’ plane perpendicular to the caustic surface at z’ = 0. The
anti-Stokes lines (solid) are located at arg z’ = 0,27r/3,47r/3. They divide regions
where the solution is exponentially large and small. If there are two solutions on
the anti-Stokes line, the one that was exponentially smaller in one region becomes
exponentially larger after crossing the line. The subscripts s and d denote whether
the solution is exponentially small or large, respectively, in a particular region. The
dashed lines Show Stokes lines, where the difference between the exponentially small
and large solution is maximal. The wavy line shows the branch cut.

at 6 = —47r/3, and no switching occurs on it. The only anti-Stokes line where the
switching between branches does take. place is at 6 : —27r / 3.

Switching between branches is accompanied by a change in phase by 7r/2. A
useful check is that it is exactly the phase factor obtained in 1D tunneling between
the incident and reflected wave [(tf. Eq. 2.13], as well as between the outgoing and
decaying wave1 [cf. Eq. 2.15]. The amplitude remains unaffected. Therefore, the
tunneling exponent (3.5) can be interpreted as a sum of the action Im S (re) for the
tail of the intrawell wave function continued to a point rC on the caustic, and the

action Im S (rcllrc) for the outgoing wave. from this point to a point on the classical

 

1An extra phase factor of 71/4 is due to the difference. between deﬁnitions of asymptotes 101,2 oc
(ZI)--l/4 and wdecaying OC Iii—IM-

trajectory rd:
’R 2 21m S(r,.|r(0)) + 21111 S(r,.,[r,,). (3.16)

Since S is analytic in 51?1,2(0)a t, this sum is independent of the intermediate point rc.

Notice that the two terms in (3.16) may have different signs.

3.2.3 Projection onto real space

According to (3.10), there. is a one-to-one correspomlence between asymptotes 101,2
of the function ¢(z’;rc) in complex z’-plane and WKB solutions D(r)exp(z’Sl,2(r))
for the wave function 1/1(r) in real space (the subscript in SL2 enumerates branches).
After projection into real space r the caustic of interest, being a surface in complex
space, will become a line (a point for a 2D potential). The relevant anti-Stokes surface
after projection will remain a surface (a line in 2D). .-\s in a complex space, it starts
from the caustic line and separates the regions where 1111 S is smaller for one or the
other of the solutions connected 011 the caustic. Only the solution with the smaller
Im S should be held in the WKB approximation. Therefore across the anti-Stokes line
there occurs switching between branches of action that describe physically different
wave packets. The relevant example would be switching from the branch describing
the tail of the intrawell state to the branch dest‘:ribing the wave packet of an escaped
particle propagating along a real classical trajectory. The particle escapes from under
the barrier at the point where the classical trajectory intersects the anti-Stokes surface
(see Figs. 3.1, 3.7 in the case of a 2D potential). In other words, the probability to

observe the tunneling particle at a point rd on the classical trajectory having real

36

momentum (branch 52 on Figs. 3.6(a) and (I))) should be larger than that to observe
it at that same point with a complex momentum (branch 81 describing the tail of the
intra well state).

We note that the position of the escape point is not generic: it is not located
at the boundary of the classically allowed region E 2 U(r), and not on the caustic
surface. Interestingly, the tunneling particle is first observed with a non-zero kinetic
energy, in contrast to the situation without magnetic ﬁeld.

Let us consider, for example, tunneling through a potential:

A,2

_32 , 2 I _ ;
(.1 +1; )+ 2111. (1 L) (z > 0), (3.17)

2
mwo

2

 

U(r) 2

which is relevant for the problem of tunneling from a. correlated 2DES on a helium
surface. This system was experimentally investigated in Ref. [34], and showed an
unexpected dependence of the tunneling rate on B. as addressed in a recent paper
[46].

We specify initial conditions at the plane 2 2 0, neglecting the effect of the mag-
netic ﬁeld on the wave function near the well. This can be justiﬁed if the characteristic
intrawell localization length 1/7 is small compared to the tunneling length L. Then,
even though the magnetic ﬁeld has strong cumulative effect on the tunneling rate,
it only weakly perturbs intrawell motion. The out-of—plane and in-plane motions are

uncoupled, and we get:
2(0) 2 o, p,(0) :17. 5(0) 21'111.w0(.1'(0)2 + 11(0)?) /2,
[5(0) 2 imwozr(0), 1231(0) 2 mining/(0). (3.18)

If we choose B along the y axis, then the. motion in the y direction is decoupled and

37

 

 

 

 

 

 

 

 

 

Figure 3.5: Two branches of the action on the symmetry axis (L‘ 2 0 as a function of the
tunneling coordinate 2: before the branching point, for the same parameter values as
in Fig. 3.1. ImS is shown before the branching point 2c in the real space. The vicinity
of the cusp 2C is zoomed in the inset to show that the upper branch is nonmonotonic.
Its extremum at zm lies on the classical trajectory of the escaping particle shown in
Fig. 3.1(b). However, the particle emerge from the barrier for z > zm and :1: 79 0.

the problem becomes two-dimensiona1. The Hamiltonian equations (3.3) are linear,
and we can ﬁnd trajectories explicitly. The syn'n'netry U(:1:, y, z) 2 U(zl::1:,d:y, 2:) gives

rise to a speciﬁc symmetry of the set of the trajectories (3.3):
t —> t", :17 —> —:L'*, y —> —;1/*, z —> .:*,S —> —S* (3.19)

The caustic of interest goes through real space at the point :1: 2 y 2 0, z 2 2:6
(2C 2 L for B 2 0) on the symmetry axis. Knowing the momentum pC at this point,

we ﬁnd that the complex plane perpendicular to the caustic is

z' 2 (z — zc) cosh a — 2'1: sinh a, a 2 tanh_1[1m Fez/(Par + wczcll,

38

where the parameter a is positive for B > 0 in this problem. Therefore, positive
:1: corresponds to the lower half-plane in the complex z’-plane. Near the caustic,
the outgoing wave is described by asymptote 1122 from (3.12). From examination of
Fig. 3.4, we ﬁnd that only one branch of action is physically meaningful for negative
2: (the upper half of 2’ plane). For positive 1‘ however, both of the branches of Im 5
should be taken into account. The one with smaller 1111 S determines the proﬁle of
the wave function. Below we describe the resulting picture.

For 2 S zc, the function Im S has two branches each of which is symmetrical in :13.
The branch 1 describes the tail of the intrawell wave function before branching. The
branch 2 corresponds to the wave “reflected” from the caustic. Their cross-sections
are shown in Fig. 3.5, 3.6. The branch 1 has a minimum at a: 2 0 and monotonically
increases with :1: and 2. As expected, the slope 8 Im S/Oz is ﬁnite at the branching
point 2C (see inset in Fig. 3.5). The branch 2 is nonmonotonic in z for :1: 2 0, with a
minimum at zm < zc. As it turns out momentum of the particle described by branch
2 is real at z 2 zm. The point z 2 2m, :1; 2 0 belongs to the classical trajectory and
is the point where the trajectory comes closest to the well (z 2 0) as it approaches
the barrier from large 2 and negative :1:, and goes away to large 2: and positive :1:
(Fig. 3.1(b)). Note that the velocity of the particle is not equal to zero at any point
on the classical trajectory. Although branch 2 describes a classical particle at z 2 2m,
:1: 2 0, it is not the exit point for the tunneling particle, because the probability for
it to have real momentum as described by branch 2 is smaller than that to have a
complex momentum as described by branch 1 (Fig. 3.3). For z,,, < 2 S zc, the branch

2 is nonmonotonic in :1:, with local mazrnnum at :1; 2 0 and with two symmetrical

39

 

1.52

 

1.515

Im S/yL

1.51
1.51

 

 

 

 

 

 

 

1.505

 

 

1.8 ~ , _

l . .
emitted ,' meldent
wave ' wave

1m S/YL

I

1.65

 

 

 

1.5

 

Figure 3.6: Cross sections of function ImS for constant :5 near the branching point zc:
(a) zm < z < 26; (b) 2 2 26; (c) z > 26. The parameter values are the same as in Figs.
31,35. The solid line shows the branches of Im S that determine the exponent of
|1/2|. The minima of the branch 2 lie on the classical trajectory shown in Fig. 3.1(b).

40

minima. These minima lie on the classical trajectory shown in Fig. 3.1(b). For
2 2 2",, the maximum and the minima merge together.

For 2 > 20, S (:13, 2) on one of the two branches is equal to —S*(—x, 2) on the other
branch (cf. Fig. 3.6). The appropriate minima of Ira S(at, 2:) continue to follow the
classical trajectory. As discussed above, 11118 is constant on the classical trajectory.
One may verify that the value of 11115 at the minima of branch 2 have the same value
for all cross-sections with different z 2 const. The exponent of [sz is determined by
either branch 1 or branch 2. It is plotted by solid line in Figs. 3.6. The switching
takes place where Im51 (r) 2 ImSg(r), that is on the anti-Stokes line that starts from
the caustic point z 2 zc, :1: 2 0 [compare to the anti-Stokes line for arg z’ 2 47r/3
on Fig. 3.4]. The escaped particle can be seen moving along the classical trajectory,
if 72/2 2 ImSg(rd) < ImS1(rd). Therefore, it “shows up” at the point where the
classical trajectory intersects the anti-Stokes line. The fact that the exit point is
located for :1: 2 0 even for a symmetric potential (3.17) demonstrates the symmetry-

breaking induced by a magnetic ﬁeld.

3.2.4 Exactly solvable non-symmetric model

The considered tunneling problem provides an insight into the dependence of the
tunneling exponent on B and the electron density observed for electrons on helium
[34], and can be applied to correlated electron systems in heterostructures [66]. Due
to the speciﬁc symmetry (3.19) it can be also solved by a standard technique by

considering (pm, .2) as coordinates of the tunneling particle instead of (x, 2). In this

41

formulation, the kinetic energy pf/2m + "1.1112232 / 2 does not include the magnetic ﬁeld.
The tunneling problem is therefore mapped onto the problem of multidimensional
tunneling without magnetic ﬁeld. To check that our approach works equally well in
a case where there is no speciﬁc symmetry, we consider tunneling in a potential with

a symmetry-breaking term (1:132.

1 ‘ 2
U(:1:, z) 2 imwga"? + (nrz + 21-7; (1 — %) (z > 0). (3.20)

Such a form of the tunneling potential accounts for the change in electron-electron
interaction as the tunneling electron moves out of the 2D layer. The symmetry (3.19)
is broken and there is no transformation of variables that would reduce the role of
magnetic ﬁeld to the potential energy only. We now show that the generic features
of the solution remain unchanged.

The Hamiltonian equations (3.3) are linear even with the symmetry-breaking term,
and can be solved exactly for the new potential (3.20). The same initial conditions
(3.18) are used to construct the wave function 1/1(r) 2 D(r) exp[z’S(r)]. The caustic of
interest crosses the real space at the point LL‘ 2 1‘6, .2 2 756 (2C 2 L, :1:c 2 0 for B 2 0)
and is marked by a cross on Fig. 3.7. The escaped particle moves along the classical
trajectory that comes from large 2: and negative 31: and goes to positive :1: and large
2. Note that the classical trajectory does not cross the line U (r) 2 E. The velocity
of the escaped particle is not equal to zero at any point, including the exit point.

As before, the function ImS has two branches, which, however, are no longer
symmetrical in :r. The cross sections for constant 2 are shown in Fig. 3.8. Apart from

the symmetry :1: —> —:1:, which is naturally no longer present, the main properties

42

 

1.3 ( l I i I I l

 

 

 

1.2 — ‘

1.1 » ‘

1 — \\ ‘

\ \ \ \ \ \

. \\ \ \ \ \ \ \\ -

0-9 ' Classically forbidden region 7
.0.6 —0.4 .02 0 0.2

X/L

Figure 3.7: The classical trajectory of the escaping particle for potential (3.20) with
a 2 0.5 and same values of parameters 112670 and won, as in Figs. 3.1, 3.5, 3.6. The
cross marks the branching point for function ImS - the point where caustic goes
through the real space. The anti-Stokes line starts from the branching point. The
escape takes place at the point where classical trajectory intersects the anti-Stokes
line. Note that the trajectory lies away from the line of zero velocity U (:r, z) 2 E.

of the manifold remain unchanged (refer to Fig. 3.6). For 2 < zc, the branch 1
describes the tail of the intrawell state. It has a minimum for :1: 2 0 and monotonically
increases with z. The branch 2 descrilms the wave “rel Iected” from the caustic and is
nonmonotonic in both :1: and 2. Let us denote by z,,, the closest point on the classical
trajectory to the well at z 2 0. For 2.", < z < 20, the branch 2 of ImS as a function of
a: has two minima that correspond to the classical trajectory. For 2 2 zm these two
minima merge together. The value of ImS at the minima is independent of 2, as it

should be. For 2: > zC, 11118 is remesrmted by two branches, one of them describing

43

 

 

 

 

 

 

 

 

 

149 I r I 1 f 4 1 17 I 1 r
1-1 l
t 1.49» (b) ,’.
V)
E
1.48 '
1.48 ~
1.47 ' 1.47 _
- -02 -01 0 -03 - -
O3 X/L 02x 01 0

 

 

 

 

 

Figure 3.8: Cross-section of function Im S for constant 2 near a branching point
(136,26): (a) zm < z < zc; (b) 2 2 zc; (c) z > zc. The tunneling potential is (3.20)
with the symmetry-breaking parameter a 2 0.5, and other parameters are the same
as in Figs. 3.1, 3.5, 3.6, so that won, 2 1.2 and 1.11070 2 1.2. Although the symmetry
m —+ —:1: is no longer present, all of the main features of the solution remain the same.
The minima of the branch 2 lie on the classical trajectory shown in Fig. 3.7.

44

the wave coming from large z (unphysical) and the other describing the wave going to
large 2 (escaped particle). The minima of the branches continue to follow the classical
trajectory.

The wave function is determined by one of the branches of action S (r) The
switching between two branches takes place 011 the anti-Stokes line that starts form the
caustic point a: 2 226, z 2 :C. The exit point is located where the classical trajectory
intersects the anti-Stokes line (see Fig. 3.7). The exponent of the tunneling rate is
’R 2 2ImS(rC1). It can be found by solving the equations in imaginary time.

To summarize, the proposed solution to the tunneling problem in magnetic ﬁeld
does not rely on a special symmetry of the tunneling potential. The characteristic

features of the action manifold, shown in Figs. 3.5, 3.6, remain unchanged.

3.3 The path-integral formulation in a magnetic

ﬁeld

In the absence of a magnetic ﬁeld, the problem of tunneling decay has a very elegant
solution in the path-integral formulation [9, 10, 16]. One of the main advantages of
this method is that the escape rate can be calculated at: finite temperatures, While the
answer for T 2 0 can be obtained by taking the appropriate limit2. Just like in the
WKB approximation, the tunneling exponent is equal to the Euclidean action, which

is calculated along a trajectory going in imaginary time with imaginary momentum

 

2However, the method heavily relies on the fact that the tunneling potential is parabolic near
its minimum. Desire to calculate the tunneling rate from a 2DES, where the tunneling potential is
strongly non-parabolic motivated the formulation of the WKB approximation presented in Sec. 4.2.

45

and real coordinates. The tunneling trajectory - “bounce”- starts from the well,
reaches the boundary of classically allowed region and returns back to the well. In the
presence of magnetic ﬁeld, the time-reversal symmetry is broken, and it is therefore
clear that not only regular W KB approximation, but also the “bounce” technique has
to be signiﬁcantly modiﬁed if it is ever to be used for B 2 0. An attempt is provided
in this section.

The general tunneling problem is formulated with non-Hermitian boundary con—
ditions - current ﬂowing from the well to inﬁnity. Consequently, the energy E of a
particle localized in the metastable state acquires a. small imaginary part that de-
scribes exponentially small decrease of the probability to ﬁnd a particle in the well as
time goes. The tunneling probability is defined as a ratio of the current behind the

well to the population of the well [64].

, 1 )
W 2j [/(Ir|1/J[2] , j 2 — [1,/2* (p — EA)1/J + 6.0] .
, 2711 c

On the other hand, from the Schrodinger equation, we know that

21 ImE [f drlrol2] = g; / dr [—1/2*v21//~ + W211” + age/21% + z/JAVII”)]

_ :2 [yep—iv — SAM; + I‘tiV - SAM/f]

2771.

= —-1Ij(r). (3-21)

We have recovered the result valid in the absence of a magnetic ﬁeld that the proba-

bility of tunneling from a state with energy E is given by
II”(E) 2 —2ImE.

The overall tunneling rate can be obtained by avr,‘raging over all states, W 2

46

—2(ImE). The fact that the tunneling rate is exponentially small implies that the
imaginary part of energy Im E is exponentially smaller than its real part Re E. This
is also true regarding the partition function Z, whose imaginary part is exponentially
smaller than its real part. This results in a simple expression for the tunneling rate

in terms of a partition function:
W : —2Z“l 2 (111113) (FE/T 2 2TZ“1Im [Z chi/T] 2 2TImln Z, (3.22)

which coincides with the answer known to be valid for B 2 0.
In the path-integral formulation, we represent the partition function as an integral

over all closed paths that return to the well in time T‘l [9, 10, 16]:

z = / (11(0) '/(0):rm) Dr(r) exp [—SE[r( .—)]) ()3 = T- ), (3.23)

where the path-integral is taken over various real paths I‘(T) that satisfy periodic
boundary conditions. The exponent is given by the classical Euclidean action in a

magnetic ﬁeld

"’2” 111 (1r 2 _c ,

The path-integral (3.23) is dominated by paths that provide extremum to the

Euclidean action, so that (SS 15 2 0. The extremal paths r(r) therefore satisfy:

(Fr , ,e (If

m—, 2 VD (r) + z— — x B . (3.25)
(172 c (IT

Without a magnetic ﬁeld, Eq. 3.25 descrilms classical t rajectories r(T) in the inverted

potential -—U (r), which are real. For ﬁnite magnetic ﬁeld both coordinate r and

momentum p along the extremal trajectory become complex. The complex value

47

of dynamical coordinates present no difﬁculties, even though the integral (3.23) is
evaluated over real paths r(T). This is because we can always shift the integration
contour into complex value of r in the spirit of the steepest descent method.

The extremal path f'(T) is also called “bounce” because it starts close to the top
of the inverted potential —U(r) (minimum of the potential U (r)), then goes far away
across the of —U (r), and returns back to the top. Along this trajectory, most of the
time is spent near the starting and ending points, where the velocity exponentially

decreases. The fast oscillation occurs over time Q" r» [Ix’l’T’mx

l (curvature near the top
of the real potential barrier. or ecpiivalently near the bottom of the inverted potential).
The actual instant of imaginary time T0 where this fast oscillation occurs becomes
arbitrary in the limit T ——> 0. \I'e will use this idea when discussing the prefactor of
the tunneling rate in Appendix A.

The fact that the dynamical coordinates take complex values along the bounce
is a consequence of a broken time-reversal symmetry. However the equations of mo-
tion (3.25) preserve a symmetry under a simultaneous action of complex conjugation

and time inversion. Periodic boundary conditions rm) 2 1(6) guarantee that this

symmetry is valid for the extremal trajectory as well:

l"(T) 2 f‘*(1)’— T), p(T) 2 —p*(.3 — T), (3.26)

Due to the property (3.26), the maximum value of exponent given by the Euclidean
action S E[f'] along the extremal trajectory is real.
An interesting feature of the “bounce” trajectory in a magnetic ﬁeld, is that

momentum, and generally velocity. does not become zero at the turning point for

48

T 2 6 / 2. The symmetry (3.26) only requires that the real part of df/dT is zero for
T 2 B / 2. This is in accord with the results for WKB approximation in magnetic ﬁeld
[see Sec. 3.2], where the velocity is generally not equal to zero on the trajectory of
an escaped particle in the classically allmvr—xl region. Heal momentum of the escaped
particle corresponds to purely imaginary momentum for the motion in the inverted
potential. Therefore, if at the turning point of the “bounce”, the momentum along f
matches that for the trajectory of classically propagating particle, it has to be purely
imaginary. On the other hand, in the absence of a magnetic ﬁeld the “bounce”
trajectory is purely real. The velocity at the turning point is equal to zero, but so is
that for a classically propagating particle.

In the path-integral (3.23), integration over ﬂuctuations around the extremal tra-
jectory f(T) determines the prefactor. It is important that this prefactor is imaginary,
since we are calculating the (exponentially small) imaginary part of the statistical

sum. The detailed analysis of it is given in Appendix .-\.

3.4 Summary

In the semiclassical approximation, the problem of single particle tunneling in a mag-
netic ﬁeld can be solved by analyzing the Hamiltonian trajectories of the particle in
complex space and time. The connection of decaying and propagating waves occurs
on caustics of the set of these trajectories. This approach does not require us to
consider either any piece of the electron potential or Iiiagnetic ﬁeld as a perturbation

and can be applied to a three-dimensional potential of a general form. It gives an

49

escape rate which is generally erpmzcntioilg/ smaller than the probability for a parti-
cle to reach the boundary of the classically accessible range U (r) 2 E. The escaped
particle “shows up” from the tunneling barrier with finite velocity and beyond the
line U (r) 2 E. Finally, it was shown that in cases where the intrawell potential is
parabolic near its minimum, the escape rate in the presence of magnetic ﬁeld can still
be calculated using the “l’)(’)unce” technirple, if we allow the tunneling coordinates to
take complex values. The “bounce”, as well as instanton, technique is a thermody-
namic method and gives the escape rate for both zero and ﬁnite temperature. Part
of the next chapter is devoted to a ﬁnite-temperature calculation of the escape rate
in cases where the intrawell potential is strongly non-parabolic near the minimum, as

is certainly true for tunneling from 2D electron systems.

50

Chapter 4

Tunneling transverse to a magnetic
ﬁeld from correlated 2D electron

systems.

A low-density 2DES is a very interesting system. Strong electron correlations that are
present in it are very different from electron correlations found in the quantum Hall
systems, in particular they are not necessarily imposed by the magnetic ﬁeld. The
unusual properties of in-plane transport characteristics [49]—[51] can be understood by
taking many-electron effects into account [52] Therefore it is natural to suppose that
measuring the out-of-plane tunneling current can also be useful in revealing electron
correlations. Such tunneling experiments deal with tunneling from the 2DES into
the vacuum, not into the 2DES. In addition, magnetic ﬁeld is applied parallel, not

perpendicular to the layer.

The effect of electron correlations on the tunneling rate may not be described
in terms of a phenomenological tunneling Hamiltonian: it is the tunneling matrix
element itself that is sensitive to the electron correlations. Therefore we have to solve
the coupled tunneling problem where the in-plane degrees of freedom change together
with the tunneling coordinate. It is possible to do that. using the approach developed
in Chapter 3, where the multidimensional decay in a magnetic ﬁeld was considered
at zero temperature. During the escape, the tunneling electron transfers a part of
its Hall momentum, which it acquired when moving out—of-plane perpendicular to
B. The momentum transfer occurs dynamically. The overall effect is very similar to
what happens in the h‘Iéissbauer effect, where the momentum of a gamma quantum
is transfered to a whole crystal, and consequently there is no frequency shift in an
absorption line.

The vibrations of electrons in the plane occur with frequencies of the order of the
plasma frequency cap, which therefore characterizes the rate of inter-electron momen-
tum exchange. The interrelation between the plasma frequency cup and the tunneling
duration Tf determines how much of the Hall momentum is transfered to the crys-
tal. The remaining part goes into the excitation of phonons, and can be viewed as
a remainder of the single-electron magnetic barrier. .-\s a result, the tunneling ma-
trix element depends strongly, and very specifically, on electron density, and also on
temperature and the magnetic field. In particular, if .ePTf >> 1, then all of the Hall
momentum is transfered to the crystal. and the suppression of the tunneling rate
by B is completely eliminated. Because of such strong dependence, it is possible to

study the electron correlations and in-plane dynamics for frequencies comparable to

52

the reciprocal imaginary tunneling time that an electron spends under the barrier.

Explicit results on the effect of electron correlations on tunneling will be obtained
assuming that electrons form a Wigner crystal. Because of strong correlations, over-
lapping of the wave functions of individual electrons is small, and electrons can be
“identiﬁed”. The problem is then reduced to the tunneling of an electron coupled to
in-plane vibrations of the Wigner crystal. As we will see, the results provide a good
approximation also for a correlated electron liquid.

In Sec. 4.1 we formulate the model. In Sec. 4.2 we obtain the general expression for
the tunneling rate in the WKB approximation for ﬁnite temperatures. In the analysis,
the discreteness of the energy spectrum of electron motion transverse to the layer is
taken into account. The result can be understood in terms of the tunneling trajectory
where the duration of motion transverse to the layer (in imaginary time) is not ﬁxed,
it has to be found for given parameters of the tunneling barrier, temperature and the
magnetic ﬁeld. In Sec. 4.3 we derive the tunneling exponent. After the elimination of
phonon variables, the tunneling exponent takes the form of a retarded action for 1D
motion. The exponent depends on phonon frequencies and the form of the tunneling
potential. The limits of T = 0 and the case of small phonon frequencies are analyzed
in detail. As discussed in Sec. 4.4 the role of in-plane conﬁnement in out—of—plane
tunneling can be effectively described in terms of one characteristic phonon frequency.
This corresponds to using the Einstein model of a crystal. For an Einstein solid,
the many-electron problem is described by a single—particle potential. The transfer
of in-plane Hall momentum from the tunneling electron to the electron system as

a whole in a many-electron formulation (_torresponds to the momentum transfer to

53

the static conﬁning potential in the effective single-electron problem. Two typical
tunneling potentials are analyzed: (i) triangular barrier represents smooth tunneling
potentials, where the tunneling length changes with tlte intra-well energy level; (ii)
square barrier represents geometrically defined barriers. where the tunneling length
is independent of the intra—well energy, at least in some broad range. It turns out

that the tunneling rate has some qualitatively different features in these two cases.

4.1 The model: tunneling from a harmonic

Wigner crystal

A 2D electron system displays strong correlations if the ratio I‘ of characteristic
Coulomb energy of the electron-elect.ron interaction (73(7m)1/2 to the characteristic
kinetic energy is large (here. 12. is the electron density). In degenerate systems the
kinetic energy is the Fermi energy rte/7n, whereas in nondegenerate systems it is
the thermal energy T. An example of a strongly correlated nondegenerate 2DES is
electrons on helium. The experimental data for this system refer to the range I‘ > 20
[53]. A classical transition to a Wigner crystal (“T“) was observed for P z 130
[67, 68]. Recently it became. possible to achieve a strongly correlated regime with
values of P ~ 40 in low-density electron and hole. systems in semiconductors. This is
expected to be sufﬁcient for Wigner crystallization in a degenerate system [48, 66].
The effect on tunneling of the magnetic field B parallel to the electron layer is most

pronounced if the tunneling length L is long, because the in-plane Hall momentum

:34

due to tunneling mwcL is simply 1')rop(_)rtional to L. Respectively, of utmost interest
to us are systems with broad and cmnparatively low barriers. Yet in experimental
systems the barrier widths are most likely to be less than 103A. Therefore, in order
to somewhat simplify the analysis we will assume (although this is not substantial)
that L is less than the average inter-electron distance ~ 72“”. In this case, since-
electrons in a strongly correlated system stay away from each other, the in-plane
electron dynamics only weakly affects the tunneling potential [69] in the absence of a
magnetic ﬁeld. We will neglect the corresponding coupling that is present regardless
of the presence of a magnetic field, and cmicentrate only on those effects of in—plane
dynamics that are specific for finite B.

The major effect on tunneling comes from a few nearest neighbors, and the pres-
ence or absence of long-range order in the 2DES does not affect the tunneling rate.
Therefore we will analyze tunneling assuming that the electron system is a Wigner
crystal. AS we will see, the results will indeed depend on the short-wavelength modes
of the WC, as expected from the above. arguments.

In a strongly correlated system where the characteristic energy of Coulomb inter-
action is much larger than the Fermi energy. exchange effects are not signiﬁcant, and
one can identify the tunneling electron. Its out-of—planc motion for B = 0 is described
by the Hamiltonian

I)?

211/
The potential U (z) has a well which is separated by a tunneling barrier from the

extended states with a quasi—continuous s].)ectrum, cf. Fig. 1.2. The well is non-

C.
CI

parabolic near the minimum, in the general case. The metastable intra well states
are quantized. We will consider temrmratnres for which nearly all electrons are in the
lowest level, with energy E9.

The magnetic ﬁeld B parallel to the layer mixes the out-of—plane motion of the
tunneling electron with the in-plane vil')rations of the Wigner crystal. The full Hamil-

tonian is of the form

H : H0 + [I]; + Hy. (4.2)
with
l _1 .,
H, = E: [m pkjp._kj + rrzwglukjumj] (4.3)
krj
and
I . .7 .
HB : Smwfzz — w(..:.\"_1'/2 Z[B >< pkj]z. (4.4)
_ k3],

Here, pkj, Ukj, and wkj are the 2D momentum, diSplacement, and frequency of the
WC phonon of branch j (j : 1, 2) with a 2D wave vector k. We chose the equilibrium
in-plane position of the tunneling electron to be at the origin. Then its in-plane 2D
momentum is p : N‘l/2 2 pk,- for B = 0.

The interaction Hamiltonian H n (4-1) does not conserve the phonon quasi-
momentum k. The Hall momentum of the tunneling electron, pH = (e/c)[B x z],
is transferred to the WC as a whole. The. term H 3 couples the out-of-plane motion
to lattice vibrations. The problem of many-electron tunneling is thus mapped onto
a familiar problem of a particle coupled to a bath of harmonic oscillators [70, 16],
with the coupling strength controlled by the magnetic ﬁeld. The distinctions from
the standard situation stem from the non-parabolicity of the potential well near the

56

minimum and from the fact that coupled by H B are the electron coordinate z and
the in-plane momenta of the lattice. These quantities have different symmetry with
respect to time inversion. In the general case [for example, where the potential energy
of the system has odd-order terms in the displacement s ukj], the broken time-reversal
symmetry requires a special approach to the analysis of tunneling [46]. The results
discussed below can be appropriately generalized using this approach.

For the model (4.2), the analysis is simplified by the structure of the Hamiltonian
(cf. [16]). For vibrations with the Hamiltonian H,. (4.3), one can always make a
canonical transformation from the canonical coordinates and momenta “kj and pk,-
to the new canonical coordinates and momenta pk,- and —uk,-, respectively. This
transformation interchanges the time-reversal symmetry of the in-plane dynamical
variables, it makes pk,- and ukj even and odd in time, respectively. Because H B
is independent of ukj and is linear in pk}, in the new variables it takes on a more
familiar form of a potential coupling which depends only on coordinates z, pkj. In this
representation the kinetic energy is given by jig/2m + ij mwijukju_kj / 2 and does
not depend on the magnetic ﬁeld. The symn'ietry under time—inversion is therefore

restored.

4.2 A many-body WKB approximation

4.2.1 General formulation

We will evaluate the tunneling rate IV in the W I\' ll approximation. The major
emphasis will be placed on the tunneling exponent. We will assume that the escape
rate is much less than the intrawell relaxation rate for relevant states, and there is an
established thermal distribution over the intrawell states of the system. This is not
necessarily true for 2D systems. Our results can be generalized to the case of slow
intrawell relaxation, see Sec. 6.2..

We consider the decay of the metastable intrawell st ates or = [{n, nkj}], with decay
rate rates W0, where phonon states are entunerated by their occupation numbers nkj,
and the index n enumerates states quantized in the out —ol'—plane direction. These rates
sharply increase with state energies Ea, whereas the Boltzmann intrawell distribution
exponentially decreases with E,,. As a result. there is a. comparatively small group of
states which mostly contribute to the escape from which the system is most likely to
escape (for fast intrawell relaxation, the relative population of these states remains
unchanged). This allows one to characterize escape by a single rate W. To logarithmic

accuracy,

W : 2‘12 l'l'(,ex1)(-—/3E,.). (4.5)

. ., . y, 2

ll o : (t’l(‘xl)[—2‘8I1(€f$€in)l'("l(€in)l ‘
Here, we introduced a vector 5 = (.3, {pk_,-}) with components which enumerate the
z-coordinate of the tunneling electron and the “coordinates” pkj of the phonons,

.38

Z z exp(—BE1) is the partition function calculated neglecting escape, l/JQ(£) is the
intrawell wave function, and (Z, are the prefactors in the partial escape rates [although
we will not discuss them here. they will be calculated for a particular tunneling
problem in order to compare the theory with the exl'wrimental data [34] in Chapter 6].

The exponents in the rates ll"), are determined [7] by the wave functions «1).,(5) at
the turning points 5, on the boundary of the classically accessible ranges (5f depend
on a, see below). It is convenient to evaluate 't;’lt(,(§f) in two steps, each of which gives
an exponential factor. The first factor, exp[—Sn(€,~. £,,,)], describes the decay of the
wave function under the barrier. Formally, it relates t;,(§f) to 2120“,“). The point 5m
is chosen close to the well, yet. it lies under the barrier, so that So can be calculated
in the WKB approximation. The second factor is 1,},(5m) itself. The resulting rate
should be independent of 5,”.

We start with the function .S',,(§. 5,"). To the lowest order in h, for systems with
time-reversal symmetry (which we restored by the canonical transformation) it is

the action for a classical underbarrier motion in imaginary time 7' = it with purely

imaginary momenta [9]
I): I 108/03,11k] 1‘ —’1 ()8 [(91)](1'. (4.6)

AS a function of the imaginary time 7', the action S(‘E. 5,“) is given by the integral of

the Euclidean Lagrangian Lu.

'T

5,.(5, gin) Z / LEdT — 111,71 (4.7)

I ()

The Lagrangian LE is obtained from the Hamiltonian (3.2) using the Legendre trans-
formation L : pz(dz/dt) — 2: uk,(dpkj/dl) — I], followed by the transition to imagi-

5f)

nary time, which gives

LE 2 L0 + L" ‘l- L1}. (4.8)
Ilere,
III (I: 2 .
L() I ‘7 E + (”(3) 14,- : H8, (4.9)

and L, is the Lagrangian of the phonons, LL, : 2k, [k], with

l 1 (kay dP-kj

Lk' = _pk'p—k' + .
J 13m. J J Qinwﬁj ([7 (1T

 

(4.10)

The classical equations of motion in imaginary time have the standard form

(I OLE 01419

 

 

. . : (l. 4.11
(17 ()5 0.5 ( )

where overdot means differentiation over 7'. To calculate the escape rate, one has
to ﬁnd the trajectory which goes from €(()) : 5m to the boundary of the classi-
cally accessible range 5, at. a certain time Tf and calculate the action So, along this
trajectory.

If the potential barrier L(:) is smooth. the wave. function and its derivatives under
the barrier have to match the \\~"I’\'B wave function in the classically allowed range
behind the barrier. The matching occurs at a turning point of the classical motion

(4.11) where the derivatives of the both wave functions become equal to zero [7], i.e.

for aSa/Bz = BSa/Bpkj : (l. i.e.
73(7)) =0, pk,(rf) : 0. (4.12)

Eq. (4.12) is also the condition of the extremum of .8), with respect to the points 5
on the boundary of the classically accessible range: the escape rate is determined

(il l

by the minimum of So on this boundary. A detailed analysis of the behavior of
multidimensional tunneling trajectories in imaginary time for systems with time-
reversal symmetry is given in Ref. [15].

Time-reversal symmetry of the. equations (4.11) in coordinates (z, pkj), together
with the condition (4.12), shows that. if the equations of motion are extended beyond
Tf, the system will bounce off the turning point. and then move under the barrier back
to the starting point. The section of the trajectory for 7' > Tf is mirror-symmetrical

to that for 7' < T],

2(7) + T) : :(rf — r), pk.,(’rf + 7') = pig-(7f — 7'), (4.13)

where 0 S 7' g 7'}. As a result. the tunneling exponent 250 can be calculated along
the trajectory (4.11) that reaches the turning point at T; and returns to the well at
277.

The time Tf is determined by the boundary conditions (4.12) and by the initial
conditions on the trajectory, which are given by t/i(,(£in). If the intrawell dynamics
is semiclassical, the dominating ('()lll'1‘ll)lll.l()ll to the overall rate W (4.5) comes from
the energies E, for which the duration of the tunneling motion 7'; = 6/2 [9]. In the

general case this is no longer true.

4.2.2 The initial conditions

We are consider the situation where, at least for low-lying intrawell states n, the
characteristic lengths 1m." of localization in the z-direction are much less the typical

widths L of the tunneling barrier, so that 7,, << L. Then, even where the effect of

61

the magnetic ﬁeld accumulates under the barrier and the tunneling rate is strongly
changed, the ﬁeld may still only weakly perturb the intrawell motion. In this case,
inside the well and close to it. the ont—of—plane electron motion is separated from the
in-plane vibrations. Respectively, the states of the electron-phonon system can be
enumerated by n and the phonon occupation numbers nkj, i.e. a = (n, {nkj}), and

the energies are
Ea 7— En + E 5k}: ﬁg 2 wkjnkj. (4.14)
k?

Usually the interlevel distances EH1 — En >> atkj, for low-lying levels.
Because of the separation of motions, we can choose a plane .2 = zin under the bar-

rier but close to the well, so that, for g a: g the wave functions t/Ja(£) are semiclassical

in

and at the same time can be factored,

[if/$71.17).“ } (E) (X 6—7,,z 8X1) [_ Z Sung (1310)] ' (415)
k}

The action Snk, determines the dependence of the wave function on the phonon co-
ordinates.

For 6 = 5,“, Eq. (4.15) gives the initial values of the dynamical variables {(0) E {in
and 6(0) on the WKB trajectory (4.11). In particular if, for z z zin, the potential

U(z) varies over the distance much bigger than l/Trn, then

 

‘2 .3» — ”2
~[l ( In) Enl] , (4.16)

m

2(0) 2 3,“, 5(0) : h = [

711.

and 7,, (4.16) is independent of the exact position of the plane .2 = 2,“.
It is convenient to write 51...,- and pkj in Eq. (4.15) in the energy-phase represen-
tation, using the phonon energy 5k,- and the imaginary time 7k]- it takes for a phonon

6‘2

1/2 of the classically allowed

to move under the barrier from the boundary (27715“)
region to the given pkj. With the Euclidean Lagrangian of the phonons (4.10), we

have for pkj = [pkjlin 3 Drew)

-0
Snkj (pk/(0)) I / (17'ij(7') — Eijkja (4.17)
' "TM
and
pkj((l) : ek,(2mek,)1/2 cosh wkjrkj, (4,18)

. , . 12 .
pig-(0) :- e_kj (Zekjmwﬁj) / smh wkjrkj

[ekj is the polarization vector of the mode. (k, 3)].

4.2.3 A three-segment optimal trajectory

To evaluate the escape rate W to logarithmic accuracy, one can, following Feynman’s
procedure, solve the equations of motion (4.11) for the vibration “coordinates” ka-(T)
in terms of z(r) and the initial energies ekj and phases wkjrkj. Then, from the
boundary condition (4.12), one can express Tkj in terms of other variables, and then
do thermal averaging by integrating the escape rate over ekj with the Boltzmann
weighting factor. Here we reverse the order and give an alternative derivation, which
provides a better insight into the structure of the tunneling trajectory. In this section
we average over phonon energies ck,- to find times Tkj. In the next section we eliminate
the phonon variables to obtain the tunneling exponent. We note that, from Eqs. (4.5),

(4.7), (4.15), and (4.17), the partial escape rate ”"0 can be written as W0 oc exp(—sa),

63

with

or,- 2n+m
:Ej l; (17' Lk](7' . )___/ (17' [43(7') + E f (17' ij(7') — 2En7'f
. (l 2

kj TI

—2 2819(77 + Tkj'). (4.19)
K?

(the term ynzin in (4.15), which is small compared to 3,, ~ 7,,L, is incorporated into
the prefactor, see Sec. 6.3.

Eq. (4.19) suggests that one can think of the optimal trajectory as consisting
of 3 segments. In the first segment. from —Tk,- to 0, the z-motion of the tunneling
electron is disconnected from the vibrations. The electron stays at z = 0, while
each vibrational mode. moves for the time Tkj, starting from the boundary ukj = 0
[cf. Eq. (4.17)]. This motion is determint-rd by the Lagrangian ij. At 7' = 0 the
interaction is turned on, and in the second segment the electron and the vibrations
move together for the time 27]. with the Lagrangian LE. During this motion, the
trajectory (4.11) bounces off the turning point (4.12), and the electron comes back to
z = 2,". After that, in the 3rd segment, the coupling is turned off again, the electron
stays at z,,,, while the vibrations continue to move for the times Tkj back to ukj = 0.
The three-segment vibration trajectory is continuous.

To logarithmic accuracy, the tunneling rate W is given by (exp(—ZSE)). The
averaging here should be performed over the intrawell vibration energies ekj [we note

that Tf and Tkj in S]; are. determined by ekj],

. T!
IV or 2:] H (lik, exp [(277 — ,8)E,, — 2 / (ITLE(T)
n k) ' 0
-(l
_2 Z/ (lTLk_,‘(T) + (2713' 'l‘ 2Tf — ,8)?ij . (4.20)
kj —Tk,-

64

The integral over ekj should be calculated by the steepest descent method. From
Eqs. (4.12), (4.17) it follows that the partial derivatives of 5;; over the times T, and
ﬁg (which depend on sky) are equal to zero. The condition of the extremum of 5'3

with respect to ekj then gives
1
no = .328 — 7,, (4.21)

This expression shows that the duration of motion Tkj is the same for all vibrational
modes. Moreover, the overall duration of the three-segment optimal trajectory of
each vibration is 2(7'kj + 7;) : .)’_ Examples of the trajectories are shown in Fig. 4.1.

For low temperatures, 13 > '27,, the direction of time along the vibrational tra-
jectory does not change. In this case the value of TR,- (4.21) which provides an ex-
tremum to the integral over 5k, is positive. The corresponding branch of the intrawell
vibrational wave function oc exp[—S ,.;( 0)] decays with the increasing pk, in the classi-
cally forbidden region pk, > (2)/NM)”; ()n the other hand for higher temperatures,
5 < 27,, we have Tkj < (1. This shows that. the extremum of the integrand in Eq. (4.20)
is reached if the intrawell vibrational wave function is analytically continued from the
decaying to the increasing branch.

For Tkj = ([3/2) — T) < 0, the “free-vibrations” term 55(0) in the Euclidean
action SE is negative, it gives rise to the decrease of the tunneling exponent. This
is the formal reason why an in—plane. magnetic ﬁeld can increase the tunneling rate
compared to its B = 0 value by coupling thermally-excited in-plane vibrations to
tunneling.

If the intrawell motion transverse to the layer were semiclassical, the sum over

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I t t l r 0.6 f ,r
”in (a) M : a” :
0.3 " J
O )-
_ 0 0 1
pH
-1 .
— i n l 0 — i - 3 i
_g. + 1,, 0 21,. % +1, ‘9 + ,f 15f ~3— +tf 21f

Figure 4.1: The optimal trajectories of the tunneling electron 2(7) and of one of the
vibrational modes pH(T) for H > 27] (a) and B < 27, (b). The numerical data refer to
the Einstein model of the Wigner crystal, pH is the p-component of the vibrational
momentum in the Hall direction z x B. The arrows show the direction of motion
along the optimal trajectory when 13 < 27,. The tunneling potential is of the form
(4.38), with dimensionless cyclotron frequency were 2 2.0, where To = 2mL/7 is the
imaginary transit time for B : 0. The phonon frequency is wp'ro = 1.0.

the energy levels of this motion E,, in Eq. (4.20) could be replaced by an integral. It
could then be evaluated by the steepest descent method, as in the case of in-plane
vibrations, with the result 7, : 1312,71,,- = 0. This is the familiar result of the
instanton theory, in which the whole system moves under the barrier from the well
to the turning point and back over the imaginary time 6 [9]. Clearly, in this case one
should not expect the tunneling rate to be enhanced by a magnetic ﬁeld.

In the case of 2D electron systems. the potential well is not parabolic and the

low-lying intrawell states are not semiclassical. Therefore the sum over E, in (4.20)

66

may not be replaced by an integral. each term has to be considered separately, and
the duration of motion T, for each energy level has to be found from the boundary

condition (4.12).

4.3 The tunneling exponent

Eqs. (4.5) and (4.19) allow us to write the escape rate as a sum of the escape rates
for different intrawell states a. To logarithmic accuracy, the overall escape rate is
determined by the maximum of the escape rate with respect to the intrawell state
77., with account taken of the thermal population of the state [here we assume that

thermalization inside the well occurs faster than escape],

IV 0( max exp(—It’,, — HE”), R, = ngip Rn[Z(T)], (4.22)

The functional 7%,,[3] is a retarded action functional for a 1D motion normal to the
electron layer. It is determined by the. functional 3,, for the nth state from which
the the dynamical variables of the in-plane vibrations have been eliminated. The
elimination can be done in a stainlard way [70] by solving the linear equations of
motion (4.11) for ukj,pk_,- with the boundary conditions (4.18), (4.12), and (4.21).

This gives

2” m dz 2 ,, 1 2 2
72,,[z] = 0 (IT, 3 ET- +f/(z)+§mwcz (T1) +Ree[Z]—2TfEn (4.23)

(we have set the energy of the intrawell ground state Eg = 0).

The term Ree is the retarded action describing the effects of electron-electron

67

interaction that we modeled by electrtm-phonon coupling [cf. 4.4],

(.02 '2Tf '7']
R0,,[z] : ——2£/ / (lTldTQZ(T1)Z(T2)X(’/'1 — T2) (4.24)
0 . 0

Here, the X(7’) = (pH(T)pH(0)) is the correlation function of the in-plane momentum
p” of an electron in the correlated 2DES. For electrons forming a Wigner crystal, it

is simply related to the phonon Green‘s function,

7n — w -T —. —w 'T
X(7') 2 EN geek) [”ij k] + (7ij +1)8 k] ] (4.25)

1 is the thermal occupation number).

(ﬁle = lexpfﬁww) — ll—

The term Ree is negative. It means that the electron-electron interaction in a
correlated 2DES always increases the tunneling rate in the presence of a magnetic
ﬁeld. Moreover, when this term exceeds (more2 / 2) f Z2dT, the tunneling exponent as
a whole decreases with the increasing B.

Two physical phenomena are (_lcscribed by the term ”Ree. One is the dynamical
compensation of the Hall momentum of t he tunneling electron by the WC as the elec-
tron moves under the barrier in the z-direction. The other is thermal “preparation”
of the Hall momentum for the tunneling electron, which is then transformed by the

magnetic ﬁeld into the momentum of motion in the z-direction. These effects are

analyzed in the following subsections.

4.3.1 Zero temperature limit

It would be natural to think that, since tunneling is accompanied by creation of
phonons for T = 0, then the higher the phonon frequency the lower the tunneling
rate. In fact just the opposite is true.

68

The effect of the electron-electron interaction on tunneling, as characterized by
”Ree, depends on the interrelation between the characteristic phonon frequency (up
and the tunneling duration Tf. The quantity (up also characterizes the rate of inter-
electron in-plane momentum exchange. When the tunneling electron is “pushed” by
the Lorentz force, it exchanges the in-plane momentum with other electrons. The
parameter prf determines what portitm of the momentum goes to the crystal as a
whole during the tunneling (note that the tunneling motion goes in imaginary time,
and the quantity Tf characterizes the time uncertainty rather than the actual duration
of a real process, see Ref. [71]). In the adiabatic limit of large prf, all electrons
have same in-plane velocity, with an accuracy to quantum fluctuations. Therefore
the Lorentz force produces no acceleration, and no phonons are created during the
tunneling. The effect of the magnetic field on tunneling should then be eliminated.

These arguments are conﬁrmed by the analysis of Eq. (4.24). If the electron
system is rigid enough, so that wijf >> 1, the major contribution to Ree comes from

T1 — 7'2 ~ (4);; << Tf. Therefore Z(T~2) z Z(T1), and we could use an expansion:

| *-*

55(72) z 3(71) + 3(Ti)(7’2 — Ti) + 5(T1)(T2 _ Tll2 (4-26)

‘

[\J

in the functional 72,... The upper limit in the integration over 7'2 can be extended to
inﬁnity, since it introduces only an exponentially small error. Then the leading term
in expansion (4.26) cancels the term (X :32 in (4.23), which represent the single-particle
magnetic ﬁeld barrier. The linear term in expansion (4.26) gives a zero contribution.

The uadratic term leads to an effective renormalization m —> m" of the electron
q

69

mass in a magnetic ﬁeld:

 

.. . 1 .1} 2 —1 “’2
m z 777. [1+ 27—n 0 dTT X(T)] z 771. [1+ (2N) a: uni] . (4.27)

Tunneling occurs as if the electron were disconnected from the phonons, and did not
experience a magnetic ﬁeld. The only effect of the magnetic ﬁeld is that the electron
mass in Eq. (4.23) is effectively incremented by a B-dependent factor. This change
is reﬂected in all of the tunneling characteristics, including the tunneling time T, =

(m*/m)1/2T0, and most importantly, the tunneling exponent becomes appropriately

R —> (/m RBzo- (4'28)
177.

This analysis is quite general and applies for arbitrary form of potential U (z)

renormalized:

 

This includes potentials where the tunneling length is well deﬁned by a jump in
the potential, as it happens in semiconductor heterostructures. Expressions (4.23),
(4.24) for the tunneling exponent still apply provided that the tunneling trajectory
for T > Tf is now defined through Z(T) : Z(2Tf — T). In addition, because z'(Tf) 75 0

expansion (4.26) has to be changed to:

. (TZ—Tl), for ’7’] <Tf or Tf <7'2
2(7'2) % 2(7'1) + 2(7'1) X (4.29)
(2T; — T1 — T2), for 72 < T, < T1
Details of the analysis for a square potential can be found in Appendix B. The scaling
behavior of the tunneling time Tf and the tunneling exponent ”R can be also checked

using the explicit answers for a triangular barrier (4.41) and for a square barrier (5.6),

(5.7).

70

In this derivation we assume that the major contribution to the integral over T
comes from times T N I/cop << Tf. Respectively, for a Wigner crystal, the major
contribution to the sum over (k, j), comes from wkj N w,,. The integral only weakly
depends on the upper limit Tf, which also provides the inverse of the lower cutoff
frequency in the sum over (k, j). For a Wigner crystal, the dependence of the mass
renormalization on Tf is logarithmic.

The tunneling rate approaches its value for B = 0 with increasing (up. On the other
hand, the SIOpe of the logarithm of the tunneling rate as a function of we depends
explicitly on top, for wc 2, top. This provides a means for measuring top.

For prf ~ 1, only a part of the Hall momentum can be taken by the electron
crystal. The rest goes into the in-plane kinetic energy of the tunneling electron,
and ultimately into creations of WC phonons. However, the major contribution to
Ree still comes from high-frequency phonons. It can be shown from (4.24) that
this contribution monotonically increases with increasing wkj [see Appendix B for a
derivation]. This is because the more rigid the electron system is, the more effectively
it compensates the in-plane Hall momentum. An important consequence is that,
since high-frequency vibrations have small wavelengths, the major effect on tunneling
comes from the short-range order in the electron system.

On the whole, for T : 0, the magnetic-ﬁeld induced term in the tunneling
exponent is positive, i.e. the tunneling rate decreases with the magnetic ﬁeld.
This can be seen from Eqs. (4.23), (4.24) by replacing Z(T1)Z(T2) in Ree with

(1/2)[Z2(’T1) + 22(T2)] Z Z(T1)Z(T2) and then integrating the function X(T1 — 7’2) =

71

exp[—wkj(T1 — T2)] over 72 [for the term 22(T1)] and over 7'} [for the term 22(7'2)]2

 

1le 2T} 1'} r
—R..lz]< ' wao/ / dTidT2[Z2(T1)+z2(T2)]e“’ki(Tl—TZ)
8N M 0 0

 

2 2Tf
Into . _ . _ . _
: c E (lT122(T1) (2 — e ““971 — e “"9“”! T2))
8N _ 0
k3

By dropping the exponentials in the integrand, we eliminate dependence on wk], and
the sum over phonons gives a factor 2N. As a result, we get an upper bound for the

many-electron term in the exponent:
1 27f
—Ree[z] < imwf/ (1T22(T), (4.30)
0

The last expression describes the suppression of the tunneling rate by B for non-
interacting electrons. Therefore. many-electron effects at T = 0 can lead at most to
a compensation of the suppression introduced by the magnetic ﬁeld. The tunneling
rate cannot exceed its value for B = 0.

Electron correlations exponentially reduce the effect of the magnetic ﬁeld on the
tunneling rate in a magnetic ﬁeld. For speciﬁc models, the dependence of the tunnel-
ing rate on B and the vibration frequencies will be illustrated in Sec. 4.4. The results

will also be compared with the experiment in Chapter 6.

4.3.2 High temperatures and small phonon frequencies

The analysis of the tunneling rate somewhat simpliﬁes in the case of comparatively
high temperatures and small phonon frequencies, where the vibrations are classi-

cal and their frequencies are small compared to the reciprocal tunneling duration,

72

wkjﬂ,wijf << 1. In this case

., . “U
728,.[3] = —2mTw(2.Tf§2, 'z' : ”Ff—1 / (1Tz(T). (4.31)
0

Note that the origin of the z-axis has been chosen in Eq. (4.2) in such a way that z = 0
inside the potential well where the electrons are localized (or that the diagonal matrix
element of 2 on the intrawell ground state wave function is equal to zero). Therefore
in Eq. (4.31) 2 79 0 for an optimal tunneling trajectory, and correspondingly the value
of Ree on this trajectory is nonzero and negative.

Eqs. (4.23), (4.31) describe also the tunneling action of a single electron, with the
Maxwell distribution of the in-plane momentum inside the well exp(——p2/2mT). The
coupling of the 2 x B component of the momentum to the out-of-plane motion gives
rise to the term —2pwc f0” de(T) in the tunneling action [cf. Eqs. (4.4), (4.7)]. The
extreme value of the sum of this term and —])2/2m,T is just equal to —’Ree[z] as given
by (4.31).

The single-electron form of the. tunneling exponent is to be expected in the limit
of small wk], because the distribution over in-plane momenta of electrons forming a
Wigner crystal is Maxwellian, in the classical limit. For small wijf the momenta do
not change over the tunneling duration, therefore only the momentum of the tunneling
electron itself is important. The above (.lerivation provides an independent test of the
derivation used to obtain the general expression (4.23), (4.24).

Let us note that the action R,.e[z] (4.31) is still retarded, it does not correspond
to a local in time Lagrangian. The functional form of RN. remains the same even

for temperatures T < wk],- provided the phonon frecpiencies are small compared to

N

73

T,"1 and we. In this case T in Eq. (4.31) has to be replaced by (4N)‘1 Zwkj(2nkj +
1). This factor explicitly depends on the phonon dispersion law, but again, the
major contribution comes from sltort-wavelength high-frequency phonons, which are

determined by the sliort-rzuige order in the electron system.

4.4 Effect of in—plane conﬁnement on the tunneling

rate

4.4.1 The Einstein approximation for a Wigner crystal

The exchange of momentum between electron in the layer is realized more efﬁciently
by higher frequency phonons, as was discussed in subsection 4.3.1. This is also re-
ﬂected by the fact that high frequencies dominate the electron-electron interaction
term Ree in the exponent. Therefore it may be natural to use the Einstein model for
a Wigner crystal and set all frequencies equal to a characteristic value: wkj 2 top. The
plasma frequency for a 2D crystal is w], : (27re2n3/2/m)1/2, where n is the electron
density.

In the Einstein approximation the Hamiltonian H, of in-plane vibrations can be

written as:

H,, = 1) Z: [Ill—lpf, + ‘lIltdSUﬂ (4.32)

n—t

n

and the Hamiltonian coupling between out—of-plane and iii-plane motions due to the

presence of magnetic ﬁeld:

I s A
H3 : image? — w(,:[B x p], (4.33)

where p E p0 is momentum of the tunneling electron. As one can see by examining
Eqs. (4.32), (4.33), the motion of electrons in the crystal becomes uncoupled. The
tunneling electron now moves in a static potential created by other electrons in the
crystal. Its in-plane motion is a harmonic vibration about the equilibrium position
with frequency (up. The problem is effectively reduced to a single-particle problem,
which mimics the many-electron one.

The single-particle character of the solution in the Einstein approximation can
be also traced from the equations of motion (4.11). Indeed, pkj oc egg-(eh- [B x z]) ,
where factors independent of kj have been omitted. Consequently, pn o< 6no[B x 2],
which shows that only the tunneling electron is moving in the plane.

If we choose B along the y axis, then the problem becomes effectively two-

dimensional, with the Hamiltonian

(1),, + runspz)2 1 2 .2 p3
' + —mw a: + —“ -+- U z . 4.34
2711.. 2 p 2712. ( ) ( )

 

H:

The Coulomb interaction l)Cl.\\‘(r‘,(_’.Il electrons in the layer not only provides the in-
plane conﬁnement for the tunneling electron, which as discussed in the Introduction,
is so important for tunneling transverse to B, but also lowers the tunneling barrier
U (2) For nL2 << 1, the change in potential energy for the tunneling electron to move
a distance 2 01.1t—of-plane is given by:

2 .2 2
, ’ (3 ’(1 .9 ’6
06(3) 2 E —— — E — a: —.:"I E — (4.35)
/-2 2 ,. 3
A. + r Ill. Tn
11.. ll

TI 7i,

_,
1

5

Using the sum rule ij mwﬁj : 2N 2,13%; = 2N 271,82 mg, where the A; are
diagonal elements of the dynamical matrix of the Wigner crystal. Therefore the

“correlation-hole” potential [72, 34] may be written as
U(::(Z) = —mt2)222 (4.36)
where the mean square frequency a; for a triangular lattice [73] is approximately:
1/2
. . 1 2
a; = [Z tofu/2N] z (4.4582713/2/777.) / (4.37)
kj

We now illustrate the role of electron correlations on the tunneling rate for two
model potentials. A triangular tunneling potential was chosen to represent a class
of problems where the tunneling length varies with intrawell energy (we will refer
to such potentials as smooth). Tunneling through a rectangular barrier represents
the opposite case where the tunneling length L is geometrically deﬁned and does not

change with energy.

4.4.2 Triangular barrier

In many tunneling experiments, the escape is measured for a ﬁnite applied voltage.
For electrons above helium surface and in certain types of semiconductor heterostruc-
tures, the potential U (z) in the barrier region (2: 2 O) is then determined by the
electric ﬁeld which pulls electrons away from the intrawell states. Although the
correlation-hole potential Uc(z) (4.36) has strong effect on the tunneling rate, its role
is similar whether or not the magnetic ﬁeld is applied. Therefore, to illustrate purely

the role of in-plane conﬁnement on the rate of of tunneling transverse to B we do

not take UC(z) into account. The effect of the correlation-hole potential Uc(z), as
well as other corrections important for small z [cf. Eq. (6.4) below], will be studied
in detail in Chapter 6, where the quantitative comparison with experimental data is

made. Counting the energy from the ground intrawell state, so that E9 = 0, we have:

N2

U(z) = 2:” (l - %) (z 2 0). (4.38)

 

Here, 7 E 71 is the inverse localization length of the ground-state wave function
1129 E 2,01 near the well, 8111 tel/Oz : —*y for z = 0, cf. the discussion before Eq. (4.16);
L is the tunneling length in the ground state for B = 0. We assume that 7L >> 1.
In order to calculate the ground-state tunneling exponent, it is convenient to solve
directly the equations of motion (4.11) with the boundary conditions (4.16), (4.18),
(4.12), and (4.21). For a triangular potential, these equations are linear. This allows

us to obtain for the tunneling exponent a simple expression

73 2 air; + 31/,,T,d(1 - 7rd) coth[wa/2 — l/pTrd] + 3

+3Trd(l/2 — 1), R, = 2m: 3112. 4.39
.1

Here, up = pro and uc : wen, are the. dimensionless in-plane and cyclotron frequencies
scaled by the tunneling duration 70 for B = O, and V2 = V]? + V3.
The quantity Tm : Tf/To in Eq. (4.39) is the reduced tunneling duration. It is

given by the equation
[(1 — Trd)I/p1/2cotl1[wa/2 — z/prm] — 113] tanh I/Trd : 1/[1/37rd — V2] (4.40)

The tunneling exponent in the. limit T —> 0 can also be obtained by solving the
equations of motion (4.11) in the (:1:,:~:) representation, according to the method of

77

 

 

 

 

 

 

 

l

2 2

 

 

Figure 4.2: The dependence of the tunneling rate at zero temperature on magnetic
ﬁeld, W = W(B)/W(O). The curves 1 to 4 refer to wpro = 0,0.2,0.4,0.6. Magnetic
ﬁeld eliminates single—electron tunneling for wc'ro 2 1 (cf. curve 1). Inset: tunneling
exponent vs in-plane frecniency w,, for 0.,ng : 1.0, 2.0, 3.0 (curves a,b,c).

Chapter 3. Because of the symmetry .1: —> —;z: of the potential on the tunneling
trajectory with imaginary time t : —z'7‘, variables pm. and z are real, meanwhile a: and

19,, are imaginary. The tunneling exponent is given by

72 = —1/27'§, — 31/p(1 — Tm)? + 3(1+ up)(1 — 7rd) + 3V2Trd, (4.41)

pr

[I/2I/p(1 — Tm) — Vf] tilllIlI/Tr(1 : I/(VgTrd — V2) (4.42)

Note that expressions (441,442) coincide with (439,440) for T = 0.

The role of the many—electron effects is particularly important in the limit T -—) O.
From (4.41), (4.42), we have that without the magnetic ﬁeld 7rd = 1 and R = 4/3
(the “duration” of underbarrier motion in imaginary time is To). In the single-electron

T8

approximation (cup : 0) the tunneling duration T, and the tunneling exponent R9
diverge for we ——> To“ [34], as shown in Fig. 4.2 on curve 1. This happens because
the effective single—electron potential U (z) + (1/2)mw§zz, which takes into account
the parabolic magnetic barrier, does not have classically allowed extended states with
energy E9 = O behind the barrier. Formally, T ——> 00 for 1/,, : 0,1/C ——> 1‘)

Even comparatively weak in-plane conﬁnement eliminates this effect. The reduc-
tion of the tunneling suppression is signiﬁcant already for small wpro, and increases
fast with increasing wpm. For wCTO > 1 and T = O, the tunneling exponent is a steep
function of the exchange rate cap in the limit of slow exchange, 001970 << 1. In the
opposite limit of the fast momentum (rm-hange, top >> 761, from Eqs. (4.39), (4.40)

fed—2. 'l‘herefore, with increasing wp beyond we, the

we obtain that R9 = —]e L 1 + we 1],
exponent of the tunneling rate approaches the zero-magnetic-ﬁeld value of 47L/3.
The tunneling time becomes Trd z 1 [i.e., 7f x 7‘0]. On the other hand, for large and
but ﬁnite wp, the slope of the logarithm of the tunneling rate as a function of mag-
netic ﬁeld provides a direct measurement of the characteristic frequencies of in-plane
electron vibrations. The tunneling exponent for zero temperature as a function of
camwc is shown in Fig. 4.2. The overall dependence of the tunneling exponent on (up
is shown in the inset.

For a given magnetic field, the delxsndence of the tunneling exponent R9 on the
frequency cup becomes much less steep with increasing temperature, as seen from
Fig. 4.3. This happens because at ﬁnite temperatures the tunneling electron may

transfer its in-plane Hall momentum not only in a recoil-free way to electron system

as whole (“zero-phonon” process), but also compensate it with a thermal in-plane

79

 

 

   

 

  

 

 

 

 

 

 

A
e -002 '- 3 ‘ ‘t
g ’ ’ ’ ’ ’ —1
74 - — ' ’
a _ 2 1 6 1 r . I I . T
m t _ Rmany-electron 'R non-interacting d
A-O.6 _ - W» _.
O »
53’ t 1 0.8 ~ -
L—l _ _.
- 1 '— 0 4 1:-1/ J -]
1 l I I t 0 i
O 1 2 3
(D T
p 0

Figure 4.3: The tunneling exponent in the ground state for a triangular potential
barrier (4.38) as a function of the phonon frequency cup in the Einstein model of the
Wigner crystal for w(.T() -_— 2. The time '0 :2 mL/7 is the duration of tunneling for
B = 0 and T = 0. The curves 1 to 3 refer to reciprocal temperatures ﬂ/To = 7, 5, 3.
The dashed line is the result of the direct variational method, with one variational
parameter 7,. The relative importance of many-electron effects is demonstrated in
the inset. Here, the difference between the full many-electron tunneling exponent and
that obtained in the single-electron approximation is plotted as a function of inverse
temperature. The many-electron tunneling exponent was calculated for wpro = 3.0.

momentum. The recoil—free process depends very strongly on in-plane frequencies,
because they determine how much of the Hall momentum will actually be compen-
sated. Compensaticm by thermal momentum is present even in a non-interacting
limit, does not have strong frequency (lepimdence. This is again very similar to what
happens in a l\/Iiissbaner effect, where the zero-phonon line disappears with increas-

ing temperature. For large pro, tum, the curves for dill'erent temperatures merge

80

together and approach the. B = 0 asymptote.

The value of R9 can be calculated independently from the functional 72,, (4.23)
using the direct variational method. Even a simple approximation where Z(T) is
quadratic in 7', with the only variational parameter being the tunneling duration
Tf, gives a reasonably good result, which is shown in Fig. 4.3 by a dashed line for
[3 = 370. Such calculation gives a good approximation for higher temperatures, and
also for lower temperatures but not too small wpro. For low temperatures and small
wp'ro the trajectory Z(T) is strongly nonparabolic, and more then one parameter is
required in the variational calculation.

The above results provide an explainition of the magnetic ﬁeld dependence of the
tunneling exponent for electrons on helium, which was observed to be much weaker
[34] than it would be expected from the single-electron theory. Detailed comparison

with the data [34] will be discussed below in Chapter 6.

4.4.3 Square barrier

In many physically interesting SyStOIIIS. the tunneling barrier U (z) is nearly rectan-
gular. This is often the case for semicomlnctor heterostructures, where the barrier is
formed by the insulating layer. If we count U off from the intrawell energy level E9

and set the boundaries at .3 = 0 and :; :‘—‘ L, the barrier has the form

U(z) = 72/2m — ‘IIMIJ222, 0 < z < L (4.43)

Here, 1/7 is the decay length under the barrier, cf. Eq. (4.16), and the mean square

frequency (I) is given by (4.37).

81

We assume that, behind the barrier (,3 > L), an electron can move semiclassically
with all energies. The picture of tunneling depends on the parameter Q = M26170
where To = mL/y is the imaginary “time of ﬂight” under the barrier for (I) = O. For
9 < 1 the particle comes out from the barrier at the point z = L where U (z) is
discontinuous, cf. Fig. 21). Then the decaying underbarrier wave function has to be
matched to an a1,)propriate propagating wave. behind the barrier at z = L. In contrast
to the case of a smooth barrier, because the potential U (z) is discontinuous at z = L,
the z-component of the momentum should not be the same on the opposite sides
of the boundary. However, the in—plane “momentum” components ukj, which are
imaginary under the barrier, still have, to be continuous. Respectively, the boundary

conditions (4.12) for the tunneling trajectory should be changed to

25(7'f) : L, ukJ-(Tf) = 0. (4.44)

In fact, the condition ukJ- = 0 gives the in-plane values of pkj for which the wave
function is maximal for z : L.

With the boundary conditions (1.4!), elimination of phonon variables from the
Euclidean action 5,; in the tunneling exponent is similar to what was done for a
smooth barrier. The resulting expression for the retarded functional 72,,[2] coincides
with Eq. (4.23), provided Z(Tf +:1r) is deﬁned as Z(Tf — :17), for 0 S a: g 'rf. In the

Einstein approximation the boundary conditions (4.44) become:

23(7)) : /,. .r(7f) = 0. (4.45)

For higher electron densities wh<~~>re $2 > 1, potential U (z) is no longer discontinu-

82

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.4: Exponent of the tunneling rate —R from a 2D WC in a semiconductor
heterostructure as a function of scaled electron density 9 = V2070 (To = mL/y).
Electron correlations increase the tunneling exponent both for B = 0 (dashed line)
and in the presence of magnetic ﬁled (solid line refers to were = 1.0). With increasing
9 the tunneling rate in the magnetic ﬁeld approaches the zero-ﬁeld line. Inset (a):
relative tunneling rate II" = W(B)/ll't’ytl) vs magnetic ﬁeld for (370 = 0.5. Inset
(b): electron potential with (bold line) and without (thin line) the reduction of the
tunneling barrier due to the effect of electron correlations.

ous and the usual boundary conditions apply
1),,(7'!) : ll. 517(7’f) = O. (4.46)

The tunneling exponent is calculatml along the tunneling trajectory satisfying that
boundary condition out of two possible (4.45) and (4.46) which gives the smallest
value. In the absence of a. magnetic ﬁeld, the transition from one boundary regime
to another occurs for Q : 1. In the presence of a magnetic ﬁeld, the transition

shifts towards larger (2. The detailed calculation is rather tedious, and is provided in

83

Appendix C. The result for the tunneling exponent in the Einstein approximation is
plotted in Fig.4.4.

For B = O the tunneling exponent is 25;; = yLKZ‘l arcsinQ + (1 — SPY/2] [
28;; = 7r7L/2Q, for 52 > 1]. l\/'Ia.gneti<‘- lield causes SE to increase and the tunneling
rate to decrease, respectively. For weak lields. the increment of SE is quadratic in B.
Transfer of the Hall momentum of the tunneling electron to the WC strongly reduces
suppression of tunneling by a magnetic ﬁeld. Fig. 4.4 shows also how SE is decreased

by electron correlations even for B : (l. in the case of a broad barrier.

4.5 Summary

In the presence of a magnetic ﬁeld parallel to a 2DES, the tunneling rate becomes
exponentially sensitive to the presence of electron correlations in the system. These
electron correlations are not imposed by the magnetic ﬁeld, contrary to the case where
magnetic ﬁeld is applied 1')erpendicnlar to the. layer. Because the tunneling rate is
exponentially small, one can consider tunneling of different electrons independently,
and therefore single out the tunneling electron. Electron correlations affect the out-of-
plane tunneling rate through an interelect ron nnn'nentum exchange from the tunneling
electron to the electron system as a whole. The mechanism is therefore similar to a
Mossbauer effect. The tunneling electron usually transfers only part of its in-plane
momentum to the electron system in a "recoil—free” way. The interrelation between
the characteristic momentum exchange rate. and the reciprocal duration of tunneling

in imaginary time I/Tf determines what portion of the in-plane momentum will be

st

transfered, so that the dynamics of electron motion under the barrier becomes very
important.

With increasing temperature, the dependence of the tunneling rate on in-plane
vibrational frequencies becomes less steep. This happens because at such temper-
atures the in-plane Hall momentum can be compensated by a thermal momentum,
and not only through a recoil-free transfer to a whole electron system. This is again
analogous to what happens in a .\lossbaner effect, where the zero-phonon line disap-
pears with increasing temperatnre. 'l‘herefore, tunneling experiments could probe the
phonon spectrum of the 21) system. if they are done at sufficiently low temperatures.
At higher temperatures, the tunneling rate can be described in the single-electron
approximation. The. dependence of the tunneling rate on temperature and magnetic
ﬁeld, however, is expected to be a. very non-monotonic function and have interesting
features, as discussed in the next chapter.

Correlated 2D electron systems in semiconductor heterostructures have been in-
vestigated by tunneling mostly for the magnetic ﬁeld B perpendicular or nearly per-
pendicular to the electron layer. ('1'. (37)]. The data on tunneling in a ﬁeld parallel to
the layer refer to high density 2Dl25s [3 1:. where correlation effects are small.

It is expected that the effect of a parallel magnetic ﬁeld will be most pronounced in
systems with shallow and broad barriers U(z). For example, in a GaAlAs structure
with a square barrier of width L : (1.1 mu and height 72/2m = 0.02 eV, for the
electron density It 2 1.5 x 10'0 cm 3 and I} : 1.2 T we have prt) z 0.6 and (.0070 x 1
(To = mL/7 is the tunneling duration for n. = B = 0). The results of Sec. 4.4.3

for square barriers, where we have taken into account the correlation-hole potential

85

(4.36), show that the interelectron momentum exchange should signiﬁcantly modify
the tunneling rate in this parameter range, provitgled the 2DES is correlated [56]. One
can therefore expect that tunneling experiments on low-density 2DESS in parallel
ﬁelds will reveal electron correlations not imposed by the magnetic ﬁeld, give insight
into electron dynamics, and possibly even reveal a transition from an electron ﬂuid

to a pinned Wigner crystal with decreasing 7L.

Chapter 5

Magnetic-ﬁeld-enhanced tunneling

This chapter is devoted to the theory of the enhancement of the tunneling rate by
a magnetic ﬁeld parallel to the electron layer. Its applications can be very useful,
because it allows to increase the tttnnelin g rate without changing the parameters of a
tunneling barrier. A magnetic ﬁeld parallel to the electron layer couples out-of-plane
and in-plane degrees of freedom. For T : 0, this leads to the energy transfer from out-
of-plane tunneling motion to the ill—plane. vibrational motion, which, in turn, results
in the suppression of the timneling rate by magnetic ﬁeld. For ﬁnite temperatures,
the direction of the energy transfer may be reversed, so that the energy of thermal
in—plane motion can be used to assist the tunneling. Because coupling is proportional
to the magnetic ﬁeld, this is a qualitative reason why the magnetic ﬁeld can enhance
the tunneling rate. ()n the formal level. the B-enhanced tunneling is a consequence
of the increase, with increasing temperature, of the absolute value of the term Ree
(4.24) in the tunneling action. Since. this term gives a negative contribution to the

tunneling exponent R, the. whole B-dependent term in R becomes negative starting

87

with a certain crossover temperature T,. and then the tunneling rate increases with
B.

The increase of the tunneling rat e with magnetic ﬁeld does not occur in all systems.
In systems where it does occur, it happens in the range that is limited in both T and
B. The range boundaries are not universal and depend on the potential U (z) and the
phonon spectrum. The main processes that limit this range are switching to tunneling
from the next intrawell level or switching to the escape via thermal activation. The
enhancement may start from B : (l or have a finite threshold in B. The later variant
occurs when escape for B = () occurs through tunneling from the next excited state
or by over barrier activation. With increasing of magnetic ﬁeld there occurs switching
from one of these processes, which determined the escape rate for B = 0, to tunneling
from the ground intrawell state, where the tunneling rate increases with B. However,
very strong ﬁelds suppress rather than enhance escape.

In Sec. 5.1 the physical limits on the lower and upper temperature bounds for B-
enhanced tunneling are discussed. The various switching processes that can take place
with increasing of T or B are different in cases where the potential barrier is smooth
(tunneling length is determined from If : (’(r)) and discontinuous (the tunneling
length is geometrically deﬁned). lixplicit results are obtained in Sec. 5.2 using the
Einstein model of a W igner crystal, in which all phonons are assumed to have the same
frequency. As an example of a. smooth tunneling barrier we take a triangular barrier,
which is relevant for tunneling in the presence of applied electric ﬁeld. Square barrier

models the potential formed by the ittsttlat ing layer in semiconductor heterostructures.

88

5.1 General properties of the transition

5.1.1 The temperature of crossover for small magnetic ﬁelds

The lower temperature bound of the enhancement domain is the crossover tempera-
ture TC. It can be determined from the small-B expansion of the tunneling exponent

for the ground state [77. = g in Eq. (4.22)],
Rg(wc) 2 129(0) + Ag(T)w3, (.0670 << 1 (5.1)

where T0 is the tunneling time in the ground state for B = 0. The role of the ground
state is special in that the barrier width is bigger for the ground state energy than
for the energies of the excited states. Therefore the effect of the magnetic ﬁeld, which
accumulates under the barrier, is most pronounced in the ground state.

The value of Ag is given by the terms oc wf in the action R9 (4.23) calculated

along the tunneling trajectory Z0(T) for B = 0. From the analysis in subsection 4.3.1,

it follows that Ag > 0 for T —> 0. The crossover temperature is given by
Ag(TC) = 0. (5.2)

For T > To the tunneling exponent Rg decreases and the tunneling rate increases
with B, for small B.

In the limit of low phonon frequencies, (Ukj << 1/7'0, TC, from Eqs. (4.23), (4.31) it
follows that BC 5 1/TC = 2702—02/23, where 2'5 is the average coordinate 2 (4.31) for
the B = 0 trajectory with energy E9, and 202— is the mean square value of z on the

same trajectory,

272,: 704/ deg(7') (E = E9).
0

89

Clearly, in this case ﬁe < 270. It follows from Eq. (4.24) that 270 is also the limiting
value of BC in the opposite case of high phonon frequencies, (Ukj >> 1/7'0. On the
whole, we have the bounds on temperature for the tunneling enhancement in the

ground intrawell state

—2
27'02é‘2 < ,Bc < 2T0. (5.3)

30
As noted above, 5 is nonzero, and generally 237% ~ 1.
It follows from the above arguments that the value of the crossover temperature
Tc = 1/5C decreases with increasing phonon frequencies, that is the crossover is de-
termined by high-frequency phonons which, in the case of 2D electron systems, have
large wave numbers and are determined by the short-range order. Note that there
is no threshold in B for tunneling enhancement for T > T C from the range (5.3),

provided the system is tunneling from the ground state.

5.1.2 Upper temperature limit for enhancement for small
magnetic ﬁelds

A threshold-less tunneling enhancement starting from B = 0 occurs for temperatures
bounded from above by the condition that the system tunnels from the ground state
rather than from excited intrawell states or via thermal activation over the barrier.
In principle, even for excited states, the tunneling rate may increase with B, but this
does not happen for simple model potentials investigated below.

If the tunneling is enhanced only in the ground state, the upper temperature

bound is often the temperature T1_,2 where the probability of tunneling from the ﬁrst

90

excited state, weighted with the occupation factor, exceeds that from the ground state,
for B = 0. It can be estimated for smooth tunneling barriers, where the tunneling
duration 70(E) for B = 0 often decreases with the increasing energy E. In fact,

the function 70(E) may be nonmonotonic even for simple potentials U (z); a detailed
analysis of this function lies outside the scope of this work, but generalization of the
results to appropriate cases is straightforward. From (4.23), for decreasing 70(E),
switching from tunneling from the ground state (71 = 1) to that from the ﬁrst excited

state (n = 2) occurs for the reciprocal temperature

E2 70(E)dE

5142:? E232_E1 ( 1;- 9)-

 

This value lies between 2T0(E2) and 270(E1). Depending on the tunneling potential,
[3H2 can be smaller or larger than BC (5.3). If a magnetic ﬁeld does not increase the
rate of tunneling from the state 72 = 2, threshold-less tunneling enhancement occurs
for Tc < T < T1_,2.

Alternatively, for B = 0 the system may switch to activated escape over the
barrier with increasing temperature for T 2 T, < TC. The threshold-less tunneling
enhancement by the magnetic ﬁeld does not occur in this case. However, both for
Tc > Ta and Tc > T1_,2 there may still occur a B-induced enhancement of the escape
rate starting with some nonzero B. We note that in the above arguments, it was

assumed that thermalization 1 inside the well occurs before the electron escapes.

 

1The Boltzmann distribution over in-plane energies is established very rapidly, on times of the
order of w; 1. In Chapters 4, 5 we assume that the Boltzmann distribution over out-of-plane intrawell
states also occurs much faster than the process of the tunneling escape. It is not always true, since
the thermalization proceeds through scattering on defects inside the quantum well (for a 2DES in
heterostructures), or on ripplons (for a 2DES on a surface of liquid helium). Extension of the results
to account for slow intrawell thermalization is straightforward, and will be done for electrons on
helium in order to compare the theory with the experimental data [cf. Eq. 6.9].

91

5.1.3 Field-induced switching between tunneling from the
ground state, excited states, and over-barrier activa-
tion

Even in the temperature range T > T1_,2 a sufﬁciently strong magnetic ﬁeld can
increase the tunneling rate, provided T > T C. This happens if the tunneling exponent
for the ground state Rg(wc) E R4,:1(wc) exceeds that in the ﬁrst excited state Rn:2(wc)
and its zero-ﬁeld value Rﬂ:2(0). In a certain temperature range where T > T142, the
tunneling rate for B = 0 is determined by tunneling from the excited state n = 2.
This rate decreases with increasing B (the tunneling exponent anz increases with B).
For some B the exponents Rn=2(wc) and anl(wc) become equal to each other. For
larger B the system tunnels from the ground state, and the tunneling rate increases
with B.

Similarly, since the activation rate is only weakly affected by B, in a certain
temperature range where escape already occurs via activation for B = 0, starting
with some B it may again go through tunneling from the ground state. This happens
if the tunneling rate for the ground state becomes bigger than the activation rate and
only happens in a limited range of B, see Sec. 5.2.1. For a special model the switching

is illustrated in Fig. 5.2 below.

92

5.2 Tunneling enhancement for the Einstein model

of a Wigner crystal

In what follows we will illustrate the general results and apply them to speciﬁc 2D
systems using the Einstein approximation for the phonon spectrum, wkj = wp. This
is motivated by the fact that tunneling is determined primarily by short-wavelength
vibrations, which have a comparatively weak dispersion. The magnetic-ﬁeld-enhanced
tunneling can be observed in experiments with smooth tunneling potentials. As
before, we take a triangular barrier as a representative for such potentials. The case
of tunneling through potentials where the tunneling length that does not change with
intrawell energy is qualitatively different: tunneling enhancement is not expected
to occur there [again, assuming that the intrawell relaxation is fast enough, with an
approximate rule that the cleaner the sample is, the slower is the intrawell relaxation].
However, switching from tunneling to thermal activation can occur with increasing

of magnetic ﬁeld for such tunneling barriers.

5.2.1 Smooth potentials: ﬁeld-induced tunneling enhance-
ment and switching from activation to tunneling

Below we use the explicit expression for the tunneling exponent obtained earlier (4.39),
(4.40) to analyze the effects of tunneling enhancement and magnetic ﬁeld induced
switching to tunneling. In the small-B limit, where we < prO— 1, the tunneling

exponent R9(B) is seen from Eq. (4.39) to be quadratic in B. The coefﬁcient Ag in

93

 

 

0.06 t t I 1 .
53?
g 0.03
E
“F
g 0
E.
-o.o3

 

 

 

 

Figure 5.1: The dependence of the tunneling exponent R(B) E Rg(B) on the magnetic
ﬁeld (4.39) for wpro = 1/3 near the crossover temperature ﬁe z 1.6770 (5.4). The
curves 1 to 3 correspond to (B — fig/To = 0.2, 0, —0.3

Eq. (5.1) can be easily calculated. From the condition A9 = O we obtain the value
of the reciprocal temperature BC which corresponds to the crossover from decrease to

increase of the tunneling rate due to a magnetic ﬁeld,

2 1/ 31/ — 3+t/2 tanhu
ﬁC——-2TO+—tanh—1 p[ p ( p) p]

5.4
wp V3 — 31/1,, + 3 tanh up ( )

 

In agreement with (5.3), 5,, monotonically increases with cap from 570 / 3 at (up = 0 to
270 for top ——> 00.

The dependence of the tunneling exponent (4.39) on the magnetic ﬁeld for different
temperatures is shown in Fig. 5.1. Above the crossover temperature ([3 < BC), R(B)
decreases with B. Then R(O) — R(B), and the tunneling probability with it, increase

with the increasing ﬁeld, for small B. The slope dR/BdB cc 6 — [30 for B ——> 0.

94

 

 

 

 

 

 

 

 

 

 

0.06 - ' ' "4 0-01
(a) 1
g .
g 2
g 0.03 ” ‘
m. 0
a .
0 . . .
0 10 2 2 20 0 2 2 2 4
(”c T 0 (0C 13 0

Figure 5.2: Magnetic ﬁeld induced switching from activation (a) and from tunneling
from the excited state (b) to tunneling from the ground state, for wpro = 1/3. In
(a), there is only one intrawell state in the potential well U (z), and the transition
to activation for B = 0 occurs for B/ro = 4/3. The curves 1, 2 correspond to
(B — ﬁc)/ro = —0.35, —0.4. In (b), the position E2 of the excited level (n = 2) is
chosen at 0.272 / 2m below the barrier top (E1 = 0). The temperature is chosen at
(5 — ﬁc)/ro = —0.16, so that for B = 0 the system tunnels from the excited state.
The observable (smaller) tunneling exponents for a given B are shown with bold
lines, whereas dashed lines show the bigger exponents, which correspond to smaller
tunneling rates.

However, for strong ﬁelds the tunneling rate decreases with the increasing B, because
the Hall momentum can no longer be compensated by thermal fluctuations.

It is clear from the data in Fig. 5.1 that, for the barrier chosen, the magnetic
ﬁeld induced increase of the tunneling exponent R is numerically small. However,
for typical R 2, 50 it can still be noticeable, although strictly speaking it is on the
border of applicability of the approximation in which only the exponent is taken into
account.

The expression (4.39) gives the tunneling exponent only for low enough temper-

95

atures where the system escapes from the ground state. For higher temperatures,
one should take into account the possibility of escape from excited states and via
an activated transition over a potential barrier. The positions of the excited levels
depend not only on the barrier shape, but also on the shape of the potential U (2)
inside the well. The analysis for a realistic system, electrons on the surface of liquid
helium, is done in the next section. Here, in order to illustrate different options, we
discuss two cases: a narrow well, in which case the ground state is essentially the only
intrawell state, and a well with a comparatively shallow excited state for which still
the intrawell relaxation rate is higher than the tunneling rate, so that its occupation
is given by the Boltzmann factor.

We start with the discussion of the case of a single-state potential well. For
B = 0 and a triangular barrier U (z) (4.38), switching from tunneling to activation
occurs here for the temperature Ta E l/ﬁa = (470/3)‘1. This temperature is higher
than the crossover temperature l/ﬂc (5.4), and therefore there is a region where the
enhancement of tunneling by a magnetic ﬁeld can be observed, as discussed above
(cf. Fig. 5.1). However, even though for T > T, the B = O—escape occurs via
over-barrier transitions, the increase of the tunneling rate with the increasing B can
make tunneling more probable for sufficiently strong B. If the activation rate is
independent of B, the overall dependence of the exponent of the escape rate R(B) =
minn[R,,(B) + ME" — E1)] oc 1n W(B) on B is shown in Fig. 5.2a. In this case,
R(0) = 72/2mT is the barrier height over temperature. Switching to tunneling and
to the increase of the escape rate with B occurs where the tunneling exponent Rg(B)
as given by Eq. (4.39) becomes less than R(O).

96

A similar switching occurs in the temperature range where tunneling from the
ﬁrst excited level is more probable than from the ground state, for B = 0. Since with
increasing B the tunneling rate in the ground state increases, the system switches to
tunneling from the ground state starting with a certain value of B. This is illustrated
in Fig. 5.2b.

In narrow-well potentials, a magnetic ﬁeld may strongly affect the wave functions
with energies close to the barrier top. As a result, new bound metastable states may
appear in a strong ﬁeld. The ﬁeld also shifts the energy levels of the existing states.
The rate of interlevel transitions may also change, since the ﬁeld mixes together the
in-plane and out-of—plane motions. The related effects may become important with

increasing temperature.

5.2.2 Square barrier: ﬁeld-induced crossover to thermal ac-
tivation

Tunneling through a rectangular barrier is special in that respect that the tunneling
length L does not decrease with increasing of intrawell energy. As a result, the
tunneling time 70(E) = —dSO/dE for B = 0 monotonically increases with energy E,
and correspondingly the energy level found from the condition 5 = 27'(E) gives not
a maximum of the function —[3E — 2SO(E), but a minimum. The maximum value,
which gives the probability of tunneling with energy E with account taken of the
occupation factor, corresponds either to the transition from the ground state or to

activation over the barrier.

97

Consider, for example, the simplest case of a square barrier with boundaries at
z = 0 and z = L, which mimics the tunneling barrier formed by a insulating layer in

semiconductor heterostructures.
U(z) 2 72/2722, 0 < z < L. (5.5)

Here, we count U off from the intrawell energy level Eg; 1/7 is the decay length
under the barrier, cf. Eq. (4.16), and we have neglected the lowering of the barrier
due to the electrostatic ﬁeld from other electrons at their lattice sites, which is a good
approximation for nL2 << 1.

Switching to activation occurs for the temperature Ta = 72/4mSo(Eg) E
y/4mL = (4TO)‘1. It is lower than the temperature Tc of the crossover from B-
suppressed to B-enhanced tunneling as given by Eq. (5.3), and therefore we do not
expect the crossover to occur in systems with a square barrier.

If the temperature T < Ta, escape for B = 0 occurs via tunneling, and its prob-
ability decreases with the increasing B. Starting with some B, where the tunneling
exponent becomes bigger than the activation exponent 72/2mT, it becomes more
probable to escape by activated transition than by tunneling. To a good approxima-
tion, the escape rate becomes independent of the magnetic ﬁeld.

The tunneling exponent that describes the escape rate for T < Ta can be obtained
directly from the [linear, in this case] equations of motion (3.3) with the boundary

conditions (4.16), (4.18), (4.44). It has the form:
R = 7L [7rd + I/CK.(Trd)]. (5.6)

where the function ”(Tad and the reduced tunneling time Trd : Tf/TO in Eq. 5.6 is

98

 

 

 

 

 

 

 

 

 

 

 

 

g _ _
é
a _
9.4 -02
Q
E. ” 2 i
-0.4 — -
t 0.156 . i . 2 ‘ 3 ‘
(D T
.0.6 1 CL 0 I l
0 1 2
(DCTO

Figure 5.3: The logarithm of the escape rate R(B) compared to its B = 0 value
R(O) = 250(Eg) E 27L, which is determined by tunneling through the square barrier
(5.5). Curves 1-4 correspond to (B — ﬂc)/ro = 3,4, 5, with To = mL/y, and 66 = for
chosen top = 1/27’0. As were increases, there occurs a transition from tunneling to
thermal activation.

found from the following equations:

I/C(C0Sh VT“, — 1)

 

n r E
( rd) V3 + V; cosh VT“, + yup coth[wp,6 / 2 — uprrd] sinh VpTrd

_ 1 113(2 — 2cosh urrd + VTrd sinh VTrd) ‘ ”3(Trd - 1) Sinh ”Tm (5 7)

you: (1 — cosh urrd)(1 -- 113/113) + I/Trd sinh VTrd

 

Here, as before, the dimensionless in-plane and cyclotron frequencies up = wpro and

12C 2 wcro are scaled by the inverse tunneling duration for B = 0, 'r0’1 = ’y/mL, and
2 __ 2 2

I/ — up + V6.

The B-dependence of the escape rate for different temperatures is illustrated in

Fig. 5.3. With increasing of either temperature or magnetic ﬁeld, the tunneling

99

exponent (5.6) becomes bigger than the activation exponent 72/2mT = 7Lﬁ/2’ro.
Then it becomes more probable to escape by activated transition than by tunneling.

The temperature of switching to activation is given by the equation Ta =
72/2ng. From (5.6), (5.7), R9 increases with the magnetic ﬁeld, and therefore
the switching temperature T, decreases with B. However, it follows from the analysis
of the above equations that T, remains lower than 1/(47'0).

The effect of saturation of the escape rate with increasing B is not limited to square
barriers, of course. For strong enough B and nonzero temperatures, the tunneling
rate becomes less than the activation rate, and the system switches to activation;
the switching may go in steps with increasing B, via tunneling from excited intrawell

states.

5.3 Summary

Coupling between out-of—plane and in-plane degrees of freedom that is realized by
magnetic ﬁeld parallel to the layer leads at low temperatures to the suppression of the
tunneling rate, and to the enhancement of it with B at higher temperatures. Because
coupling strength is proportional to the magnetic ﬁeld, one may be able therefore
to control the tunneling rate in a broad range, just by changing temperature and
magnetic ﬁeld, without changing parameters of the tunneling barrier.

The overall escape rate as a function of B and T is expected to display a number
of other unusual features. These include switching from activated escape to tunneling

and vice versa, and switching between tunneling from the ground and excited states.

100

These switchings have been analyzed for simple but realistic models of the tunneling
barrier.

Sufﬁciently strong magnetic ﬁelds will always suppress the tunneling rate. There-
fore, the enhancement of tunneling occurs in a limited range of magnetic ﬁeld. Other
competing processes, such as tunneling from higher intra—well states, and over-barrier
escape, limit the range of temperatures for B—enhanced tunneling. For a given geom-
etry of the tunneling barrier, this effectively sets the limits on how fast the escape
rate can be enhanced. However, in some cases, such as for high mobility samples,
low in-plane scattering rate probability results in the suppression the rate of tunnel-
ing from higher intra—well levels and the rate of over-barrier escape. This happens
because the occupation of higher levels depends on a scattering amplitude into these
states and no in-plane momentum from the ground state and large in-plane momen-
tum. In other words, there is no equilibrium distribution of states inside the well2.
Because the competing escape processes are suppressed, the B-enhanced tunneling
occurs for higher temperatures, where the effect is bigger. In particular this happens,
for electrons on helium, where the electron mobility is N 108 Vs”1cm"2. If there were
equilibrium distribution inside the well, then the over-barrier activation would be a
dominated process even for T < Tc, and the enhancement of tunneling would not

occur at all. Contrary to this, B-enhanced tunneling is indeed observed.

 

2T his can also happen in classical systems, if they are highly underdamped.

101

Chapter 6

Comparison with experimental
data on tunneling from helium

surface

In this Chapter tunneling experiments are discussed. The only tunneling experiment
known to me, where tunneling occurrs from a strongly correlated 2DES in the presence
of magnetic ﬁeld parallel to the electron layer was done for electrons on helium. Such
systems are advantageous from the point of view that several major parameters that
control tunneling can be easily varied. In addition, the electron mobility is very high.
The tunneling rate, obtained from the model with no adjustable parameters, is in
qualitative and quantitative agreement with the experiment [34] in a broad range
of ﬁelds, electron densities, and temperatures. In particular, the results explain an

exponentially strong deviation at low temperatures of the tunneling rate from the

102

predictions of the single-electron theory.

6.1 The tunneling potential for electrons on he-
lium

The formation of an electron layer on a liquid helium surface occurs because of two
forces. On one hand, there is an attractive image force on electrons because they
polarize the dielectric. The effective charge is Ac 2 %(:+;11)l, where e z 1.057 is the
dielectric constant of helium. On the other hand, a very high afﬁnity potential for

helium (a: lev) prevents electrons from penetrating into the dielectric. The corre-

sponding Schrédinger equation for a model potential
Uim = (6.1)

with the appropriate boundary condition t/2(0) = 0 can be solved exactly. This so-
lution [74, 75] gives the energy levels E, = 7,2, / 2m, where the intrawell localization
length of the n-th level is 7,, = 71/77. = Am/n, (y E 71). The ground state is described

3/2zexp(—’yz), with 7‘1 = 76Aand energy E, % 7.5K.

by a wave function 1121(2) = 27
The smallness of E1 justiﬁes the use of the inﬁnite barrier approximation. The av-
erage distance to the surface for an electron is (z) = 3/2'y z 114A. This problem
was analyzed more precisely in Ref. [74], taking into account the ﬁnite value for the
afﬁnity potential, but the results yield only small corrections to the above expression.

In experiment [34], electrons were injected from above into a cylindrical cell of

height 2.5 :l: 0.05 mm that is half ﬁlled with liquid helium. A negatively biased guard

103

ring of diameter 18 :t 0.05 mm prevents electrons from escaping to the sides. The
helium surface was kept close to the middle of the cell: d 2 ch (with the uncertainty of
i0.025 mm), where d and h are distances from the layer to top and bottom electrodes.

A voltage V, applied between the top plate and the grounded bottom plate creates
the external electric ﬁeld —Vt/(d + h/e) that can either additionally conﬁne electrons
or extract them from the layer. The tunneling rate is measured as the difference in
the number of electrons before and after the extracting pulse of the electric ﬁeld. In
addition to the applied electric ﬁeld, there is an electric ﬁeld from charges induced in
the t0p and bottom electrodes by electrons of the 2DES. As a result, the total electric

ﬁeld becomes density-dependent [76]:

V: d—h

'5” Z d+h/e ‘4” I e I (€+1)(d+h/e)

 

(6.2)

 

In the presence of electric ﬁeld, the ground state energy changes. This change can
be found by the ﬁrst-order perturbation theory in electric ﬁeld. The ground state

energy becomes:

__1’1_ __7_2 _3_
E1— 2m ]e€y|(z) — 2m (1+ 27L) , (6.3)

where the tunneling length L = 72/ (2m[e£i|). The energy shift is small as an inverse
of 'yL, which is a large parameter of the theory. Large 7L is also necessary for the
adiabatic approximation to be valid. For electrons on helium this condition is well
satisﬁed with ’yL ~ 30.

The tunneling potential is also affected by the Coulomb interaction between elec-
trons. The corresponding “correlation—hole” potential [69, 72] can be obtained by
keeping only the lowest-order terms in the ratio of the tunneling length L to the

104

—1/2

inter-electron distance n , as discussed in paragraph 4.4.1, and has the form

—mc0222, where a?) oc 723/2. The conditions 7’1 << L << nil/2 are typically very
well satisﬁed in experiment, with the decay length 1/7 = 1 / Am z 0.7 x 10“6 cm,
L ~ [Eg/efil z 72/277r|e£j_| ~ 2 x 10‘5 cm, and 71‘1/2 ~ 10‘4 cm [in the estimate of
L we used that [E9] >> legil/y,m032/72, and that IeELI/y 2, a].

We can now write the total tunneling potential and separate terms that are small

as (VD—13
U(z) = #— [U0(z) + $ZU1(2)] , z > 0 (6.4)
Uo(z) = 1 — g — $1,727,? (32 (6.5)
2 3 3_2 22:
U1(Z)—_(—Z/—L)+-2—+§w TOE (6.6)

Here, the energy is counted from the ground state energy. The linear term in Eq. (6.4)
describes the electric ﬁeld that pulls electrons out of the layer. The parameter To =
2mL/7 is the imaginary tunneling time in the limit 77. —> 0. Although the image

1

potential o< 1 / 2 provides the major contribution for 2: ~ 7‘ , it becomes small deep

under the barrier.

6.2 Exponent of the tunneling rate

To compare the predicted dynamical effect of the electron-electron interaction with
the experimental data [34], we use the Einstein model of the WC, and set all
the phonon frequencies wkj to be equal to the characteristic plasma frequency

top = (271'627L3/2 /m)1/ 2. The numerical results change only slightly when this fre-

105

quency is varied within reasonable limits, e.g., is replaced by the root mean square
frequency 0'). Note that the correlation-hole potential is determined by the mean
square frequency 112 = (4.4562713/2 / m)1/2.

The magnetic ﬁeld dependence of the tunneling rate for different T is calculated
from Eqs. (4.11). The actual calculation is largely simpliﬁed by the fact that, deep
under the barrier, the image potential —A/z in (6.4) can be neglected. The equations
of motion (4.11) become then linear, and can be explicitly solved. In what follows we
will use dimensionless frequencies VC 2 were, up = wpro, 17 = 11270, and 1/2 = V: + V3.
The coordinates :1: and z are given in units of the tunneling length L, and momenta

p, and pz are given in units of 'y/ 2. The tunneling trajectories then take the form:

 

 

 

 

A A
3(7) 2 -—:\—:— sin A17 + T12 cos A17 - T:- sinh A27 — Tﬁ cosh A27;
12 + A2 A2 — V2
192(7) = /p 1(.42 cos Alr — A1 sin A17) + 2 p (A3 sinh Agr + A4 cosh A27);
”CAI /\2Vc
V2 + A?
z(r) = pu A2 (A1 cos /\1T + A2 sin A17)
c 1
A3 — 113 , __2
2 (A3 cosh A27" + A4 srnh A27") — 1/ ;
110A2
19,,(7) = A1 cos A17 + A2 sin A17 + A3 cosh A27 + A4 sinh A27- — 1162(7). (6.7)

Here, the eigenvalues A12 of the matrix derived from the equations of motion are

given by:

 

A? —_= [2172 — V2 + (/(2272 — V2)2 + 81721;; ,

 

[\DIH [\DIV-J

 

A3 = [1,2 — 2172 + \/(2172 — V2)? + 8172ng .

106

The constants A,, 2' = 1, 2, 3, 4 are determined from the initial conditions (416,4.18):

_ 21/6 + I/psc(0)(t/2 -— Ag) coth[wa/2 — uprrd]

 

 

 

 

A _

1 A? + A3
A _ 417211C — (I/2 — A§)(2uC — 123310))

2 _ A1(/\i + A3)

21/C + 1/,,,.2:(O)(1/2 + A?) coth[wa/2 — uprrd]
A3 = - 2 2
A1 + A2

A _ ~4172uc + (t/2 + A¥)(2Vc — 1432(0))

4 — A203 + A3)

According to the boundary condition (4.12), at the exit point :c(rrd) = pz(r,d) = 0,
which allows to ﬁnd rrd and 23(0). In particular, as it follows from expressions for
33(7) and 122(7) on the tunneling trajectory (6.7), this is equivalent to requiring that
A1 sin A17“, 2 A2 cos A1 Trd and A3 sinh A27“, 2 —A4 cosh /\2Trd.

The tunneling exponent is then given by:

2 2
R: L2,.+—,——— +—- +1—— .
7 [ Td 172 (1 A?) A11/C ( Ag) VCAg] (6 8)

 

 

Expression (6.8) is plotted in Fig. 6.1 for the quantity W(B)/W(O). The correction to
R(B) —R(0) from the image potential and other terms of U1(z) in Eq. (6.6) is ~ 1/7L.
This results in changes to the theoretical curves that are smaller than the uncertainty
in R(B) — R(O) due to the uncertainties in n and Si in the experiment [34]. Note,
however, that the potential U1(z) has to be taken into account when comparing the
tunneling rate W itself with experiment.

As seen from Figs. 6.1, 6.2 the dynamical many-electron theory is in good qualita-
tive and quantitative agreement with the experiment, without any adjustable param-
eters. At low temperatures (T = 0.04 K), the many-electron tunneling rate is bigger
than the single-electron estimate [34] by a factor of 102 for B = 0.25 T, see Fig.6.2.

107

 

 

 

 

 

 

 

 

 

10 _- —————————————— .1"
Iol - "’ "—*—_,—;"‘|- X j 026
Em 1 ‘ ‘ At A l A 4 0.22
‘\ [:1 I x T
3 ~ 6 D x i x >2]: 0 18 T
0.1 [— + + - ' x
. ° + : l
0.01 f o , 0.14 a
0.001 f 0 + -
- T=0.04K 0'1
0.0001 ‘ * ‘
0 0.05 0.1 0 .2

.15 0
3202)

Figure 6.1: The relative rate of electron tunneling from helium surface W(B)/W(O)
as a function of the magnetic ﬁeld B for the electron density n = 0.8 x 108cm‘2 and
the calculated pulling ﬁeld SL = 24.7 V/cm (solid curve). Solid lines show how the
theory compares to the experimental data [34]. The errorbars show the uncertainty
in the theoretical values due to the uncertainty in the parameters of the experiment.

For this temperature, the tunneling rate is well described by the T —> 0 limit, cf [56].
The B-dependence of the tunneling rate is very sensitive to temperature. It becomes
less pronounced for higher T, and the role of dynamical many-electron effects becomes
less important, too. Interestingly, the theoretical data on the ratio of W(B)/W(O)
become less sensitive to the experimental uncertainties in the cell geometry (which
determines 8 1) and the electron density n for intermediate temperatures T N 0.14 K.
This is because the corresponding errors in W(B) and W'(0) compensate each other

for such temperatures.

108

 

rTir vrrrlrr
/

/

l

0.01

rrrr

0.001 . \\ 0 1

[Yr

I
1 41 111

 

 

00001 O 1 1 I 4 0.65 11 1 m 1 0.1

 

1320.2)

Figure 6.2: The rate of electron tunneling from helium surface W(B) as a function
of the magnetic ﬁeld B for the electron density n = 0.8 x 108cm"2 and the calculated
pulling ﬁeld El 2 24.7 V/cm (solid curve). Solid line is the theoretical calculation
for T = 0. The experimental data are taken from [34] for T = 0.04 K. For such
low temperature, predictions of ﬁnite and zero temperature theory are very close to
each other, as can be noted by comparing with Fig. 6.1, where the ﬁnite-temperature
curve is given. The error bars show the uncertainty in the theoretical values due
to the uncertainty in the parameters of the experiment. The dashed curve is the
calculation [34] for T = 0.04 K without inter-electron momentum exchange.

The crossover to magnetic-ﬁeld enhanced tunneling occurs for temperature T c 2
0.19K, for the parameters in Fig. 6.1. The expected increase of the tunneling rate
with B for T > Tc is shown in Fig. 6.1. It has indeed been observed in the experiment
[34]. The analysis of the experiment requires to establish whether, for temperatures
of interest, escape actually occurs via tunneling. To that end we note ﬁrst that, as

it follows from a direct variational calculation, the potential U (z) (6.4), with the

109

parameter values speciﬁed in Fig. 6.1, has only one metastable intrawell state.

If the intrawell relaxation were fast enough, the temperature of the crossover
from tunneling to activation T a for B = 0 would be given by the condition that
the tunneling exponent R9 be equal to the activation exponent (Um,x — Eg)/T [here,
Um,x is the maximal value of the potential U (2)] This would give T, x 0.15 K.
However, activated escape requires that the in-plane thermal energy of an electron be
transformed into the energy of its out-of—plane motion. This involves a large transfer
of the in-plane momentum ~ [2m(Um,,_x —— Eg)]1/2. The electron-electron interaction
does not give rise to such a transfer in a strongly correlated system, since the reciprocal
inter-electron distance is TLl/2 << [2m(Um,,x — Eg)]1/2.

The major process which gives rise to the momentum transfer is scattering by
capillary waves on the helium surface, ripplons [53]. Electron-ripplon coupling is
weak [77]. As a result, the prefactor in the activation rate, which is quadratic in the
coupling constant, is small. For B = 0 it is N 72T2/ha [78], where 0 is the surface
tension of liquid helium. For temperatures T < 0.25 K this prefactor is less than the
prefactor in the tunneling rate (hyz/m) exp(—2) by a factor < x10‘5. Therefore the
crossover from tunneling to activation occurs for higher temperatures than it would
follow from the condition of equal tunneling and activation exponents.

For the parameters in Fig. 6.1, the rates of activation and tunneling escape be-
come equal for temperatures slightly higher than 0.26 K (for B = 0). Therefore the
experimentally observed increase of the escape rate with B is indeed due to the dis-
cussed mechanism of B-enhanced tunneling. The smaller experimental values of the

relative escape rate W (B) / VV(0) for T = 0.26 K can be understood by noticing that

110

the activation rate is close to the tunneling rate for such T, and since it presumably
only weakly depends on B, the overall slope of ln[l/V(B) /W(0)] should be smaller than
that of the theoretical curve which ignores activation (approximately, by a factor of
two).

To include the activation processes in the overall escape probability, W = VVtunn +
W’act, the probability of the over-barrier escape was calculated following the logic of

the paper [78]:

_ m 2 (“ET —E,, k T
Wact -— CEUCBT) Ea (1+ 3“???) 8 / B , (6.9)

where the numerical factor C = 1.42 differs by a factor of 4 from those in Ref. [78].
The barrier height gives the activation exponent Ea = Umax — E9. The result for the
escape rate for T = 0.26 K is shown by the dashed line in Fig. 6.1. This calculation
neglects the change in the probability of over-barrier escape with magnetic ﬁeld.
Although the rate of activation escape indeed only weakly depends on B, there are
several factors that could affect the over-barrier escape in a magnetic ﬁeld. First, the
ﬁeld may “push” the ground state upward in energy, by mw§[(zz) —- (2)2]/2, for a
weak ﬁeld (the averaging is performed for the ground state). A second factor is the
change, by the magnetic ﬁeld, of the wave functions with energies close to the barrier
top. For B = 0.4 T the magnetic length l = (he/(38)”2 is ~ 0.6 of the distance from
the helium surface to the barrier top position (A/ |e£i|)1/2. Both factors decrease the

activation energy for over-barrier escape.

111

6.3 The prefactor

The dependence of the potential U (z) (6.4) on 11 gives rise to the density dependence
of the tunneling rate W(B) even for B = 0. We can calculate the exponent and the
prefactor in W(O) by matching the WKB wave function under the barrier for 1/7 <<
2 << L with the tail of the non-WKB intrawell solution (here, L :2 h272/2mleé'il is
the characteristic barrier width). In the spirit of the logarithmic perturbation theory
(LPT) [79], the wave function of the ground state inside the well and not too far from

it can be sought in the form
120(2) 2 const x zexp[—A(z)] (6.10)

[we explicitly take into account that the function 029(2) has a zero in the ground state].

Near the well (z << L) the ﬁrst term in (6.6) provides the major contribution
to the potential. The solution can be found by considering the last two terms in
the potential U0(z) (6.5) as a perturbation 6U(z), which is polynomial in 2. Using
the anzats (6.10 for the wave function, one gets a Riccati equation for the function
f (z) E dA/dz:

d
d_j;+.:.(f_7)—f2+72:2m(E—6U). (6-11)

The function f has to equal to 7 for z = 0. The general solution has the form:

2

f E 'y +/ dzl2m(E — 6U(zl))% exp[—2'y(z — 21)] + Bz‘2 exp[2yz]. (6.12)
C

Since we are looking for the localized state wave function, f has to go to zero as

2 -—) 00. Therefore, coefﬁcient B = O and integration constant C : 00. The energy

112

E can be found from the condition that function f remains ﬁnite as 2 —> 0:

fooo (122:2(SU(z)e‘27z

E :
fooo (12226—272

 

(6.13)

From expression (6.13), it si clear that the contribution to the energy E from different
orders in z of potential (5U are independent and additive, so that for the perturbation

6U(z) = C2“, the energy is given by

E = (2pm. + 2)!/2, (6.14)

and the function f has the form

p—l
—i p—i (H 'l' 2)!

Near the well, the linear term oc 8i dominates the quadratic one in (6.5). To the ﬁrst
order in 8 1, the exponent A(z) of the wave function can be obtained by integrating

over 2 the function f from (6.15), with 11 = 1 and C = —|e£i| E —72/2mL:

A(z) z 72. (1 — 2151:) . (6.16)

The correction to A (6.16) is small for 2 small compared to the barrier width L.
We note that the exponent A(z) has an overall functional form which differs from
that of the commonly used [53] variational wave function 10(2) OC 2 exp(—7yz), with ’7
being the variational parameter.

The expression for A (6.16) matches the small-z/ L expansion of the action S of
the WKB wave function under the barrier for L >> 2 >> 7‘1. This allowed us to ﬁnd
the prefactor in the WKB wave function and in the tunneling rate. The resulting
tunneling rate is shown in Fig. 6.3. It fully agrees with the experiment (see also

113

 

 

 

 

 

 

E l l l l T T I l . L
A :
8 1 _ + _
CD
E
0.1 _,
f i
0.01 r -
I _._
0 2 0.6 1
-2
n (108 cm )

Figure 6.3: The rate of electron tunneling from the helium surface W(O) for B = 0
as a function of the electron density. The dots show the experimental data [34]. The
pulling ﬁeld 8; for n ——> 0 is calculated for the parameters used in the experiment to
be 26.7V/cm.

Ref. [80], where a good agreement was obtained between measured and numerically

calculated tunneling rates without magnetic ﬁeld for electrons on helium).

114

Chapter 7

Conclusions

We have developed a semiclassical theory of tunneling decay in a magnetic ﬁeld. We
show that, as in the case without the magnetic ﬁeld, the tunneling exponent can be
found from the classical equations of motion. However, in contrast to the B = O-case,
decay of the wave function under the barrier is accompanied by oscillations. Related
to this is the conclusion that the particle appears from under the barrier with ﬁnite
real velocity, and behind the boundary U (r) = E of the classically allowed region.
In the presence of a magnetic ﬁeld it is no longer sufﬁcient to ﬁnd the wave function
just at the boundary of a classically allowed region: it is necessary to ﬁnd both the
exit point (or any point on the trajectory of the escaped particle), and the absolute
value of the wave function along it.

From the technical point of view, the semiclassical solution of the tunneling prob-
lem in a magnetic ﬁeld is based on the analytic continuation of the wave function to
complex space. In the semiclassical approximation this corresponds to considering

trajectories with complex coordinates, momenta and time. The action S (r) calcu-

115

lated along such tunneling trajectories will have both real and imaginary part, and
this would describe both decay and oscillations of the wave function. The condition
for the particle to become classically observable is that its coordinate and momentum
become real, i.e. Irn r=Im p20. The real part of momentum (or velocity) is not
necessarily equal to zero, and gives the initial momentum of the escaped particle.
This momentum was accumulated due to the magnetic ﬁeld during motion under
the barrier. As we show, the set of complex under barrier trajectories has caustics.
The semiclassical approximation breaks down at the caustics. Different branches that
correspond to decaying and increasing solutions, and to incident and outgoing waves
match at the caustic, and we show how to choose the appropriate solutions to account
for the boundary conditions.

We have also analyzed the single-particle decay problem in a magnetic ﬁeld for ﬁ—
nite temperatures. We have modiﬁed the known bounce technique for the problem of
decay from a parabolic metastable potential well. In the presence of a magnetic ﬁeld,
the bounce trajectory becomes complex. However, as we show, the corresponding
action remains real, and the prefactor in the partition function is purely imaginary,
although the structure of the related eigenvalue problem is totally different (the eigen-
values become complex).

If the tunneling potential cannot be approximated as parabolic near its minimum,
at least in one direction, the “bounce” technique no longer works regardless of the
presence of a magnetic ﬁeld. Such a situation is relevant in tunneling from 2DES,
where the out-of-plane motion is quantized and the tunneling potential is strongly

non-parabolic in the out-of—plane direction (either singular, as it is in the case of

116

electrons on helium, or discontinuous, as in the case of 2DES in semiconductor het-
erostructures).

Of central interest for 2D science are the effects of electron correlations. We
show that tunneling in a magnetic ﬁeld provides a unique tool for revealing and
investigating these effects. They lead to an exponential increase of the tunneling
rate as compared to the predictions of the non-interacting electrons picture. The
mechanism responsible for such an increase is similar to a Mossbauer effect, where a
gamma quantum can transfer its momentum to the crystal as a whole. In our case the
momentum that is transfered to the electron system as a whole is an in-plane Hall
momentum acquired by the tunneling electron during its motion out of the layer.
In the non-interacting picture it is this in-plane Hall momentum that leads to an
exponential suppression of the tunneling rate in a parallel to the layer ﬁeld. However,
complete transfer of this momentum almost never occurs, as it requires that the rate
of inter-electron momentum exchange (plasma frequency) be much larger than the
inverse characteristic imaginary time of motion on the tunneling trajectory. When
these two quantities are of the same order, only a part of the Hall momentum is
transfered in the “recoil-free” way, which results in a partial, yet very substantial
compensation of the suppression to the escape rate introduced by the parallel ﬁeld.
Similarly to a Méssbauer effect, there is no need for a long-range order in an electron
system for our mechanism to work. The effect should be seen even if electrons form
a strongly correlated ﬂuid without the long-range order.

At high temperatures the tunneling rate is expected to exhibit a number of new

unusual features. One of the most interesting of them is that a parallel magnetic ﬁeld

117

may enhance, rather than suppress the tunneling rate. Such B-enhanced tunneling
does not happen in all systems. Competing processes, such as tunneling from higher
in energy intra-well states or activation, limit the range of temperatures where B-
enhanced tunneling can be observed. On the other hand, sufﬁciently strong magnetic
ﬁelds will always suppress the tunneling rate, which set the limits on magnetic ﬁelds
that enhance tunneling. Nevertheless, the possibility to increase the tunneling rate
without changing parameters of the tunneling barrier is not only unexpected, but
also very useful. For example, in the development of quantum cascade lasers, high
tunneling rates are necessary to achieve an inverted level occupation, meanwhile the
parameters of the tunneling barriers are essentially ﬁxed by other requirements.

The B-enhanced tunneling leads in turn to several types of switching effects,
such as switching between tunneling from different intra—well states, and from over-
barrier activation to tunneling, and vice versa. These switching processes may differ
qualitatively for barriers, where the tunneling length changes with intra-well energy,
as opposed to barriers, where the length does not depend on the energy. Both cases
have been analyzed in detail using simple models of a triangular and a square potential
in order to describe the ﬁrst and second situation, respectively.

The results have been compared to the tunneling data from experiments on
strongly correlated electron systems formed on liquid helium surface [34]. The ana-
lytically calculated tunneling rate and its evolution with ﬁeld and temperature are
in full qualitative and quantitative agreement with the experimental data, with no
adjustable parameters. This proves that tunneling experiments with a magnetic ﬁeld

parallel to the layer can be used in order to reveal strong electron correlations and in-

118

vestigate in-plane electron dynamics in a 2DES. The measurable quantity is the auto-
correlation function of the in-plane momentum of an electron in a strongly correlated
2DES. Of particular interest are low-density 2DESs in semiconductor heterostruc-
tures. Until now correlated systems in semiconductors have been investigated by
tunneling mostly for the magnetic ﬁeld B perpendicular or nearly perpendicular to
the electron layer. We show that for typical densities used in low-density systems,
and with barriers grown with standard techniques tunneling experiments designed to
probe in-plane correlations would require easily attainable values for magnetic ﬁeld

and temperature.

119

APPENDICES

120

Appendix A

The instanton method in a
magnetic ﬁeld: analysis of the

prefactor

To ﬁnd the prefactor in Z, we consider paths that are close to the extremal trajectory
f(7) + 6r(7), and show that the integral (3.23) provides an imaginary contribution to

Z. The action S E along these paths can be expanded as:

 

1 6233
R: . _ — d I i ' ’ .
SE Sg[r(7)] + 2// 7d7 6r,(7)6rj(7’)6r (7)6r,(7) (A 1)
To evaluate the integral (3.23), we would like to diagonalize the operator for 6255,

and therefore have to look at the eigenvalue problem:

 

.. d ' 6st ' -—/\ A2
/_ r 5.,(.)5.,(.1)¢Wl— me) ( .1

l
27‘

We need to investigate some prOperties of the eigenvectors 112(7) before we could use
them to expand the ﬂuctuations 6r(7) around the extremal trajectory f'(7). In the

121

presence of a magnetic ﬁeld the operator itself in (A2) becomes non-Hermitian. For

example, in the case of a uniform B it may written as F (7) 1/Jn(7') = An ¢n(7) with

. d
' C.,--B— A.
+me63k ldT ( 3)

m d2 6-- + 02U
d72 U agar,-

F,,-(7) =
The eigenvectors of the problem (A.2) form a biorthogonal set, meaning that eigenvec-
tors {W} are orthogonal to eigenvectors {42”} of the Hermitian conjugate operator:
F [4)” = A;¢n. Taking into account the symmetry (3.26) of the extremal trajectory

f(7), and counting the imaginary time from the middle point 6 / 2, we ﬁnd that the

operator F belongs to the class of ”PT-symmetric non-Hermitian Hamiltonians:
Fl(7) 2 F(—7) : F*(7). (A.4)

In our case, variable 7 is a space coordinate, so that inversion ’P corresponds to 7 —>
—7, while time reversal T is a complex conjugation. Therefore we have come upon
yet another example of PT—symmetric Hamiltonian. Recent physical applications of
non-Hermitian Hamiltonians include depinning of vortices in type-II superconductors
[81] and growth of populations [82]. Bifurcation of the initially real eigenvalues [83]
into the complex plane with increasing of an imaginary external ﬁeld (in our case -
magnetic ﬁeld) indicates the delocalization transition.

The symmetry (A.4) is a strong property, and in particular it shows that 1,0,,(7) =

(V (7). Then the orthogonality relation reads:

n

73/2
[62 d71/)m(7)1/J,,(7) = Andmn, (A.5)

where An is a normalization factor.

122

Because the operator F is non-Hermitian, some of the eigenvalues An will be
complex. Let us compare the spectra of operators F and F i. In general, they are not
the same, but in our case it will be true. We start with the eigenvalue equation for

operator Ff:
MUN/17.0) = Alt/4(7),

perform the time inversion, and then use the symmetry (A.4) to get

F(T)¢*(-T) = AWN-T)-

The last equation does not mean that all eigenvalues An are real. Instead, it shows
that the spectrum of operators F is identical to that of operator F l. The eigenvalues
An are either real, or are present in pairs of complex conjugated numbers.

We are now in position to expand the ﬂuctuations around the extremal trajectory

r(7) — f(7) in terms of eigenvectors '42,,(7):

19/
MT) = Ail/2 2611111710)» 0n = 1451/2] 2 dT6r(T)¢n(T)3 (A6)
43/2

Note that by multiplying an eigenvector 1])” by a phase factor em does change the
phase of the normalization factor A,, by 201. However the coefﬁcients of expansion
{en} remain unchanged. The second-order correction to the action is diagonal in 0,,

variables,
SE z slat-(7)] + Z AnCi/2. (11.7)

The statistical sum is therefore given by a Gaussian integral in variables {ca}. One

123

could formally rewrite (3.23) as

—1/2
Z Z [/ / Dr(7)Dr*(7)e_SE[’(T)]—SE['.(T)]]
r(—ﬁ/2)=r(3/2) r‘(-13/2)=1"(t9/2)

—1/2
oc e”SE[f(T)] U [/ den/dc; exp[—A,,c?, — Ancﬁ] (A8)

The integration over variables on is easy to perform, the prefactor will be propor—

tional to
Z oc HAgl/2e_SE[f(T)], for An 75 0. (A9)

Integration in (A8) over variable c,- that corresponds to A,- = 0 will be done below.
Since the spectrum of operator F contains complex eigenvalues in pairs, they provide
positive contribution to the prefactor. We now turn our attention to the real part of
the spectrum of operator F. The real negative eigenvalues are of utmost importance,
and there should be an odd number of them, in order for S E[f(7)] to determine the
exponent of the imaginary part of statistical sum Z.

First of all, one of the eigenvectors is known - it is 1/21 2 df/dr. This can be

checked by a direct substitution to the eigenvalue equation (A.2):

 

T (snaps-(TI) (17’ _ 6r.(7)

 

d

.4 2 -.
/T I (533 dl‘]_ ($53 :0. (A10)

.1
Therefore, 11), corresponds to A1 = 0. In the integral (3.23) we integrate over various
paths 6r(7). According to transformation (A.6), the shift in 6r(7) due to a change in
coefﬁcient c, is given by

Ar(7) = Ail/2Acl¢1(7)

On the other hand, the shift of the middle point of the “bounce” may be regarded as

124

one of the possible ﬂuctuations around r(7)

Ar(7) = ___dr(7 _ T0)A70 = —A1—1/2¢11(7)A70

dTO

Assuming that 5 >> 1 (we consider the limit of low temperatures), the position of
the middle point can be anywhere in the interval (—6 / 2, 6 / 2). A shift in the middle
point results only in an exponentially small change of initial values. By comparing

the last two formulas, we ﬁnd that Acl 2 A70, and therefore

.3/2
del =>/ dTQZ
5/2

The integration over cl that correspond to A1 = 0 has thus been performed. No
divergence occur in the prefactor (A.9) due to the presence of the zero eigenvalue. It
is to be excluded from the product in (A9) Instead, we obtain a very natural factor
of B = T“, which eliminates a linear dependence of the tunneling rate (3.22) on
temperature.

Presence of a zero eigenvalue is not related to magnetic ﬁeld, and occurs without
the magnetic ﬁeld as well [9, 10]. For B = 0, the ﬁrst eigenvector 1/21(7) = drc1(7)/d7,
where the path rcl(7) is real and corresponds to a classical trajectory in the inverted
potential —U (r) The fact that the eigenvector ¢1(7) gives the ﬁrst excited state can
be seen by noting that it has one, and only one, zero at the middle point of motion
along the “bounce” trajectory: 11,.(0) = 0. Without a magnetic ﬁeld, operator F is
Hermitian, so that the oscillation theorem is valid, and number of zeros enumerates
the eigenstates, with the ground state eigenvector not having any zeros. Therefore

there is one and only one negative A0 < 0, which makes the prefactor imaginary.

125

Let us follow the evolution of eigenvalues An as we turn on the magnetic ﬁeld. As
we know the eigenvalue A1 = 0 is not shifted, other eigenvalues, however, will change.
We also know that complex eigenvalues are present in complex conjugate pairs, and
therefore have the same real part. Because the overall number of eigenstates remains
unchanged with increasing of magnetic ﬁeld, pairs of real eigenvalues ﬁrst have to
merge together. Only when their real parts are equal, the imaginary part may appear.

The state with A = 0 does not mix with any other solutions. Suppose the opposite
were true, so that two different states ’l/Jl = df/dr and 1/22 satisfy the condition

F 1,1212 2 0. We can subtract these two equations from each other to obtain that

d . . -
3; [112.112. + «1.11). — ichl¢1 x «121] = 0.

Because the eigenvectors are zero at boundaries at i6 / 2, the following is true as well

‘1/127’1 + 161712 + Z'611c‘1b2f11b1 X B] = 0.

Remembering that 1/21 2 df/d7 and using the equations of motion (3.25) for f we
ﬁnd that 11211212 — VU 1122 = 0. This is a ﬁrst order equation, and one solution is
known to be df/ d7. It is also possible to show that the components of 1,122 which are
perpendicular to 1b, are equal to zero. The eigenvectors 1p, and 1112 differ only by a
constant.

We have just shown that state of the zero eigenvalue A1 = 0 remains non-
degenerate. Therefore, it separates the negative eigenvalue A0, which corresponds
to the ground state, from the rest of positive eigenvalues, and prevents the ground

state from mixing with higher levels.

126

After complex conjugate pairs have formed, they could, upon further increase of
a magnetic ﬁeld, change the sign of their real part from positive to negative, while
having non-zero imaginary part. Then it is possible for the imaginary part to become
zero again with increasing of the ﬁeld. However, even if this were to happen, it would
add an extra pair of real negative eigenvalues, and therefore the overall number of
real negative eigenvalues would remain odd.

To summarize, we know that integration over paths close to the extremal path
r(7) provides an exponentially small imaginary part to the statistical sum Z. The
exponent of the tunneling rate W = 2TIm Z / Re Z is given by the Euclidean action
S E calculated along the extremal trajectory r(7) that satisﬁes the equations of motion

(3.25).

127

Appendix B

Many-electron inﬂuence functional

Ree[Z] at zero temperature

The scaling behavior of the tunneling exponent given by Eqs. (4.27) and (4.28) is quite
general and takes place for all tunneling potentials provided characteristic frequency
top > 7!. Below, the derivation is presented for a square barrier. Such potential is
special because the velocity 2(7f) at the ﬁnal point along the tunneling trajectory is
not zero. The symmetry (4.13) is imposed. Under these circumstances it is easier

rewrite the original retarded kernel Ree[z] as an integral from 0 to 7,:

729,.[2] = —111§/0Tf/OTl d71d722(71)z(72)[x(71 — 72) + X(2Tf — 71 — 72)] (B.1)

At zero temperature, the kernel x(7) = 711(2N)“l 2:ij exp[——wkj7]. It is clear that
different frequencies component are independent, therefore let us examine a kernel
x(7) 2 mo) exp[—w7].

Because of the exponential dependence of the kernel x(7), the major contribution

128

comes from 7 = 0. For the ﬁrst term in (B.1), this corresponds to 71 — 72 < w”, and

we can use the expansion (4.26) as before [for convenience I rewrite it here]:

zen.) z 2e.) + awe. — 71)+%5(71)(72— .,)2

For the second term in (8.1), both 71 and 72 have to be close to Tf. Therefore both

2(71) and 2(72) have to be expanded near 7f:

Z012) % Zle) + 2(7001.2 - T!) + $500012 — Tf)2 (B2)

A straightforward integration gives the following answer for the inﬂuence functional:

2
777.61) c

2N

 

Ree[z] = — Tl (1722(7) +w;].22(7)2(7) — 2(7 )2 (7 )w—2 J (B.3)
U [A f2 f k

As for a smooth tunneling barrier, the term o< 22(7) cancels the single-electron mag-
netic barrier in (4.23). If potential is purely square, then 25(7) E 0 [hard to imagine,
but to just to show that the scaling results (4.27), (4.28) hold in this case]. Mass
renormalization 1S then due to the boundary term or 2(7f)z (7 7,,) since for constant 2",

we can always write it as an integral

Reelzl= ~"meg/OT! d7 [2(7) )‘Ingzzzm (B.4)

On the other hand, if 'z' is not identically zero on the optimal trajectory, then the
same answer (8.4) is achieved by one integration by parts.

Frequency-dependence of the inﬂuence functional is studied next. It is shown
below the larger the phonon frequency wkj, the larger is the absolute value of the
contribution from this mode to the many-electron inﬂuence functional Ree[z]. For

T : 0, the function that we sum over phonon frequencies in Ree[z] is given by:

2Tf T1
f(w.71)=/ (171/ d722(71)Z(T2)w6""‘T‘—“’) 03.5)
0 0

129

We will show now that this function is monotonic in terms of the frequency parameter

(.2. Indeed, its derivative with respect to w is:

2T] Tl
f1,(w,7f) 2/ d712(71)/ d72z(72)(1— w(71 — T2))€—w(Tl—T2)
0 0

Using the symmetry 2(27f — 7) = 2(7) of the tunneling trajectory, we can change
the upper integration limit from 27f to 7f. Because for 72 S 71 5 7f, tunneling
trajectory 2(7) monotonically increases with 7, we have 2(72) < 2(71). Therefore one

can substitute 2(71) by a smaller quantity 2(72):

TI TI
£30017!) > 2/ dTi/ d72z2(72) [(1 — W(Tl — 72))e—W(T1-72)
0 0

+(1 — w(27f — 71 — 72))e—w(27f'71’72)] (86)
Changing the order of integrations in order to integrate over 71, we get:
T] 2
11.1w. 7.) = 4 / d72z2(72)(71 — 72h?" “tr-12> > o
0

In this way, we know that terms corresponding to higher phonon frequencies provide
larger contribution to 7299M. In addition, the density of phonon states also favors
larger wkj. To summarize we can conclude that at least for T = 0, the tunneling
rate is determined by high-frequency phonons, which in the case of Wigner crystal
correspond to short-range vibrations. The out-of—plane tunneling probes the short-

range order in a 2DES.

130

Appendix C

Square barrier: calculation of the

tunneling exponent.

In the Einstein approximation of a Wigner crystal, the tunneling problem is formu-

lated for one-particle with an effective tunneling potential:

1 2 112 2 1
H = %(px + wcz)2 + 21:; + 2;; + imwga:2 — mgr/:2 (C.1)

 

 

In what follows, we will scale the coordinates by the tunneling length L, meanwhile
the momenta will be scaled by h'y/ 2. This results in the scaling of the energy by
it”? / (4m), and the action S by th/ 2. The frequencies found in the Hamiltonian
(C.1) will be scaled by combination hy/(ZmL), which indeed has the appropriate
dimension of 3‘1, and is half the imaginary tunneling time for B = 0. The dimen-
sionless frequencies are therefore deﬁned as up,c : 2mep,c/h’y and 17 = 2mLcD/h'y.

The Hamiltonian in these dimensionless variables takes the form:

9:
2

1 1
H = 5(1):; + ch)2 + + 2 + 51/5232 — 172732

131

The corresponding equations of motion are linear:

dpz__21722_yc_l£ dp$_ V23? d2_ . d3:
d7 _ Cd7’ d7 _ p ’ d7—pz’

 

 

and have real and imaginary eigenfrequency of vibrations:

 

A; = [V2 — 2172 + \/(1/2 — 2172)2 + 817211;] /2

 

Af 2 {—1/2 + 2172 + \/(u2 - 2172)2 + 8521/15] /2
Using the initial conditions for the trajectories (4.16), the tunneling trajectories can

be found to be:

 

 

 

 

A A A
1(7) 2 —_):1 cos A17 + :12 sin A17 + "T: cosh A27 + -/\—: sinh A27;
1x2 — A2 V2 + A2
112(7) 2 p 2 [A3 cosh A27 + A4 sinh A27] + p 1 [—A1 cos A17 + A2 sin A17]
VC/\2 All/C
2 A2 /\2 _ 2
2(7) 2 — V _+ 1 [A1 sin A17 + A2 cos A17] + 2_ V [A3 sinh A27 + A4 cosh A27]
21/2126 21/211C

p$(7) : —1/62(7) -—- [A1 sin A17 + A2 cos /\17 + A3 sinh 1/27 + A; cosh A27]

where constants A1, 2' = 1, 2, 3, 4 are given by:

V306 — V2).’Eo — Zone/\f. 1210:3093 — V2)

 

 

 

 

A1 = A1”? + Ag) , A2 = (A? + A3) coth[wp(B/2) — VpTrd];
1/2CII0(V2 + V) —- 2w A2 1/ 220(112 + A2)
_ p 1 C 2 , _ P l _

Here, 7rd 2 7f/(2mL/h7).

Until now, the boundary conditions at the exit point have not yet been taken into
account. The tunneling potential (Cl) is special in that respect that there a two
possible boundary conditions depending on the parameters of the Hamiltonian. For

17 > 1 and B = O, the boundary condition is

2247,53) = M?) = o, (0.3)

132

where the superscript denotes that the imaginary tunneling time was found using

condition ((3.3), or equivalently from:

 

A1 V2 + A2 VP

£112 _ A; [A—Z- cosh A27(d) + coth[wp(5/2) — V127 r(d 1)]sinh )0de )] COS A1755)

— _[§- COS A173) UCOthle(B/2) VPT rd 1)]Sin )‘1Trd )1 COSh )‘2Tr(d) (C4)
1

The corresponding tunneling exponent is given by 5);) = 2753).

Upon either increasing of the magnetic ﬁeld or decreasing 9, there appears another

tunneling trajectory, which satisﬁes the condition:

2< 53’) =1 x< 5.?) =0, (0.5)

In the explicit form, the equation for 73) is

21721751/C(cosh A2753) — cos A175?) [A1 sinh A278) _,\2 sin A1752 )

—\/§17 coth[1/,,(ﬂ/2 — 7f:))](cos A1753) — cosh A2753)”
2 [(1/2 + A9)” (sinh A27r(d) coth[1/p(ﬂ/2—( rd ))]+ 11);:- coshAg 7( 23))

1

x [A (172+ A2)sin)\17r()§ — Agog — 1/ 2)sinh A2753) — 17292 + A3] ((3.6)

The tunneling exponent calculated along the trajectory with boundary condition

(CS) is given by
St” = 2753’ + pz<r£§’)/2. (0.7)

In the range of parameters where both trajectories exist, then the exponent of the
escape rate will be given by the smaller of 5);) and 5):). Switching between the
regimes where the tunneling exponent is determined by one or the other boundary

133

conditions leads to singularity in the derivative of tunneling exponent, as can be seen
in Fig. 4.4.

In summary, tunneling through a square barrier have been analyzed taking into
account the “correlation-hole” potential from the electrons remaining in the layer.
The tunneling exponent have been found exactly by solving the equations of motion
(C2). The result describes tunneling at ﬁnite temperatures, including the T = 0 case

as an appropriate limit.

134

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