PQLARON Emmi. EN THE 0mm... ammam’zas as mm gewecomcmas fimsfis €00 f3“ DQ’QNN a»? Div.» El MICBLGAN STATE UNIVERSE? Robert .3; Heck 1968 mum '— N: 'Qiliifiia‘l 3118” University This is to certify that the thesis entitled POLARON EFFECTS IN THE OPTICAL PROPERTIES OF POLAR SEMI-CONDUCTORS presented by Robert J. Heck has been accepted towards fulfillment of the requirements for Ph . D. degree mm Major professor Date 25 June 1968 0-169 ABSTRACT POLARON EFFECTS IN THE OPTICAL PROPERTIES OF POLAR SEMI-CONDUCTORS V" by Robert Ji Heck The optical absorption coefficient of a polar semi-conductor is calculated for light energies comparable to the width of the forbidden energy gap of the material in an effort to determine some effects of the electron— phonon interaction. It is eXpected that the effects will be significant in polar materials because of the strength of the interaction. A simplified version of the Kubo Formula is used which reduces the problem to the calcula- tion of the one-particle electron Green's function. A perturbation approximation is made to determine this function. The result is an absorption coefficient that has anomalous structure occurring at an energy of one phonon above the band edge and a "tail" region of states to which electrons can be excited below the band edge. An improvement is made on the perturbation eXpansion for the Green's function which results in the appearance of structure at energies of two phonons above the band edge. POLARON EFFECTS IN THE OPTICAL PROPERTIES OF POLAR SEMI—CONDUCTORS By Robert J. Heck A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Physics 1968 ACKNOWLEDGMENTS I wish to thank Professor Truman O. Woodruff for directing me into a very vital field of physics and for his eXpert and patient help in suggesting this problem and many different ideas in approaching it. I also wish to thank the National Science Founda- tion for their support of this work. Table of Contents Page Chapter I. Introduction . . . . . . . . . . . . . . 1 II. The System Hamiltonian . . . . . . 2 A. Table Of Coupling Constants . . . . 2 B. Derivation of Interaction Term. . 3 C. Discussion of Eigenstates of Non-interacting Term. . . . . . . 5 D. Discussion of Effect of Interaction III. The Formula for Absorption. . . . . . . A. Description of Standard Approach. . m-x‘lxlm B. Discussion of Kubo Approach . . . . C. Simplification of Kubo Formula. . . 14 IV. The One-particle Green's Function. . . . .' 24 V. The Absorption Curve . . . . . . . . . . . 32 VI. A Two-phonon Diagram Correction. . . . . . 47 VII. Conclusion . . . . . . . . . . . . . . . . 60 Figure One! The Band Scheme . . . . . . . . . . . . . 62 References. . . . . . . . . . . . . . . . . . . . . . 63 I. Introduction The use of Green's function techniques for perform- ing calculations of the preperties of many-body systems is now almost standard. The Green's function formalism has the advantage that it does not require finding the wave functions of the system, which in themselves are rarely of interest anyway. The one-particle Green's function1’2’3’4’5 yields information about quantities such as the free energy and density of states of a system and the self-energies and life- times of its quasi-particles, and the two-particle Green's 1,2,33435 , p information about transport properties as function well as the ground state energy. In this paper we apply Green's function techniques to the calculation of the inter- band optical absorption coefficient of a polar semi-conducting crystal for photon energies comparable to the band gap. This problem is of interest because the interaction between electrons and longitudinal optica1«phonons is significant in these materials and might be expected to lead to observable anomalies in the absorption Spectrum. II. The System Hamiltonian Associated with the optical-phonon modes in a polar crystal is a dynamic field of dipoles with which the electrons interact. A measure of the strength of this interaction is the coupling constant ©(Zflé4(i _._‘_ Y.“— ”a. 1: 2J3) 659° 6:5‘)(fhmtbda) D , where m is the free electron mass, mz is the effective mass of a carrier in band I (In our investigation we are interested in transitions from the valence band to the conduction band so the subscript 5 will refer either to the conduction band, £=c, or to the valence band, £=v.), e0° and es are the high fre- quency and static dielectric constants reSpectively, and mo is a phonon frequency. Values of the coupling constant for various polar substances are shown below. Table I LiF 5.2 KCl 5.6 Cu20 2.5 PbS 2.5 NaF 6.3 KBr 5.7 MgO 2.3 InSb .014 NaCl 5.5 KI 4.6 ZnO .85 GaAs .06 NaBr 5.0 AgCl 1.7 CdS 1.2 NaI 4.8 AgBr 1.6 ZnS 1.3 The interaction energy between an electron and a di- pole field is > > Hm.‘ ”f0” Dm'PW . > where D is the electric displacement vector of the electron and P is the dipole moment per unit volume of the field. From this expression we can see why it is the longitudinal mode rather than a transverse one which interacts most strongly with the electron: integration by parts gives /0 Pd’rz—e/QE VP/V‘ 3 where D=eE= -eV @. But V P=O for a pure transverse mode. The D(r) field of a system of electrons having the wave function w(r') is. + 3 ’ C 9/0“) WI“)/ ,0” > ' ‘1”! 0 ~ > For longitudinal phonons vxP = 0 so that in this case P may be written as the gradient of a scalar potential field > m(r), P = vn(r)/4w. Therefore HM.: e, M? d'r’v.(9’(rgzr-/vj WW0» 477' Ir—r/ = - 6%” 43rd?“ PM?“ WI“) WI“) / r47 =/a/3r’ )Vffl") WI”) 790’“) . (2-1) The first step was an integration by parts; in the last step we used the fact that l -v,3 773;,- CIN‘I‘OL/W' . The dipole field is related to the longitudinal optical phonons by > = 1, 4.19.)“ Her PM (2.37%)2 dE/P 7/?! (bis +648 )’ where bk’bk respectively create and annihilate a phonon of > momentum k, and since 4vP = Wm, ”(Wm)"? ; 7t ( M: 5’7} (6.565):_:fl Q where We Inserting the expression for m in equation (2.1) we obtain the interaction term in the Frohlich Hamiltonian6 HM; 9776 w. J. <13!“ 9V2}? Wr) (77? ”if“ 2,3 He/ f x (516%,: 5/. 8“")? (2.2) The total Frohlich Hamiltonian is kinetic energy of electrons + kinetic energy of phonons + [WM [/09 51,0743, + Hm. (2.3) where V(r) is the periodic potential of the crystal lattice when the ions or atomic cores are in their equilibrium pos- itions. If we consider the system to have only one mobile electron, the linear combinations of products of electron wave functions with phonon wave functions which make up the eigenfunctions of this system are said to describe "polarons" The eigenstates (Bloch states) of the electrons for the Hamiltonian without the interaction term (2.2) are determined, of course, by the structure of the crystal. For convenience we consider a simple structure like a cubic crystal so that quantities like the conductivity will be a constant times the unit tensor. This isotropy implies that the surfaces of constant energy in k Space are Spherical. We further assume that the electrons in the conduction and valence bands have the energy dependence on k, the quasi- momentum, shown in figure one, i.e., we assume parabolic dis- persion curves with the minimum of the conduction band and ) 2 2 the maximum of the valence band at k = 0. So E(k) = $55- 2 2 and E(k) ='EEE— + A for valence and conduction electrons c respectively, where mv and mc are the band masses of the electrons, and A is the forbidden energy gap width (note that mv is negative). But what are the energy levels of the interacting system? If we assume H int. can be treated as'a perturba- tion then the energy levels will be those of the phonons plus the band electrons plus correction terms obtainable from perturbation theory which are presumably small com- pared to the first two terms. In addition the Hamiltonian will not mix states of electrons from differentbands except in very high orders of perturbation theory. So we can still talk about bands, but of polarons not electrons, and the density of energy states will resemble the non-interacting density of states for electrons in having a forbidden gap. However the band gap will be "fuzzed out" by the interaction, and the density in the bands will be changed. In the calcula- tion of Optical prOperties of the system, this difference turns out to be important. III. The Formula for Absorption To perform the standard calculation for the absorp- tion7 one takes the first order, time dependent perturbation theory result8 for the probability of a harmonic external potential of frequency‘m to cause a transition in the system from the state J to the state m after a time t: 9 > r 2 I 1 __L . /)°‘ x where hth = Em - E3, the energy difference between the > . upper and lower states A is the external light potential, > and p is the momentum Operator. To get the total rate one multiplies the above by the probability that in thermal. 'equilibrium m is unoccupied and J is occupied, Nchn) - NF(EJ) and sums over m and J. Instead of summing one could intro- duce density of states functions and integrate. This would be a difficult procedure in our case because we would have to calculate the matrix elements and energy levels of the interacting system. Instead we will use a more direct and more exact formal procedure which will require us to find only the one-particle Green's function of the interacting system (but without the exciting light's potential in the Hamiltonian). This formalism due to Kubo9 is described below. Our Specific goal is to calculate the interband absorp- tion of light by a semi-conducting polar crystal having the previously mentioned simple structure. The absorption coefficient is defined by the equation I = I oe'“x which relates the intensity I at a distance x into the material to the intensity Just inside the surface 10' Since the absorbed light energy is being taken up by an increase in the kinetic energy of the charge carriers it is not sur- prising that the absorption is related to the in-phase part of the conductivity; the relation is: ($747269, Re g-Igo e is the real part of the dielectric constant which can be related to the imaginary part of 0. So if we know 00») we can find all the optical propertiesll. The Kubo formula for conductivity is: 61-90%») = 470816.23; +#w/di/d’rd3r” 41(wt—le-l"+/e’-l") g ,> I x 6 9%) (f 9.160,}? (ng} 9 (3.1) where t = x - x ', N is the average electron density, 9(t) O O is a step function, < > means averaging over a grand canonical Tre'p(H ' “N)o ensemble i.e. (O) = and Tre-R(H - UN) > I ’ (x) = .t’. x- f * — 7‘ y :71 Z(V W») Va) Wx) WM} 2. /m ) Am Wm KW) ; R(x) is the vector potential of the light. The Kubo formula can be regarded as a consequence of 9, which relates general the fluctuation-dissipation theorem susceptibilities to correlation functions of the related physical response. In this case 0 is the susceptibility and 5(x) is the response. Actually the fluctuation-dissipation theorem can be proved by a generalization of the following derivation of Kubo's formula: lO "' > One begins by calculating J = Ter, where o is the 12 density matrix , to first order in the external field, using O = it [H + AH, o] where AH is the external field part Of the Hamiltonian i.e. AH = - %-5 dx A(x) - jkx). The origin of the two 3's in in the Kubo Formula is clear at this stage. It is important to mention that the 3(x) and 5(x') Operators in the Kubo formula are Heisenberg Operators which are developing in time under the full Hamiltonian Of the system but without the externally applied field Of the exciting light. It is then physically reasonable that <5Ix)3(x')>, which is a measure Of the system's ability to sustain a current without a driving field, should be related to the conductivity. net The quantity I 9(t) e dt can be related to a two-particle Green's function as follows: «Ark. I" ogk" V, > . fed/(Eganfix’fl > 6”": c C o/z‘oz’rd’r’ .— '1- ‘- ' —.¢:(90V‘ 4:16;,“ " "($773): 43rd3r’dt ea) amte c 11 x < { Ev; who) Wat, W9) Wx ’21 - [View MAM $0M) Wxfl - [(774 WWW). W? 72‘ WW]. f [Mmza sum) W7x917j‘/’(XZ7}> > (3.2) We have neglected the terms containing A in the definition > of 3 since the Kubo formula only takes the first power Of > A into account. Also, using the long-wave length approxi- ik'r mation, we have replaced e by 1. We can do an inte- gration by parts on each one Of the last three commutators which makes it equal to the first. The quantity (3.2) becomes --e__a t“ L/ a! r d r ’a/r exam? WHW’KXMV, W) W) We choose for our representation of the wave functions the > Bloch functions Mk £(x) Of the non-interacting Hamiltonian 1.2.3.4,5 (this gives the zero order Green's functions 12 a ver 8 m e orm .e. _ + y 1 pl f )1 Myrggakflfffl >917?” 121016?) where a+ (a ) is an Operator in the second—quantized k,£ 14,2 representation which creates (annihilates) an electron Of > momentum k in band E. Equation (3.2) then becomes, ) > «:57- fdtezt) i no: (whit) when) 777" hh,K2 .Jhflg x (magnate aé‘ifcfi.m)l\/ “(16.91/39 =f ‘1 fig/(X) f2? 43X E ”‘<’“'I’°‘I"'lfi‘)‘3*rfz The delta—function indicates that we are neglecting Umklapp processes. The quantity + + but / 6d) ([6113) den/:31, 0149/. akgogylye cit: K #203 can be Obtained by calculating the two-particle Green's function 13 Ms) =-‘-’<7‘0. So if we can construct a function which is analytic in the upper half Of the complex (ii-plane and which takes (on the values K(O)n) at the points imn, we have found Kr(m). The above remarks apply to any double-time functions defined analogously tO K and Kr’ e.g., consider G0; to); am 3’3“”? [(226%) >Quail > dt‘ 14 I and II, fi(h,w,,_)= e‘“"’<7(94.ma;(a))>dr ~t\~ then (Mk, icon )=b. (Ran) . Now we have transformed the problem to the calculation Of a two-particle Green's function, but we will go further and reduce the two-particle function to a product of one— particle functions by an approximation which is valid for our 13’1”. The reason for this step is that better particular system techniques are available for calculating one-particle functions than for two-particle functions. TO perform this reduction we note first that transitions in which £1=£2 and/or £3=£4 refer to intra-band transitions which we are not considering. Then the diagrams for K(t) take the form f I I 1 0 t +- 5 + /"‘\ ' e— 4V 1 x I knit ,. [731,3 - .. I - ~ '1‘ on ’p q ‘ ’0' s I s , Q ’l a \ 1 I o \ L _1/ \ g ‘ It i ' l ,' t + ,- ,. + ’,- I ,.~ + \ a c I ‘. ll ‘ . 4 _4_ ‘. I 1 11 L_i ‘ V \‘ , + - fl- .- where the solid lines are electron propagators, the dotted lines are phonon propagators, and the vertices are interact- 15 ions in which R and Z are conserved, i.e. 3 ,// ‘ k)-1 ’///3 Also the propagators are diagonal in k and fl i.e. nbo. The proof Of the theorem apparent- ly depends on showing that matrix elements of interactions such as the one taking place in our simple diagram are small. According to the Feynman rulesl’5 for evaluating this dia- gram, the following factor enters into the amplitude for this process: 2 d”: dfi + r. f a kl (h‘ HG.) Ir. 91!”) “13¢ 91v” ) gift} 1:!“ t“ + -~ / W in 82:9» it!» a, we [’7‘er (We have chosen the electron at r1 to be in the conduction band and the electron at r2 to be in the valence band). Due to the fact that electrons in the valence band are pre— dominantly closer to the ions or atomic cores than electrons in the conduction hand, there may be some basis for saying this integral is smaller than one like \///22;:f77 zigkéig) zz:;}<>ségl£;:2’/¢<7/‘ril7“2%dH7C1%rl which is a factor in a diagram such as 17 since there should be_more overlap Of the functions in the integrand Of the latter expression. However, I know Of no con- vincing general proof Of the inequality, so that the "asymptotic theorem" may not really be a theorem, although I expect it to be valid in many cases and as we shall see later, it leads to a reasonable result. If we follow Bonch-Bruevich and assume the "asymptotic theorem",.we see that in the set Of diagrams for K0») we can neglect those which connect the two solid electron propagator lines with a dotted phonon propagator line. Then K(md becomes the Green's function for independent propagation Of an electron in band £1 from O to t and an electron in band £2 from t to O, i.e. K(t)"’>b‘ (k1,£1,t) {fl (k1,£2,-t) or t . m.) = i e‘ “”247 (A 2.7:) /J (t .12. :t) «if /T .. L Lao-r: o’th-’ "t - a e " 7“; g30 where Gr (f) “*3 (szs),s”?o>]_> 5W) . Similarly there exists a function Ga(w) such that Ga(imn)= )3 (an), mn>0, where 6.6:) = 2'. < ELI/(t), Wool) 8(—t) . Gr(a)) is analytic in the upper half-plane; Ga(cn) is analytic in the lower half-plane; so the summation becomes: 19 T; b ([0. ,fl/lwnyflOQ/LIWNI‘UM) -—> [Gr/“(Apnea 6%(t,,/z,ni;n1; dw’ 65%: where the integration is over this contour: r Mod—=0 \ + 7 W), fi U The contour avoids all the singularities of the integrand :‘Lwh except those at w=(2n+l)imT (there are other singularities since the first function in the integrand changes from Gr to Ga at the line Im mfl=0, and the second function in the inte— grand changes from Gr to Ga at the line Im.wfl=-iwh); so the residue theorem assures us that the integral equals the sum- mation. Since the contributions to the integral associated with the arcs at infinity are zero, the four horiaontal lines are the only part Of the contour that require consideration. 2O , 4e K(W)~ fdw’ Gr (higw’u'wn) ‘°‘ 6"“.1/ X [Gr(él)lll WI) " GA (knll; “9].!- QL(b'J (U waxed") €fl(w’—xlwn)+’ X [Gr(k.,12,w’)- Ga (6%., 601)]; : fifth/{Gr (leuizingiwh) tame-rUQ/Uw’j "PO emu)! +1 + Ga(kul,)w’v£wn) sow 6;.(12UZi/w’) 9’5“»),- £Nn)+/ 4(- (We have used the fact that G r(‘°)=Ga(‘°) which follows from the definitions of Gr(t) and Ga(t).) Since / “(w/'véwn) e oo’-.C(2n+’)“ '. 13°" '7 C < " a C’ k (W) -" Q/dw/{Q—r—Qfiiiz.’ WiiWn> Q'M Gk (kn/UN) °° 636011; 21 f + Ca (h,l,,w{-£wn) flm Grféuzzl 0),) j 0 6%(wl-ZWn)+I Then since Im Kr(imn)=1m K(mn) we find: «D’mkflufi = 2/ {flmGr(b,l,,t«/j 'QmGrap'll’zww’) I 9M. Gr(h,l,)w4%0r “Puke/j} ’ Bw’ 6 H and finally letting (n+(D'--->a)' in the first term we Obtain: M Kf‘ad) .':' afz’w/ 9m Gr(’?u11)w9 J'WuG‘f‘ (kill!) “(.09 Y (”Hui-afi-‘nflwlfl ’ where nf (60' )=1/e”'+l . 22 Thus the Kubo Formula (3.1) becomes «(do -= 4.1.7, Pa and) = 91%" d3/e 015 2 66¢ ce 1 firm .efc v 1.13% Nz(i,l.13na(/$,2,¢J Jim 6,031. E-w)(n.(E-w)—n.(£)) 33310:,41‘3; fd’kdEgfim Gr(/;,ClE-w) Rw€€lfifrm x 41m. GJLME) + MGJEEW) 41% 6,0299} ( mas—w) 47,452) . The first term will give practically no contribution because Im Gr(fi,c,E-m) is peaked at a distance 2A—u from Im Gr(1%,v,E), as we will see later when we get an explicit expression for Gr' 23 At this time we make the replacement V1VJ"'>1/3 x81 Ilellv(k)|2, which is valid for a cubic crystal. Also we assume that v(k) is slowly varying with k so we can take it_out of the integrand. Then we obtain the final form of the expression for a: an); .-..— ge‘/Uct’)/" oz’le d5 .0... Grays-i9 cebwmw XJ’M (#0295) (he (E'“)‘nF (5)) (3-4) In the limit as the electron-lattice interaction goes to zero, AMGJEJH 72'3” {(E-éié: —a at; +4.) , and the eXpression becomes the one which is well-known for inter-band absorptionls: a(m)~(hm~A)l/2. Less trivially the formula for a begins to look like the one described earlier as the "standard approach" if one Observes that Im Gr(l,k.m) is the density of states for a given k and m.in band Z. In that formulation one has a product of a squared matrix element be- tween initial and final states, the density of states at the initial state, the density of states at the final state, and the thermal probability that the initial state is occupied and the final state is unoccupied. Then one sums this quantity over 24 all initial and final states€o is the density of states): 3 «a.» “Z/dlf; 0/31, 0/5.- 45; H2P(k‘.)EA:) 200v. 5X77, (OW/(0) Conservation of energy and momentum causes the above to re- duce to: and) «fa/’1» 0/5 M‘POQE) P(l;,E-w)(>7,=(£)- Fm] which agrees with our final form of the Kubo Formula after the substitution of p(§,E) for Im Gr(fi,E). It is shown in text- books On Green's functionsl’e’3 that Im Gr is indeed the den» sity of states of the normal modes of an interacting system. IV. The One-Particle Green's Function The main task left is to calculate the one—particle Green's function. It is the sum of all the connected selfe energy diagramsl’S; it is impossible to sum all the diagrams, but it is possible to sum this subset: > + ' ' > + 00...... 25 where the solid lines are electron propagators, dotted lines are phonon propagators, and the vertices are interactions as in the two particle diagrams. To generate this subset one calculates the self energy, 2(kyw) corresponding to this diagram: v and inserts it into the Dyson' equation laminar i/A'Mcrzfl’vww) > where?) o is the free electron propagator Fax—ELF! (k I) + ,1 O ’ ' and Eo(k,£) is the energy Of a free electron in the lth band. The Dyson equation can be written alternatively as fl : Ao+b92b , which can be solved for a given 21 in a perturbation sense by iteration, giving: fi‘rfla‘fflojnod’hoffioz 230+" '3 which explains how our subset is generated by Dyson's equa- tion. According to the Feynman rules 2(1th =<£_%T #3 Quiz. _ . . ,. ’w") qu‘tc‘ 4;. 7723’- ’4””"“’")D “9“") . 26 if we let temperature be low, T. “Mfg/w Emmi)»: cfi; fldw’ W M §(__J___ _ _I__ 2 1 54.342? w’réwo . AAWJn’w) * JP‘Q’I- Achi-M :Zth, gg"integration: The poles of the integrand are at w’: 1'in and ’ '10“th +4.; IP-M" +4231 c—Mu) Zrng The last pole is in upper (lower) half-plane if 2 = c(v), whichever is the case we close the contour to include only one pole as in the diagram. Ar—V ‘Cdo / M. ‘ €164” tifigliklijkiu .— 4 . a ‘Lflwn +LK19-klz’c'a 'A‘UQ$ K 2WW =<3 27 Z ’ " - ((8.1,6A943“ / ‘ 3- "E; , _ 1 ’ ’ 7- ' 777' e (H ¢tM+Awf§§anHflJLC+M IL (upper sign for 2:0, lower for £=v). Introducing spherical coordinates and letting the polar axis lie along the p vector we Obtain, Z=§:3_Jo‘/:clkc(6.oén9 ‘I1T is" .c' 1: wk 1 A- w. “A M... + or}: (193. 13-24% m9) ZJML “ = 6.22.2. as fee a sass-n4aM—t‘cmzxfi/etft’ We #1: to 451m: two-451,cm{tgm)(1o,gjfpk)j erZZfEO’lZFZ u/KEZé!égl/«CJhthEX1obf21cfln=+llfegafigxafiflur ‘7‘17’6- fi' 6’ o I? I 4'. tat/hTde‘AJ‘CICI'M'fiéthf‘k) ~f“2(_/g A A-(fibz/ o ’2 ,4-[7?p.i£fl 28 ’4 = 3% (Atwnlhwo ‘AJ.€,C+M) Since the integrand is Odd, the above integral equals /e(.ék/ A—(‘P+k)/ “fix \8 all‘ \s a h. 7% it 52.5; T ht \ «FIR? + g, M +(7'+1’)/‘ AM Mr-U/ - 4W3 “(W0 . .\/7x+’P vat—40 o . "JA— 4"? _\Jz.n1y (position of poles in complex k-plane) If we close the contour in the upper plane the first and third terms in the above integrand have singularities that must be avoided. For the first term the contour is: 29 n“ < -Q-----.o-- -{/ t.‘ 0 We always aVOid the k = O pole because our derivation Of the interaction Hamiltonian implicitly assumed that the summation > > over k did not include k = G. Since the integrand vanishes on the arc and is analytic inside the contour the first term equals: M =6->c> W M x Afiff‘g‘w + are >06 ’7 ‘Hffl—‘fp o r . mJ Moor/M) +1». ‘2 Mega)“ Fe Wan-70 9" m eéfifi-P where Miisbthekkeyhhole—like part Of the above contour. .c' {oi—77') 30 The last integral has a uniformly continuous integrand so we can exchange the order Of the integration and taking the limit ase—> O. The only part that may give trouble is Meet) 6,0.“68‘9 seams-w €£hé€w V3'7°+€e‘9 \IX-p But .amm 6M: Oso the last integral is zero. The first 6-?0 two integrals add up to 52f 6”?de 2 a save/0 ° Pecfl+fl1o O “Pin/74'"? ranch/h— M‘Ur -2°//,°° warm/M 40/ +11TLzéw/t/ZHD/ A similar calculation shows that the singularity at JA'+ «9 in the third term Of 2 gives ’(‘PVfiJQNZWW +QTI’4'..@.\ NIH?!) ; 31 SO 5 We < was Mei—£1) The last term approaches the log of unity, which is zero, so that 2 (70,1, («t/h) ’ A. C __:__Nom2 fin/(:£¥(Afiwi+twp-A62 (+14)) 31$ at“? 7° (2W(Ltwn+1wo‘ArSIL+M)YZ-I-P Letiel-JoMI- 4.7% , then 21335 ”(1):! 1w“): VJNvfifP “A Skull. ’Z(7;.fi,wn) ZYHL and C (£1w)-t/Xw-L19-AJQC+M-L(1‘1,-iw) 12Yfl4 forwin the upper half complexw—plane. Note that Gr(l'.ldn):)j(wh) as required. 32 V. The Absorption Curve At low temperatures the factor Nf(E4&))-Nf(E) in the Kubo Formula equals unity'ii'ku)is of the order of the band- gap. The Kubo Formula for the absorption (3.4) is now, «(w/ -= eel/Um/z'l/fl‘ d? ész C6"2mw l A-kw-ggv +M +1.:’93h,®n;'§"£=(E-m+twom))z+£ .Ilm in . 2E; n ‘”§-45 "I 2 '{z-‘(E ch‘A'f“ )1+ ( ., )"2~£ ‘(5.1) We have gone to Spherical coordinates and integrated over 9'1 [/15 ‘r‘zléz- A'+M. +41%? 1% the angles. This merely introduced a factor of 4Tr,since the integrand did not depend on the angles. If E>Rw-KWsM the log in the first term in d has the form gm ia+b/ ’ LCI- £> where a,b are real. 33 f &/ 1:25:33] : 2w(a3'+ b1) +LW'a/b- kph-g) 11326-075) -= nib/“"04 . Then the first factor in a( is, 9m | /E hW+4€‘t—:é1—9~?vt,taw|((2m"(E-(-Fw+tu-B+M))J5) 2m, 1% i The 1L6 factoriis there because Gr is the analytic continuation cfifllb irnthe upper half of the complexcu —plane. Using the fact i that I -= P "' +¢TTJ(X) 6+0 X+k5 where P means Cauchy principal value we find the above factor is 116 (E m- 6.. ”1e 4%? MY 3.1%: FwMWo-I-M» va k By a similar analysis we find that the second factor becomes 34 17' JG? ILhz-A+%‘?ck “WK“;- ‘?<4+t_‘_h‘— M+g.:'¢_‘e"ta.&m ( Z—Vflgfiwna-EWU i—WM ‘2 The second term is zero unless W. 21% - ' .5 mm M Ai—fxw >licu+13mv «+11%; tom( Qfiw—F ))) :7R0J-R&Ub £12“ +A-M Zita/«((‘QWAWOT 5W») 2"": Ire ;>'hflkr‘kfiuo‘14. . The third term is zero unless ”>300 +2.15 M44 33%}- tam< zmv< kw‘E‘KaJo 74))? V T 36 >A+1Rw°"M . The above restrictions determine the k limits after the E integration is done, however, it is necessary to simplify the above expressions to determine these limits so we will neglect the terms which are small because they contain gV and go. This causes a small error in the determination of the ranges over which the three terms in the eXpression for the absorption are non—zero. The integral becomes “Malta” 17" @219 Jfli’f- fiA-tw-av-aggfififlngj la 27m, (1,37 tw-A—Xwovz 2mg - 13% &m(h( 2-0.3 (kw "We 51/2) 37 ) Ia (-2_Yl"V( t w 1V1: 010))»: + CUQ A?" x 2%(tw-A-twofi‘a— L? 37 M . ‘ 1 'L xhw—A—Lk +3127... L331 ’0,“ {($1t‘k Mtwo» ( 2m 1m u )L k +iafg‘x L/(zfl‘(kw+52_‘12:—twe—A));:+/e‘ k c Q // (Z-M‘(}§w-A- half) db k$m I 2 ‘L 1'; + W/R‘DM Jaw -A-’a_lg“ 3%: Lnggnfzmv(t%+%:));7+ ( II ) .. 1 m: L L ’°_A qp+k + ;?:;§’aa\(2tt (It, “+27%;‘R )3)“ ’9 38 + Qm I g +A twig? +~ vino,“ (2% «fl +A Fed-+11%» 2+Ie+ 2. ( ,. We I. m +kflgj £)J3- Neglecting the smaller term will then cause unphysical be- havior when the values ci'a)are such that this value of k is in the range of integration of k. This happens at Kw = A and kw: 4“ MLvnm‘ “do ' C II 39 These integrals will then have the form: dx x A/(Aw’) +x I ; -_'_ nah/{mmk‘m % (A+6><‘)"=—— -x 3‘ (mar-fix] (Ck +D)"- (KM-D +8 W! C‘ I W'%é«"-”;( “8 D‘M) M (90 ‘A/B)/ (WA/:3“) I —-A-' _._L_ 7W «an '7» ‘/ ——-— I B 2 ("'13) z (WA/3W ‘ Considering the first term of (5.2) again, we need to know the zero of the function in the argument. To find it approx- imately, we expand the tan.1 terms around ‘2‘: (Whine. “WNW a s which is the zero of the algebraic part of the argument, and take only the first term. With these approximations, which take advantage of the smallness of gV and go, we can evaluate all the integrals and get the absorption explicitly, but first we examine the form of Im Gr to see what the analysis says about "Qlfl' 40 the density of states. Only the delta-function term is non-zero below E=A+ two . If we neglect the tan":L terms, we would have the density of states of the free particle, d( E“ ”Sam 'A 0(1,C+M) and we would have reached the familiar result for the absorption that «~(tUJ‘A)I/a' 3,5 This tells us there are no states below E=A ford/ac and no ‘~—« absorption untilthA, both of which we knew. But the tan-1 terms cause this edge to have a tail i.e. states below the edge, “ which is the fuzzing out we eXpected. As we shall see later, this allows absorption at energies less than fiW—‘T—‘A . At kw:A+T\UJO the other two terms in the expression for the absorption become non-zero. This is the energy at which real phonon emission is possible. Below this energy only virtual phonon states exist so the quasi-particle (dressed electron) is stable; above this energy the quasi—particle can decay, so that its lifetime is finite. In Green's function analysis this finite lifetime is associated with a complex self- energy which is Just what appeared in our analysis, i.e. 2 went from a real to an imaginary quantity at Rwi'A-FRWO. The reason something similar to this does not occur at multiple phonon emission energies, e.g.Fw3A-i-QIWD, is that the dia- grams we summed only contained one phonon intermediate states. Carrying out the integrations one obtains forCX I 41 b<- fit tt/Whfl H773 (kw-A +31% v a 7) Mcmv - Ce‘fi—mw “me3 (1‘ mmfmA))"1 x 7.3..” ((.2M:(W_ (mlyys ’Wifw':):ta:a))y7 (/ t MYRMV (11 ”‘49) I. / +2 3.21”. pm" (PM W MVW‘AFKMW' 1'mf-m _ q, WM:MV W‘A) (1 WE.” AW “ MM“ ) (5-3) when 13w.) < 13+“ (dc As stated this is real for values less than kw=A because the tan"1 factors are positive and real quantities for AuKA. 2two+a>tw>kwo+A, 4“ ,. 2+2 RM ce'fi-WU m W X .= afili/QWWL [ ”fit“ ‘- x A (ZTrgvflw-A-Q'Kwowv +( 1Mv(t\w—A chsfi/ C 1/ )2' ( ” >% (‘2' mm"; ( 1—“ (Kw-A'W°>)* WA) + 1mm ' :mVQJO k mi; WW+ 1 - .. fx/ /m‘m':‘Y‘:’lc 'K 2 M WW 4.31”” A) r, K {*4 Yn'thmV l .4 (5w W'AszJ—ga‘fi.) ('3' Mg WW I... ) 7%;me + ZYYWCU O "Ex—"we 43 '/ 7. firm 2m__:Wo> (" ir-‘Mkwfl-A‘woy .. k1 mC-mv Xtdm‘( (’2 'Fn" <’— 1‘72”]va ('2%(tW’A-RWo)>a + 0i- :LZMKAL(5-m’""vw>1'CWNUWJ“A two»:)? J vabdok ( \ ‘ E‘— ('%"(‘K&)*A"Rwo)))k" vawo + 332% ;__hp_¢mv JAPMWw-Mw41:)+(1—'£—V((tw--A-JW) 1‘1 (W‘c‘mv) (23' -._—(’Rw Wu) )két‘ 131v (kw-whim“ 5’2. mgmv (fink (fiw.A-‘A'wo kw A + 2% (:00 dim -A> (m | W K“ Wig-WW wk -A 1V" M 4-- MJYW t “a )E 7;;VA: aw, lefi WA) i 44 33L, 2 (w A)(?-.£'Zi’g(tw-two—A))"L '7”:va )( 60:3 (gmgmv -".'l.< _l '/ . mi?" "N 5 WMw—A-MT 2%(kw-A—Rwo) k (1%: kw— A-‘waoflk" ( I + W 2;...ng mm» 2’— ML Ltkwktwo) ), H 45 + git (~kiwiw((:=-E (*Wtémwlfi' «(AwAnubfl " )V’< II )2). 3 fl.) mv (2M((‘w.A- Wtwd» +‘KUO-‘A + (’7— m‘w°) | ; {gm-rm '5‘ mw ’WO 1‘); Mflv) chjL-vm | ' - - We ’ 2- Mcmv 1; Mymc (kw A)( If, )4.— / 1. t ‘5 Mt. Mu (.k/‘l. mc— -mV+ ETC/W 2h m::mv+2£’l_ Zsfitwkw Z Yflch+ 2m OJ “Fwd-A ‘ ° (I 4'ng (2m:(1{w-'A'7Cwo» »)1 46 fly “331‘< 11. Wk .__W\_V Afég‘mf (to) 'A'1Rw°))z’(z%(tw‘4'hwo))g F—c—‘MMV II )‘42’-< I! )‘/1,‘ ‘ z mc‘ v 11/?" mmv (zmvf(tw’A“1\WO))+Tw-A "Fly 1—6{ ‘10 o+A ‘WA’Q I ' 4 X 1%.“W‘c’W‘v (IMVG‘WJA Kw) '5, —MJ .6“ A) ”my; l 6' 7"ch )2) vac ) at W‘ g-Mu 1M w- I MC‘MV MV ‘ . ‘ “WAX?" dtw +4 m» x( 4 ”4ng __ a 4 4; RW‘A (ZMV (tWo‘I’A km) If M W? '2 2m____v f “Wfiwo A 'RLO) p (' “(’W ") L(tw-A-xw.)))( 2%(1awmwg 47 I -%\;_l W‘ A (2%c(tw,A-RWO))¢ ( {mflmtwy tw))( (2 mv(0+7§w°“w))€ (H flvxg’/ 1% “(W‘A'wamr Zjfigég¥tuh’1&0) - ‘+ ( Sla) (5-4) VI. A Two—Phonon Diagram Correction Our previous results for the inter-band absorption have been derived using an approximation in which processes having more than one phonon "dressing" the electron at one time are not included. These processes should affect the self-energy of the polaron and show structure at multiples of phonon energies above the band-gdge because of the possibility of real multiple phonon emission at those energies. To investigate the way multiple phonon processes affect the absorption calculation and results we will calculate the Green's function for a two- phonon process as an example. The process correSponds to the following diagram: According to the Feynman rules, the self-energy corresponding to this diagram is 25:: gilt3’ fJAO’L?C!:fK¢Xhh£:k01. CU» (200124@;¢qv+ +QJO‘.TU”4qu - (2") mm» 72/ (zm‘fi‘ 4. JL.<-+M) / 2W X C(wo+L(W/’W3-) 4— wo-ri-(wrwz Lth‘ mt—AJQIC'fM 2-“ The poles in the complex baa-plane are at flak-fa), and _ _l_ I ' 1. 1.. . " 1:“th ‘AmV-A J“) The last pole is in the lower half plane for i=0 and the upper half plane for £=v. If we close the contour in the upper half plane for £=c and in the lower for £=v, the result of the integration is: 3 3 —-—'——-— + --’-—-' 1, Z N dpdfa. dW, (W0+L(W‘Uq) WO‘MW‘M) 27,200 lr’F/‘L /f f/ 16"“)! UL 1M C+M)(x ‘02,”. + x0)!" EFL“. 2MC X (43%, $£w.-¢fl£’—AJL<+M) 2mL 49 (upper sign is for i=0, lower for £=v). The poles in thegh - o __ . o 1 '1. ' plane are at (M: (Hf-4WD 'L' (4- okwo‘~k_Pa—«Ac5£.<+lu) ’ tfi ‘2vnA , . 1. 'L . HEC— 4* fi/zmc-LA 51,C44.M) . The last two poles are in the lower half (~a-plane if £=c, and the upper for E=v. Once again we close the contour in the upper half-plane for E=c, and the lower half-plane for £=v. The result of the Q5 integration is: 2 ~ W {@3123} 170—- mem‘ (ztw:two-§§A;2, > to lie along thegpfdirection for the pgintegration and along > p for the pl integration we see that the above becomes! 2 ~ am; 791722,? dflm <9, M etc/9’ 49... (fitiarfimeom‘whmw 9») X x ( 4'. t w 3. two— CF" - A Jz,cm)“(£tw:2%u2yflat-£9 Q’Wj ZVVLL :znmL 50 4mm— amp/ya. 70,, 44V Mlflaffimelf (m); Fulo‘fliA 51min“); zml X A/fiz}?~1'273flmezl 7T! (4} tw I 2 two‘filfl—IIAJ 12 ,c—l-M) o 2W The p2 integration has the form “1"X)62)? A 2;? f ail (Zn [AMI-AM -x/)(C« /P+x/ '34 [fix/1 (XL’ ’34 )1. P 28‘ .= .1. [4X(“MA/Mo(Mm—M794!) P x (.1... + I .- __?-_... ) (”J-$6); (KW-54% (X‘- B4:) 53 ‘ffijtfiflvl A-flg/ Am w; W r iii) “ti—‘5: + ficflwflfial’fiJP-‘fico -(AIA+¢5¢I‘&IA'\/T3e/)wn/1‘+félflaw—Jib] W3: . Since the integral along the semi-circular part of the contour is zero, the only difference between the integral over this contour and the integral from-go to go is the key-hole-like part that avoids the branch out of the log function, so we must subtract its contribution. 54 e j (1% P+ ' 9° ‘( ’ 77' J))(AH¢+A+P5€M-J) -f («5" PHI 6 (WW/Ma /70+A+P 66(8-11‘7' _ M'Io 4:(JWI'T’) /_L. W19 ___.L__ (239' 1((Alfeiw-Q Mg): +(f / A+F€‘(JI1T)B [2) CM?“ 6 ’9} 5/) 55 m: , . f Alé eaeda (“6+Ltgfl’al7g4’q4'éfitfimbf J—vr 3&4 7“A-e 5‘19”, The last term goes to zero asé"0 ,L and the first two add up to: 3'” 2p _ w ‘ .0 {abfjn/r-M PI [79 A+ ”((104 Ffir. 4 ' "' —‘—— - .1... A‘F-‘LJECY— A‘P’Vg’c A'lo+\/§/¢)\/;) 5.91723 )"W p]- gfl/flMP’J-K m A W34. “Far-Q), “+9 P(431+7-P+7:1;;p) 56 (ii-(3‘54. +A*‘P+\/'5/c) - WBJOZAWM-P/ _ _ I W‘M‘Mfia MW) : x: A A J— +—-L- AM“ gill [$34K A-xfiée A+\/§/¢)+ all/‘5’”) WM + filAifé‘) +__«__£"/%/ +fl/fia/ m t“ m” We —JEé/4P(Wfl-P/ wenzz-A‘P/ 6 A-P- J34 A+V5JF $410+»?! -%4/40-A+Pl)] . A-rffié- P A'CBE'P 57 The last four integrals look deceptively simple, however, there appears to be no anti-derivative involving only ele- mentary functions for this type of integral. What we can do is to expand ln(p+x) through the linear term in p; then we can integrate, but our answer is good only for small p '(the polaron momentum). The integralsbbeommei 3.11 .. o. d9 I ((A’HCA'V'é/c‘”) (AfiANE-fl) =940(a€w/A- -)P Ian—[A Vac: -PI+IMM L/MAH/E Pl) H: \/—>2_ Vfiit. = 91(AA’CM/A-VE/chfiw4fln/Mfad) J3: So the total contribution from the key-hole part of the contour is: I, “g“ gnu 47:3,: m new sac) #2 gm 6" 79.. 3,6 +%G%,4AIA«IA‘=B/cl] ‘ Them 5%; cmCw:+x/—/:4/A-xl)(Cgl’/a+xlwen/7m!) w (“EB/cf. =93" L/ A+__/_3é 8%... [A W 111—52 fifljggc w Aflefi - )A'V'Br 30 fix: PA/E’fi [(754 J 2m¢ A" (— kffiiw IIIND‘A6£)Q+M))P 59 I3: (ttwo“tw+A 6.0,( 4M) ’C'fi Z): . ZMJ Consider the term ~ MA+_JE/C\_1M(:§’“(tw:ntwo-A5e,¢+u))i+ A‘fir— ( II )%—+ +(M 3(4-Mo‘tx (41+ A 5% C+M))/1’/ "( I/ )4, 9 we have a situation here similar to that which we found in the study of the one-phonon diagrams when kWZA-k-‘ROOO'M i.e. the log function causes 2 to become complex above and real below that value.-This indicates we have sucessfully taken account of the possibility of double phonon emission by the inclusion of our two-phonon diagram. 60 VII. Conclusion What do the previously described calculations signify? Let us review the principal assumptions which went into the calculations we have described and the results to which they led. It was assumed that the only interaction taking place in the system was between electrons and longitudinal Optical- phonons and that the interaction was weak enough so that a perturbation calculation was possible. This means we neglected the effects of impurities, otherttypeSprophonpnsnoand.fihe coloumb interaction between electrons. It was found that the absorption curve near the absorp- tion edge has a parabolic component (kw-AV; (as is the case in a simple independent electron band model) with some complicated additive corrections which have the following characteristics. Below the band edge we find a tail region of absorption,;equation (5.3), like an Urbach taill6. At energies of one and two phonons above the edge we find there should be an onset of structure, equations (5.4) and (6.1), such that the absorption curve has discontinuous derivatives correSponding to absorption due to creation of phonons. A calculation of the size of the effect gives a magnitude in 61 the neighborhood of .01 inverse centimeters for a material like InSb at an energy of A+gtw°. This is near the limits of resolution, although at that energy the parabolic part has a magnitude of 2 inverse centimeters. The physical arguments presented make these features appear plausible, and the set of one phonon diagrams should give us a very accurate descrip- tion of the structure betweenwt‘uoand A+1fiw°since we have included every process that could give an imaginary Z in this range of energies. There are reports in the literature of experimental findings that support our results. Structure has been found at longitudinal Optical phonon energy intervals above the 17’18 and radiative band edge energy in photoconductivity recombinationlgemeasuremeats. Such structure has also been 2 reported in actual absorption spectra by Ascarelli O and Larsen and Johnson2l. The former found it in AgBr and the latter in InSb. 62 Figure One The Band-Scheme 50(1). 10. 11. 12. 63 References A.A.Abrikosov, L.P.Gorkov,aand L.E.Dzyaloshinski, Methods 'Ea11,1963) L.Kadanoff and G.Baym, Quantum Statistical Mechanics (Benjamin, New York, 1962) V.L.Bonch-Bruevich and S.V.Tyablikov, The_green1§_fiungtion M§§h9§ 1n_SEatist}0al_Physics (North Holland Pub. 00., Amsterdam,'19627 ----------- D.N.Zubarev, Soviet Physics USpekhi, 5,320 (1960) T.D.Schultz, Quantum Field Theory and the Many—body Problem (McGraw-Hill, New York, 1964) H.Frohlich, in Polaron§_and_§xcitons edited byC.G.Kuper and G.D.Whitfield (Oliver and-Boyd, Edinburgh, 1963) F.StVrn, in Solid State Physics edited by F.Seitz and D. 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Let. 16 655 (1966) 11111111711711 1!] A H E" u "M" 3 9 2 1 3 Wm»