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"Li-I." ..,. :. :0: . ~_‘--- ..‘ J - I . , 3 ‘r ".3 ' I 3 ... ‘ .u .. p I o... 53;: '."' m &‘~..',-I".f~t.‘...'-- ...~, lbb-o.. a... ”DIE-.g ‘ 3-2.“ .....ov.0 - I H I . “' 0 ~ ' " I. 0 . 0 ' 0 0: O ‘V - .’ ’ O . ‘-‘OUOI ’W": "' .' - X XML) X><§<§> Fig. II.4 The staggering between two basal oxygen planes shifted relatively by b/3. The three possible gallery cation sites (m1, m , m3) 2 are shown. Open circles occupy a rectangular lattice. The relative positions between basal oxygens and interlayer water molecules at site m1 are also shown. An intercalant sits on the top of three water molecules and is covered by the other three water molecules(not shown). 18 There are three different gallery sites for exchangeable cations for b/3 staggering (illustrated in Figure 11.4). It has been found that cations such as M92+ and Ni2+ are likely to be retained in the tetrahedral cavity sites (m1) (see figure II.4, relative positions between water molecules and basal oxygen plane are also shown.). And for Na+ and Ca2+, the cation positions can be all three sites(m1,m and 2 m More detailed discussion about this topic will be given in Chapter 3" III. Another important aspect is that in the vermiculite with charge density of 2e- per unit cell, the lattice structure formed by divalent intercalants is rectangular as shown in fig. 11.4. In reality, the lattice arrangement of divalent cations can be thought of as a rectangular network with site occupancy disorder. Fluorohectorite (FHT) is also a trioctahedral 2:1 layered silicate. Its structure is basically the same as vermiculite (Figure 11.1). There are three differences between them. First, in FHT the ion substitution happens in the octahedral sheet instead of in the tetrahedral sheets. Secondly, FHT has a charge density of about 1.6e- per unit cell formula which is less than in vermiculite. Thirdly, the oxygens in the common plane between a tetrahedral sheet and an octahedral sheet and not shared by both sheets are F- instead of (08)-. The interlayer cations can also be replaced by other intercalants as in vermiculite. Chapter III The dc magnetic susceptibility of vermiculite intercalation compounds with 3d metal ions as intercalants III.1 Introduction The magnetic properties of lower dimensional systems including layered metallic superlattices and layered magnetic insulators are particularly interesting relative to the behavior of the bulk hosts. An example can be LaZCuO4 which has been attracting considerable attention recently because of its relation to high-Tc materials. Layered silicates such as vermiculite provide natural layered structures on which one can carry out the same kinds of studies as well. The advantages of the latter have been stated in the overview (Chapter I). Although there are recent studies of the magnetic properties of vermiculite (ver) with several 3d metal ions and 4f metal ions as intercalants3° (namely Mn2+, C02+, Ni2+, Cu2+, Dy3+, and Er3+) cointercalated with water, there are still significant questions remaining. First, Ni2+-ver shows a Curie-Weiss behavior in the high temperature region and the Curie-Weiss temperature is positive ( ~ 10 K) indicating that the intraplanar exchange interaction between Ni2+ ions is probably ferromagnetic. But the magnetic susceptibility does not diverge for temperatures down to 4.3x indicating there is no phase transition corresponding to ferromagnetic ordering for temperatures from 4.3K up to 300K. Secondly, other compounds such as Coz+-ver, Mn2+-ver and Cu2+-ver exhibit negative Curie-Weiss temperatures suggesting an antiferromagnetic interaction between these cations. But again previous 19 20 authorsao have failed to observe any phase transition corresponding to antiferromagnetic ordering. In other words, all these systems behave paramagnetically at temperatures near 4.3K. In order to clarify the behavior of these interesting compounds we have conducted more careful studies on the magnetic susceptibility for samples with higher intercalant concentration and under different hydration conditions. Specific compounds studied here are Mn2+-ver, CoZ+-ver, Ni2+-ver and Cu2+-ver in which the magnetic intercalants are all divalent and their spins are 5/2, 3/2, 1 and 1/2, respectively. As one should expect, there are two major factors which influence the magnetic properties dramatically. First is the distance between two nearest neighbor intercalants which depends on the cation concentration. Due to the fact that the magnetic exchange interaction will decay exponentially as the distance between intercalants increases, we should have a cation concentration as high as possible. Thus, we have developed a new method to synthesize powder samples with higher intercalant concentration than previously obtained. The second factor is the presence of water molecules in the gallery spaces which will affect the magnetic properties dramatically as well. Therefore, we have conducted more careful studies of the dc magnetic susceptibility under three different conditions: hydrated, dehydrated and rehydrated. III.2 Experiments III.2.1 Sample preparation 21 The original sample starting material for our studies is trioctahedral Mgz+-vermiculite (Llano, Texas) which has been ground into very fine powders with particle size of a few micrometers. In order to purify the gallery, the exchange sites are saturated with Mgz+ ions to replace other impurities ions such as Ca2+ or Na1+ in the gallery. Chemical analysis by using Induced Coupled Plasma (ICP) method shows the following unit cell stoichiometry (315.6A12.4)("95.7A10.24Feo.oasTlo.03’020(°H’4°Mgo.99°(320’7.6 which is abbreviated as x'Mu°(H20)u, where it stands for the host layer matrix and M stands for the exchangeable metal ions in the gallery. Here Si4+ and A13+ ions are randomly located at the centers of the tetrahedra. There are some Al3+,Fe2+ and Ti2+ ions in the octahedral layer as impurities which can not be replaced. To ensure complete exchange of gallery M92+ ions with the desired transition metal ions we have carried out a two-step ion exchange procedure. In the first step, all the M92+ ions in the gallery have + been exchanged with alkyl-armnonium ions such as (CH NH ions, by the 3’3 reaction of the powdered parent compound with excess [(CH NH]C1. The 3’3 next procedure is to replace (CH NH+ ions by the desired magnetic 3’3 ions. This is done by putting intermediate product into suspensions with the desired ions. These two ion-exchange procedures are complete and can be confirmed using their x-ray diffraction patterns (Figure III.1). Due to the different sizes of MgZ+, (CH3)3NH1+ and 3d metal ions, the c-axis basal spacings for these compounds are quite different 4- + (the basal spacings are 14.5A, 12.8A and 14.5A for M92 -ver, (CH3)3NH1 - + Fig. III.1 The x-ray diffraction pattern for M92 -ver, (CH and Co2 eye. 22 I fl€”-Vur : !’ !' .F : - . 1 . 1,1 A J L - - . l . A! (suggest-Vu- :3: a? 5! E‘U .9!- sold: 3: . 2 . 1 Alli..L.l [.11 v_ za 5 0—0 V, C) H 3?.” 29 (degrees) NH+- 3)3 ver, +-ver at ambient conditions. The vertical line is the guide to These pattern were obtained at room temperature using Cu(Ka) radiation. 23 ver and M-ver, respectively. Here M stands for the desired magnetic ions). By monitoring the c-axis basal spacing from x-ray diffraction spectrum we are able to see the completion of the ion exchange reactions. The purity of the final compounds has been further confirmed by infra-red spectroscopy and by chemical analyses such as ICP. Four'ctifferent vermiculite intercalation compounds have been synthesized with different cations such as Mn2+, 002+, Ni2+ and Cu2+ ions. At ambient conditions, these compounds contain two layerslof water in the gallery spaces surrounding the cations. By heating these compounds to 600 C one can remove water completely.31 At that temperature, the host layers undergo dehydroxylation and this process is irreversible. It has been found from thermal gravimetric analysis (TGA) that most of water is removed at temperatures above 150 C (Figure III.2). Therefore, we have annealed annealed our samples at between 170C and 180 C in vacuum for more than 12 hours. Figure III.3 shows the x-ray diffraction spectrum of the hydrated sample at ambient condition and that of a annealed sample. The basal spacing has decreased from 14.5 A to 10.5 A. All of our powder samples have been pressed into pellets for easy handling. III.2.2 DC magnetic susceptibility measurement and techniques for data analysis The DC magnetic susceptibility ( X(T) ) of these magnetic intercalation compounds has been measured from temperature of 2 K to 300 K by using a Superconducting Quantum Interference Device (SQUID) magnetometer from Quantum Design. The applied magnetic field H was 100 24 73 a D E .0 3. 4-3 .1: U '8 '. i: -- o 100 200: 300 400 500 600 T (C) + + Fig. III.2 The weight loss of Co2 -ver and M92 -ver versus temperature measured by the TGA method 25 t + ( Niz+-Ver )— i '1 r 'I A > r. - 1| ‘ g i I ‘ \ b. | ’1 \ g; I ’\__.e ,______§_ .fi ,. —¢- .3 ' - - -iliri - i - >s : p 4‘ .. 3 : ll Co‘*-Ver Q s +3 h .3 r- E E _ ’rfl. . .rrl-..r -- . o 1 3 4 2 q(1/A) Fig. III.3 The x-ray diffraction patterns for C02+-ver and Ni2+-ver before annealing (solid lines) and after annealing (dashed lines). The c-axis basal spacing are about 14.5 A and 10.5 A for both compounds before and after annealing, respectively. The source radiation is Cu K O a 26 Gauss. And the error bar in the susceptibility measurement is extremely small ( about 1/1000 of the magnitude measured ). Since there are some residual magnetic impurities such as Fe2+ ions in the octahedral layers and they contribute to the total magnetic moment, we have first measured the susceptibility of the parent compound Mg2+-ver to find out the contribution from these impurities. Figure III.4 shows the susceptibility X of MgZ+-ver versus temperature. As one expects, this system behaves paramagnetically indicating that the impurities from the host layers are isolated and do not interact with each other. By fitting the data with the Curie formula32 X = x +-jj1—- (111.1) where X0 is the the contribution from the full shell electrons and C9 is the Curie constant. We have found X0 to be -o.91x1o'7 emu/gram which is obviously the diamagnetic contribution from the full shell electrons. The next term is from the magnetic impurity from the host layers. The Curie constant Cg of MgZ+-ver has been found to be about 1.331(10-4 emu K /gram. The Curie constant is related to other parameters as follows32 - (111.2) where NA is Avogadro's constant, M the molar mass of the unit-cell stoichiometry X.M(H20)x, ”B is the Bohr magneton, k8 is the Boltzmann constant, and Pe is the effective magnetic moment defined as32 ff 27 2000000 1500000 1000000 x"1 (emu/s) 500000 Fig. III.4 The inverse susceptibility x-1 of hydrated MgZ+-ver versus temperature. The open circles are the data points and the solid line is the least square fit to the Curie law. 28 Peff=g(JLS)/m (111.3). From Cg obtained above and the concentration of Fe2+ ions in.MgZ+-ver ( obtained from the unit cell formula ) , we have obtained the effective magnetic moment to be 5.21113. This is close to the value found in the literature for Fe2+ ions32 ( 5.4 us). Since the impurities are very dilute and located in the octahedral layer, we ignore any interaction between them and the magnetic cations in the gallery and subtract out the paramagnetic signal from them by using the parameters measured frtml MgZ+-ver. We have measured the dc susceptibility for Mn2+-ver, C02+-ver, Ni2+-ver, and Cu2+-ver under three different conditions: hydrated, dehydrated, and rehydrated. The dehydration procedure has been carried out using the methods described earlier. The dehydrated samples were transferred and sealed in polyethylene bags to prevent hydration. After the measurement had been done for dehydrated samples, they were exposed to air for more than 6 hours in order to regain water into the intralamellar gallery spaces. Their masses have been measured in different hydration stages to monitor how much water has been removed and reabsorbed. The experimental data have been analyzed by least square fitting the higher temperature portions of the susceptibility curve to the Curie-Weiss formula (also see ref.32, p712) 'r _ 9 (111.4). 29 Therefore, the Curie constants C9 and Curie-Weiss temperatures 9 have been obtained for each case. The relations in equations (III.2) and (III.3) are maintained. Thus the effective magnetic moments Peff and the Lande 9 factors for different ions under different conditions have been also calculated as for Fe2+ ions in the Mgz+-ver. III.3 Results of the DC magnetic susceptibility of vermiculite with magnetic intercalants III.3.1 Mn2+-ver The Mn2+ ion has electronic structure 3d5 and spin 5/2. Five d electrons occupy all five d orbitals and the total spin moment is 5/2. Figure III.5 shows the inverse of the magnetic susceptibility, x, per gram sample versus temperature for a pellet sample under three conditions: hydrated, dehydrated; and rehydrated. The sample lost about 16% of the initial weight after annealing and regained all of the weight back when it has been rehydrated. The impurity signal has been subtracted by using the parameters obtained from Mgz+-ver. As one can see, the inverse of X for the three different hydration linearly depend on the temperature. This demonstrates that the system is paramagnetic for temperatures down to 2K ( which is the lowest temperature limit of the SQUID we use here) regardless of whether water molecules are present in the gallery or not. By least square fitting to the high temperature portion of x, We have obtained C9 and 9 for various conditions and in addition we have obtained the effective magnetic moments, P eff’ and 9 factors from Cg. 30 15000 1 F 12500 t Mn“—Ver C X Hydrated x 10000 r- 0 Dehydrated 9 o i a Rehydrated x ° ’53 r y 7500 - 9 ° 8 I . 3 9 T . o 5‘ 5000 b 5! 7‘ . a . es » 9 . 9. r 9 2500 ~ “g 1 “9° o._._..JL11111.-.1_L_1_41111.L o 10 so so - + Fig. III.5 The inverse susceptibility X 1(T) of Mn2 -ver under different hydration conditions. 31 X0 Cg 9c at low Tc (emu/g) (emuK/g) (K) temp. (K) Hydrated 1.59x10'5 4.40x10'3 0.08 Parama . None Dehydrated 1.86x10'5 5.2311103 -0.80 Parama .. None Rehydrated 1.6lx10'5 4.2511104 -0.10 Paramah None (Continued) Peff (uB) Lvalue Hydrated 5.81 1.95 Dehydrated 5.87 1.98 Rehydrated 5.71 1.93 Table III.1 Parameters for Mn2+-Ver from a least square fit to the Curie-Weiss formula. 32 These physical quantities are summarized in Table III.1. As one can see, the Curie temperature 9 is close to zero and is scattered for the hydrated compounds. The most important feature of table III.2 is that the effective magnetic moments for different hydration conditions are essentially the same. This implies that the magnetic properties of a 2+ Mn ion in vermiculite are apparently not affected by the change of its hydration environment. III.3.2 Coz+-ver C02+ ions have the electronic structure of 3d7and spin 3/2. The DC susceptibility X for different hydration states of this vermiculite is shown in figure III.6. So is the inverse of the susceptibility x-l. We can see a Curie-Weiss behavior for the higher temperature portions for all cases. In lower temperature region, both hydrated and rehydrated compounds are paramagnetic in agreement with other authors’ results for hydrated samples3o. In the absence of H20 molecules, the susceptibility X shows a peak at around 5K while X decreases with temperature. By fitting the higher temperature portions of )((T), we have obtained C9 and 9 for the different hydration conditions and calculated the effective magnetic moment Fe and 9 factors. They are listed in Table III.2. The Curie- ff Weiss 9 values are negative indicating that the nearest neighbor interaction is antiferromagentic. Another feature one should notice is that the Curie temperature 9 clearly depends on the hydration conditions of the sample. When water is present in the gallery, 9 shifts from about -7K to about -2K indicating that the interaction is weakened. The 33 EH TE Co“—Ver 'x {5! .R D Hydrated is 3’ r 0 Dehydrated 5 T g i a: X Rehydrated x i 1.43% .- go i 3 II a e n .1 [#1 111111141 1.1.L : x . x D r x a . n O . ii 0 e ' . ° \ ’ s ° 2: r ° 0 i" 9 0 V R T : Egoo X r 90° :- o p ’41 l 1 11. -AL-L 11 11 0 10 20 30 40 T (K) Fig. III.6 The dc magnetic susceptibility X(T) and its inverse x-1(T) of C02+-ver under different hydration conditions. 34 X0 C g 9c at low TC (emu/g) (emuK/g) (K) temp. (K) H drated 7.1351110“6 2.38x10'3 -l.98 Paramag. None Dehydrated 3.424): 10'6 3.4911 10'3 -7.07 Antiferro. ~5K Rehydrated 5 .7 22x 10'6 2.37 x 10'3 -l.43 Paramag. None (Continued) Peff (HB) g value Hydrated 4.28 2.21 Dehydrated 4.80 2.48 Rehydrated 4.28 2.21 Table III.2 Parameters for C02+-Ver obtained from a least square fit to the Curie-Weiss formula. 35 effective moment increases from 4.28 "B to 4.811B as water is removed 4. from the gallery. Peff for Co2 in the dry sample is consistent with the literature value (see ref.32, p658). 111.3.3 Ni2+-ver A Ni2+ ion has 8 electrons in the 3d shell and has spin 1. The susceptibility X for a hydrated Ni2+-ver is shown in Figure III.7. It also shows a Curie-Weiss behavior for higher temperatures for different hydration conditions. Figure III.7 also shows the lower temperature portions of the inverse susceptibility x-l. The curves corresponding to hydrated and rehydrated samples behave in the same way as temperature is reduced to 2K in agreement with previous work30, indicating that hydrated/rehydrated compounds behave paramagnetically. The inverse of the susceptibility X-1 for dehydrated case levels off for T < 4K. Although the signal fluctuates, a dramatic difference between the dehydrated case and hydrated/rehydrated cases still can be seen. We have fit the higher temperature portions of x for all cases and all parameters are summarized in table III.3. The Curie temperature 9 is negative for the dehydrated compound which indicates antiferromagnetic exchange interaction between N12+ ions. And it becomes positive as water is admitted in the gallery. This phenomenon demonstrates a dramatic change of the exchange interaction from antiferromagnetic to ferromagnetic. The effective moment also increases when water is taken out of the gallery. 36 i z: ' Ni"-ver r%fl D Hydrated ) X A 1 ‘3: Q . '3‘ ° Dehydrated '3 r u g . 3°00” 8 X Rehydrated X i a . .. E 3 o 5 3 . as g a A 4 A l L A A l . A_L l i g- x F i n L x ' n I x ° 3 L 9 o a r B 9 ° :9 F 9 ° I R 0 >< r 9 2 ° . y I ’ l A l . .1 l J a A 4 O 10 20 30 40 T (K) Fig. III.7 The DC magnetic susceptibility )((T) and its inverse X-1(T) of Ni2+-ver under different hydration conditions. 37 X0 C 8 et at low Tc (emu/ g) (e mu K/ g) (K) Temp. (K) Hydrated -6.26x10'7 1.22x10'3 4.13 Paramag. None Dehydrated -5.20x 10' 7 1.5911 10'3 -0.66 Antiferro. ~ 3K Rehydrated 1.0471 10' 7 l . 19x 1 0'3 1.66 Paramfiag. None (Continued) Peff (NB) & value Hydrated 3.07 2.17 Dehydrated 3.24 2.29 Rehydrated 3.03 2.14 Table III.3 Parameters for Ni2+-Ver obtained from a least square fit to the Curie-Weiss formula. 38 III.3.4 Cu2+-ver A Cu2+ ion has the electronic structure of 3d9 and spin l/2. Since the spin is small, one should expect so-called quantum fluctuation which may quench any magnetic phase transition caused by relatively weak interactions. The dehydrated sample has lost about 16% of its initial weight and regained back this weight after it was exposed in air. Figure III.8 shows the susceptibility X versus temperature for Cu2+-ver under different hydration conditions. All three curves follow a Curie- Weiss law for temperatures down to 2K regardless of wether water is present or not. We have fit the higher temperature portions of X for different hydration cases and the results are tabulated in table III.4. As can be seen, the effective moment increases from 2.01.1B to about 2.1511B when water is removed from the gallery. The Curie temperature is negative for all cases indicating an antiferromagnetic interaction between cations. III.4 Remarks In this chapter, we have present the DC magnetic susceptibility data of hydrated and dehydrated vermiculites with four different transition metal ions Mn2+, 002+, Ni2+, and Cu2+ as intercalants. Data have been analyzed by using Curie-Weiss formula. Therefore, the Curie- Weiss temperature 9 and Curie constants Cg have been obtained from the least square fitting. Furthermore, the average 9 factors have been extracted from the Curie constants. 39 0.00020 ~x Ln ° +3 (5 Cu -Ver ,9 g D Hydrated is 0'00015 g x Dehydrated ' s A "1'? ° Rehydrated «0 'é \ i3 '5 X 0 0.00010 - V , <3 x .9 ‘ 0 . ‘31:. 0.00005 # .‘n. x + 6° 5 U 0 o X g X g ' § 0 0 a L 0.00000 “‘*L4='AL4e-ALJ...11..-. 0 20 40 00 80 100 1‘00 Fig. III.8 The DC magnetic susceptibility X(T) of Cu2+-ver under different hydration conditions. 4O X0 C a 0c at low Tc (emu/ g) (emu K/ g) (K) Temp. (K) Hydrated 2.18x10'5 5.22x10'4 -l.16 Paramag. None Dehydrated 2.3911105 6.9711104 -1.16 Paramag.. None Rehydrated 2.2011 10' 5 4 .97 x 10" 4 -0.77 Paramag. None I (Continued) Pefl' (RB) g factor Hydrated I 2.01 2.32 Dehydrated 2.15 2.48 Rehydrated 1.96 2.26 Table III.4 Parameters for Cu2+-Ver obtained from a least square fit to the Curie-Weiss formula. Chapter IV Crystal field, exchange interaction, and phase transition in vermiculite intercalation compounds with 3d metal ions as intercalants IV.1 Introduction In chapter III, we have presented the experimental data of the + 2 , Ni2+, and Cu2+ as vermiculite intercalation compounds with Mn2+, Co intercalants. As mentioned in the Overview, the paramagnetic properties of a single ion are manifested in the high temperature portion of the susceptibility. An important quantity is the Lands 9 factor which is related to the electronic structure of the ion and the crystal field environment. Although the paramagnetic properties of various ions in different crystal fields have been studied by using crystal field theory, there are no such detailed studies on, our specific CIC compounds. Therefore we present in this chapter a very detailed investigation of the paramagnetic behaviors of the four ions in hydrated and dehydrated vermiculite. Furthermore, these four compounds exhibit different behaviors when temperature is very low. Thus a discussion on the magnetic exchange interaction and phase transition in these CIC compounds will be also given to understand our experimental results. IV.2 Lande 9 factors and anisotropic Hamiltonian From the effective magnetic moments, we have calculated the Lande g factors for various ions under different conditions by using equation 41 42 (III.3). Because our samples are powders, we can only obtain average 9 values. They are summarized in table IV.1. As one can see, the g factor for Mn2+ ions is very close to 2 for all three cases. This is consistent with the results of many other Mn2+ complexes which have been investigated previously.33 The reason is that a Mn2+ ion has electronic structure of d5. The ground state of a free Mn2+ ion is 68 which has no orbital momentum. In the solid, the ion behaves like a single s electron and exhibits no splitting regardless of the symmetry of the local crystal field. The g factor is always 2 for an orbital singlet since the expectation values for L vanishes. Another result is that this system is isotropic because the moment derives from the spin only. Mn2+ ions are considered to form a classical Heisenberg system since they have large spin moments (s=5/2) and interact with each other isotropically. The 9 factor for 002+ ions varies significantly between hydrated/rehydrated states and dehydrated state. When water is present, the 9 factor is about 10% greater than 2. This indicates that in the ground state of a Co2+ ion, the orbital angular moment is quenched but the spin-orbit coupling still contributes to the magnetic moment by mixing in higher excited states. When water is taken out of the gallery, there is about an additional 13.0% increase in the 9 factor. There are two factors which will affect g: the symmetry and the strength of the crystal field. A change of Symmetry will result in different energy spectrum. A change of the crystal field strength can result in less or more mixing of energy levels through L-S coupling. For Ni2+ion, the 9 factor is about 7% higher than 2 when water is present in the gallery indicating the effect of spin-orbit coupling as 43 Mn2+ C02 + Niz + Cu2 + Hydrated 1.95 2.21 2.17 2.32 Dehydrated 1.98 2.48 2.29 2.48 Rehydrated 1.93 2.21 2.14 2.26 Table IV.1 The Lande g factor calculated from the effective moments of different ions under different hydration conditions. 44 well. When water is removed, the 9 factor increases by an additional 6% to a value of 2.29. This result is close to what has been observed in other Ni2+ complexes such as Ni2 2.25.34 + (H20)6 which normally has 9 value of For Cu2+ ions, the 9 factor value is 2.2810.3 when water is present and increases by about 8.7% when water is removed. The study of Cu2+ in many complexes has been reported35 and the g values are more or less the same as what we have observed here. As we can see, the 9 factors of these magnetic intercalants in dehydrated CIC are always larger than that of hydrated/rehydrated compounds (except Mn2+). This can be caused by either the change of crystal field strength or the symmetry as mentioned. To address the theoretical origin of the behavior of the 9 factors for these cations in the gallery is quite difficult for following reasons. First the localized wave functions of the electrons in a solid, which are the spatial Fourier transform of the Bloch's waves, are the Wannier functions which may not be the same as the atomic wave functions. They are atomic functions heavily perturbed by the crystal environment and are difficult to obtain (one conunon numerical method is the Hartree-Folk self-consistent method, see ref.32, p344). To first order, one can approximate these functions by atomic wave functions and apply crystal field theory to obtain the energy spectrum and other properties by using symmetry argumentsae. This method is quite successful to obtain the energy spectrum and explain the paramagnetic behaviors of various magnetic compounds with localized spins. Therefore, we will employ this method to analyze our data here. 45 Secondly, the cation locations and their local symmetry are essential to their magnetic properties and need to be understood. In this particular clay silicate, there are two situations to be addressed: with and without water in the gallery. As mentioned before, vermiculite has a stable phase with two layers of water at ambient conditions (normally referred as the 14.55 phase). Many detailed investigations have been carried out to find out the arrangement of these interlayer water molecules and their relation to the host layers and the intercalants. The first postulation about the interlayer water arrangement in vermiculite was made by Hendricks and Jefferson” (1938) in which water molecules were thought to be in planar hexagonal rings with the oxygen of each water being tetrahedrally bonded to four neighboring oxygens through hydrogen atoms. No provisions for cation sites were made at that time. Later studies of the x-ray diffraction spectrum by Mathieson and Walker (1954):”, Grudemo (1954)39, and Mathieson (1958)40 have demonstrated that the water molecules and exchangeable cations occupy definite sites in the interlayer spaces. In 1966, Shirozu and Bailey"1 studied the interlayer water molecules in a Mgz+-ver single crystal from Llano (Texas) and confirmed the conclusions by Mathieson and Walker except for the delineation of different layer stacking sequences. Recently, de la Calle et.al. showed consistent results of the interlayer water arrangement and stacking sequences of a Mgz+-ver from Santa-Olalla, Spain.4'2 They have found the random b/3 shift between adjacent host layers in their hydrated sample. The structural aspects for vermiculite are as follows. There is an a/3 shift within each layer along the x-axis parallel to the layer (see figure 11.2 for the definition of the unit cell). And alternate layers 46 are displaced relatively to one another by +b/3 or -b/3 along the Y- axis. In the sample Shirozu and Bailey have studied, the c-axis stacking sequence is regular (+b/3, -b/3, +b/3, -b/3,...) with occasional mistakes. .And in De La Calle's sample, there is no regularity for the b/3 shift. The stacking view of two basal oxygen layers on top of each other with a mutual b/3 shift has been illustrated in Figure II.3 in Chapter II. The exchangeable 1192+ cations lie in a plane midway between adjacent host layers and have a plane of water molecules on each side. These cations and water molecules take the form of an incomplete octahedral sheet with only a portion of the sites occupied. Exchangeable cations lie under or above the A13+/Si4+ substitutional sites. Studies by Shirozu and De La Calle further showed 'that cations probably reside at one of the three possible sites: 1111 sites in Figure II.4. The relative positions between basal oxygens, water molecules and cations are also shown in Figure II.4. Exchangeable cations are surrounded by six water molecules and the crystal field is dominated by these water molecules. There are no detailed studies on the staking sequence of vermiculite after water is taken out of the gallery spaces. It is reasonable to assume that the stacking sequence is unchanged because the b/3 shift is energetically favorable (the basal oxygens avoid each other in this staggered stacking configuration). Therefore, one can suppose that intercalants still reside in m1 sites as water is taken out. In this case, they are coordinated by six basal oxygens and the symmetry of the crystal field is unchanged. Let us first discuss hydration/rehydration compounds. Bleam has shown that the static electric potential from host layers becomes 47 insignificant when an intercalant is more than 2 Angstrom away from the host layer.43 Therefore we only have to consider the potential resulting from the water molecules themselves. The detailed arrangement of a cation and its surrounding water molecules is shown in figure IV.1. Basically there are six water molecules surrounding close to a cation to form a distorted octahedron. The symetry of the crystal field is C and can be approximated as an 0 3V field with a trigonal distortion h along one of the C axes which is the c-axis of the sample. 3 The Hamiltonian for a single ion in a solid can be written as 2 2 2 H 3 _ .E._ 2 V 2_ Z .32. + .l. 2 9 + V + 15°; 2m i r 2 rij c 1 1 1:3 9 9 = Hion + Vc + ALOS (IV.1) where the first term is the kinetic energy, the second term is the Coulomb potential, the third term is the electron-electron interaction term, Vc is from the crystal field, and the last term is the spin-orbit interaction (A is the coupling constant). For the first transition group, the electron-electron interaction is normally stronger than Vc and the spin-orbit coupling is the smallest term in (IV.1) (see ref. 44, p58). Therefore, we start from the ground state of H and treat Vc as ion a perturbation. The spin-orbit coupling will be treated as a further perturbation. + The detailed mathematical calculation of energy spectrum for a Ni2 and a C02+ ion in an Oh field with trigonal distortion is explained in Appendix I. We have chosen the c—axis, which is one of the C:3 axes of the octahedron, as the axis of quantization. The ground state of a free 48 L1 éH E \E H “‘5' O b '0" H (a) H""O / ' H ‘9 H HE R (b) Figure IV.1 Schematic illustration of a gallery cation surrounded by six water molecules at ambient conditions. (a) side view. (b) top view. 49 + + Ni2 and Co2 ions is 3F and 4F, respectively. When they are put in the crystal field, they behave as a single f electron with an orbital degeneracy of 7. The corresponding atomic energy level splits into three sublevels denoted as A , T 29 , and T , which are shown in Figure 29 lg AI.1.. The subscript 9 corresponds to those irreducible representations of the 0 group which are invariant under the inversion operation. The h trigonal distortion causes both the T29 and Tlg levels to split into a singlet and a doublet. A Ni2+ ion has the electronic structure 3d8 and it can be treated as a two-hole case. On the other hand, a 002+ ion has the electronic structure 3d7. It can be treated as a half-filled shell with two extra electrons outside. It has been shown in Appendix I that 2+ 2+ the energy spectra for a Ni ion and a Co ion are just off by a sign. Two important parameters are the strength of the 0 field and the h trigonal distortion, Dq and Dt. They are defined as follows Dq = Constant -——:£—-— (IV.2) 7/45 2 Dt = constant (IV.3). Here and are defined as follows m * m 2 — I f3d (r) r f3d(r) r dr, m - 2,4 (IV.4) while f3d(r) is the radial part of the 3d wave functions. 4. Without the trigonal distortion, the ground state for a 002 ion is an orbital triplet (4T , where the superscript indicates the spin 19 degeneracy) with wave functions 91, 9 and 93 defined in equation 2' 50 (AI.2) «of the appendix I. It happens that the 9 value for this triplet is very large (g---4.333).44 This is because the spin-orbit coupling causes a very small energy splitting of this level ( of the order of the spin-orbit coupling strength 100' which is about 180 cm.1)44 which results in a large 9 factor (the Lande g factor can be far from 2 as in O with 002+ where the 9 factor is about 2.5 along one direction and 6.0 along another.52). When the some compounds such as Mg(CH2COOH)2'4H2 trigonal perturbation is switched on, the situation changes. The triplet ground state T then splits into a doublet and a singlet (see 19 figure AI.1). The new ground state is the singlet with wave function / 3 o -3 93 = -1%- ( 13 ;:§:- 13 - 13 ) (1v.5). 10 This ground state gives a 9 value of 2. Therefore, the trigonal distortion transforms the ground state of 002+ to a singlet and reduces the 9 factor to 2. Our observation is that g is about 2.21. The difference comes from the mixing between ground and excited states by the spin-orbit interaction. The first excited state is the doublet from Tlg with wave functions 81 and 82. And the energy difference between the ground state and first excited state is ‘i—Dt. For an orbital non degenerate ground state with spin-orbit .9 coupling, the 9 factor is related to a second rank tensor K (see ref.24, p122) as follows: gij=2(5ij-Anij), (IV.6) 51 where )1 is the spin-orbit coupling strength and 513 is the Kronecker delta function. The tensor A is defined as: A = Z <0|Liln> ij n O 0 (1v.7) where |o> is the ground state, |n> is the excited state, L1 is the angular momentum operator, E0 is the ground state energy, and E: is the excited state energy. To first order, we only consider the mixing between the ground state (83) and first excited state (81,82). Other excited states are located far apart and only gives small corrections to the 9 value. Furthermore, we ignore the splitting of 81 and 82 caused by spin-orbit coupling by assuming that this splitting is much smaller than the splitting between the ground state and the first excited state. By using relations such as Lx=(L++L_)/2, Ly:(L+-L_)/2i and expressions of 9 , and 83 in Appendix I, we have obtained the A tensor as follows 91' 92 . 9/(8AE) o 0 K = o 9/(353) o (1v.a). o o 0 Therefore, the g tensor is 2-9ACO/(4AE) 0 gCo = o 2-9100/(403) o (1v.9), 0 O 2 and the average 9 is S2 31 "' a _1_ 8 __ Co gco E 3 911 2 203 (1v.10) where AE = ‘33— Dt and )‘Co is the spin-orbit coupling strength which has a value of -180 cm-1 (spin-orbit coupling constants for various transition metal ions are given in table 4.1, ref.24). Our experimental result (9 = 2.21) for Co2+ reveals that Dt = 857 cm-1. This value is much greater than the spin-orbit coupling constant llCo' . However, we will show shortly that it is much smaller than the 0 field strength. h For a Ni2+ion, the ground state is an orbital singlet, 3A29, which does not have any orbital angular moment ( i.e., the expectation values for Lx' Ly and L2 are zero). The 9 value for such a state is 2. Because of spin-orbit coupling, the ground state is mixed with excited states such as T29. Using the same procedures as for C02+, we have obtained the A tensor as follows . 4/AE o 0 K = o 4/AE o (1v.11) o o o where AE = 10Dq - '%' Dt. Therefore, we can obtain the g tensor for Ni2+ as follows . 2-81Ni/As o .p g x o 2-81Ni/AE o (1v.12) which gives an average 9 1 16A 9“. = E '3' g.. = 2 - 3A3 (IV.13). 53 Our results for Ni2+ ( ENi = 2.16, and ANi a - 324cm-1) reveals a AE of 10,800 cm-l. Notice that AE contains two parameters Dq and Dt. The trigonal parameter Dt has been found to be 875 cm-1. Therefore, we find that the Oh field strength from water is 10Dq 8 13,000 cm-1 which is larger than the trigonal perturbation Dt. It is also larger but of the 4. order of what has been found in other complexes such as Ni2 (1120)6 ( 1ooq is about 10,000 cm-l).34 The field strength parameters from water molecules can be further verified by the average 9 factor from Cu2+ ions in hydrated/rehydrated vermiculite. A Cu2+ ion has 3d9 structure and can be treated as one 3d hole case. The energy spectrum for a 3d hole in an octahedral field is given in the literature and wavefunctions are given in Appendix II. The ground state ( 2eg ) is an orbital doublet. As shown in Appendix II, the trigonal distortion down shifts the ground state energy but will not lift the double degeneracy. The spin-orbit coupling AIRS does not lift that degeneracy either. The Lande 9 value for such a state is still 2. Our observation from susceptibility measurements for Cu2+ is that g = 2.28. Again the difference is caused by the mixing between the ground state and higher excited states. Although there is a complication arising from the fact that the ground state is a doublet, we treat each Cu2+ ion as if it occupies each component of the ground state with equal probability (see Ref.24, p135 for detail). By mixing one component of the ground state with the excited states, and by using the same type of calculation as for Co2+ and Ni2+, we have obtained the g tensor as follows S4 1 2-21/(1ooq+ot)-4l/(1ooq-2Dt) o o 9 = 0 2-61/(1onq+ot) o (1v.14). o o 2 And the average g is _ 81 41 = 2 - cu - C“ (IV 15) 9Cu 3(lODq+Dt) 3(lODq - zot) ' ‘ + + - - Using the parameters from Co2 and Ni2 (Dt=850cm 1, 100q813000 cm 1) and ACu = -830cm-1, we obtained 36“ = 2.252 which is consistent with our observation for hydrated Cu2+-ver. Table IV.2 summarizes above calculated field strengths. The experimental result for Cu2+ and the calculated values are also compared in that table. At this point we conclude the following. First, the fact that the 9 value for Coz+ ions is slightly larger than 2 can be explained by a small trigonal distortion on the Oh field. Secondly, the field strength 10Dq is of the right order of magnitude and is further confirmed by the 4. Cu2 data. The distortion strength Dt is about 1/15 of the Oh field + + + strength. Thirdly, marked anisotropy for Co2 , Ni2 , and Cu2 is evident. When water is removed from the vermiculite gallery, as we discussed before, the intercalants will most probably stay in sites m (see 1 figure II.3 in Chapter II) where they are coordinated by six oxygens from basal planes of host layers. We can still treat the local symmetry as an octahedron symmetry with distortion along one of the 03 axes. The resultant energy spectrum is the same as with water present but with + 0K. Our observation for Mn2+-ver is consistent with this theoretical conclusion. For C02+-ver and Ni2+-ver compounds, although they are anisotropic systems, they don't order either and need further theoretical understanding. For Cu2+-ver, the failure to magnetically order may be traced back to quantum fluctuation effect of an ion with small spin moment (see p704, ref.32 and ref.53). For dehydrated compounds, Mn2+-ver and Cu2+-ver do not exhibit a phase transition for temperatures from 2K to 300K. The 9 factor for a Mn2+dxn1 is still close to 2 and ions interact isotropically while water is present in the gallery. And the quantum fluctuation in Cu2+-ver may still quench the magnetic ordering. In contrast, there is a peak in the susceptibility curves for Coz+- ver and N12+-ver compounds indicating the possibility of a phase transition occurring. As discussed in section III.3.1, the spin-orbit coupling will cause anisotropic 9 factor for Co2+ and Ni2+ in a distorted octahedral field. This anisotropy will also cause the spin- spin interaction at different lattice sites to be anisotropic (Ref.24, p250). Consider a pair of magnetic ions at lattice sites a and b, the Hamiltonian is H = H + H where H is the unperturbed term and H is 0 l 0 1 the perturbation which is defined as follows 9 9 9 H = J s -s + l L Os + l L -s (IV.16) where Jab is the exchange interaction between two ions, la and Ab are the spin-orbit coupling strength for each ion, 1. and S are the angular and spin momentum operators. By using Brillouin-Wigner perturbation theory (Ref.24, Appendix 2), one gets the effective Hamiltonian up to second order in the spin-orbit interaction and first order in the exchange up 29 9 29 9 999 - .. in s 0103 1 sh K sb + n-saxsb (1v.17) where the second rank tensor K is defined in section IV.2. The last term is called antisymmetric exchange interaction with . 1 , , a b <0 oblf.a bIn 0 >0 - - = -* —' W n na nb, where na'b i 2 :::3 30-30 (IV.18) na'b n where a,b stands for two different lattice sites, |0> is the ground state, and In> is the excited state. As Moriya pointed out,26 such term will vanish if there is a inversion center between two spins and will exist otherwise. This effect forces spin a and b to lie in the plane perpendicular to it). Thus, the exchange interaction will be anisotropic and yield an x-Y system or even an Ising system. Figure IV.2 shows the distorted Kagome lattice ( only one hexagon is shown). As one can see, there is no inversion center between two cations located two adjacent 1111 sites. One thus should expect an 62 ‘> -> Y Y' —> x I Intercalant / O Figure IV.2 The Kagome lattice (only one hexagon is shown) with in plane distortion. Each tetrehedron is rotated 6 degrees clockwise or counterclockwise. 63 anisotropic exchange interaction between these two intercalants. Furthermore, the crystal field in dehydrated compounds is weaker than in hydrated/rehydrated compounds. This gives rise to additional anisotropy in the exchange interactions and may cause stronger correlations between spins in the Co2+ and Ni2+ compounds. Finally, we consider the peak position( To) in X for Coz+-ver and Ni2+-ver ( about 5 i 1 K and 3 t 1 K, respectively). As we recall, Co2+ and Ni2+ ions have spins of 3/2 and 1, respectively. And Tc represents the transition temperature at which the magnetic ordering occurs. To gain insight into the phase transition in these two CIC compounds, we make the following assumptions. First we simplify the lattice as an two dimensional rectangular lattice, which is the possible lattice arrangement of the interlayer intercalants for the half-filled Kegome lattice30 ( there is one divalent ion per unit cell 2 x b in figure II.2), with anisotropic exchange interaction J1 and J2 along two directions (see figure IV.3). Secondly we treat the interaction to be Ising-like since the interaction between two intercalants in C02+-ver and Ni2+-ver are anisotropic. These assumptions are the simplification of our systems. The Hamiltonian is Hex = - 2Jl 2 S2 i,jsz i+1,j - 2J2 2 8z i,jsz i,j+l 1:3 1:3 --2J 3220 0 -2Js2 20 0 (NH) 1 z i,j z i+1,j 2 z i,j z i,j+1 ° i,j i,j where i,j correspond to a cation at ith raw and jth column.r The 2D Ising system has a closed analytical form for the transition 64 ~01 J column b , I I I a I J 1 1m l'OW )1 [1’ J2 / [ Figure IV.3 A rectangular Ising system with anisotropic interactions. J1 is the exchange coupling along rows and 12 is along columns 65 temperature48'49 2 2 2|J |s 2|J |s Sinh(——L—)°Sinh(—‘2—) = 1 (IV.20) kBTc kBTc Ji and J2 are the exchange interaction between two electrons at two lattice sites and they are the same in C02+-ver and Ni2+-ver. There is an unique solution of Sz/Tc in above equation for a fixed J1 and J2. Therefore, Tc should be proportional to 82. Thus a system with larger spin moment will have a higher transition temperature. This is consistent with our experimental observation. Furthermore, the SZITc ratio for Co2+ and Ni2+ is 0.4510.1 and 0.3810.1 respectively. They are almost equal within their errors showing that our observations are consistent with the theoretical predictions. IV.3 Concluding remarks In this chapter, the average Lande 9 factors for different intercalants under different hydration conditions have been obtained and analyzed by using crystal field theory. We have estimated the crystal field strengths for hydrated/rehydrated compounds as well as for dehydrated compounds. Substantial decreases in crystal field strengths have been found when water is removed from the gallery. Furthermore we have vertified the physical origin of the anisotropy and discussed the effect of spin-orbit coupling on both the paramagnetic 9 factors and the spin-spin exchange interactions in these compounds. 66 ‘The effect of water on the magnetic interaction in these compounds has been investigated and discussed. All hydrated/rehydrated compounds do not exhibit magnetic ordering consistent with previous workao. Water molecules weaken the antiferromagnetic interaction and destroy the ordering. we have observed for the first time the antiferromagnetic 2+ 2+ ordering in Co -ver and Ni -ver when water is removed from the gallery. Mn2+-ver is found to be a Heisenberg isotropic system under either hydrated/rehydrated or dehydrated conditions. Therefore, there is no phase transition for temperatures down to 2K. On the other hand, the + exchange interactions in C02+-ver, Ni2+-ver and Cu2 -ver are 2+ 2+ anisotropic. Co -ver and Ni -ver exhibit signatures of a phase transition at very low temperature ( ~5K and ~3K, respectively) without water in the gallery. The spin dependence of the transition + + temperatures for dehydrated Co2 -ver and Ni2 -ver has been explained by a very simple 2D Ising model. Although Cu2+-ver is an anisotropic 2+ 2+ system as are Co -ver and Ni -ver, it does not exhibit magnetic ordering for temperatures from 2K to 300K. This phenomenon may be related the fact that it is a quantum fluctuating system. Chapter V Diffusion and absorption of inert gases in mixed ion systems v.1 Introduction Heteroionic clays with different cations in the gallery 'are particularly interesting for studying access phenomena in 2D microporous systems by probing the conduction and absorption of atomic and molecular species. In percolation experiments on real systems, the dynamic physical quantity measured is, for examples, the electrical conductivity or the mass conductivity as a function of disorder. For the systems of interest here which are ternary intercalated layered silicates Al-xBx-Y' where Y is the host layer, and A and B represent the cation with different physical properties such as size, the mass conductivity of a third guest species is studied. The basic picture is quite simple and can be described as follows. The third guest is denoted as C in this thesis. Suppose that the gallery structure is such that species C can easily diffuse though the A B -1' system and the mass conductivity is l-x x finite when x30. On the other hand the gallery space does not allow any C particles to move through for x31. With increasing x there exists a critical value, which is the percolation threshold xc, at which the mass conductivity for C is zero and remains so for xc < x < 1. Therefore, by studying the permeability of the host to the guest species C, one can in principle probe interesting physical phenomena such as anomalous (diffusion at the threshold concentration xc which has attracted considerable theoretical and experimental interest in recent years.56 67 68 In addition to diffusion studies, one can also study the access phenomena by measuring weight uptake and the absorption isotherm. The physical quantity measured here is Wc(x) (- M(x)/MT(x), where M(x) is the absorbed mass of C for a given x, and MT(x) is the total absorption mass that can be obtained when all available interlayer surface is occupies by C). There is an xc above which there are no percolation channels through which C can pass to the available spaces in the gallery. The exact form of Wc(x) and the value of xc can be determined from percolation theory. It is very important to understand the structural aspects of our ternary mixed ion system. The porosity in pillared lamellar solids such a Bx-Y is crucial for percolation to happen. The method used to s Al-x characterize microporosity in terms of the sizes and the spatial distribution of the cations is as follows: Assume that the cation A and B and diffusing molecule C can be represented as ellipsoids of revolution with diameters (er’hA)’ (2rB,hB), and (2rc,hc). The topological constraints on the access of a guest C in the the gallery spaces will depend on the geometric property of the pillars and the adsorbate through the following five parameters: 6AA 8 (a - 2rA)/2rc, 5 s (a - (rA+rB))/2rc, 5 AB - (a - 2rB)/2rc, 5 s hA/hc and 88C - 88 AC hB/hc. Here a is the distance between two gallery cations. Any given geometric condition for the diffusion of C inside A Bx-Y is completely l-x determined by the five member set of the above specified parameters (porosity parameters). Define the porosity parameter set S as S = [31:82:83.841851 ‘ {Bi} (V-1) 69 and assigning s1=+1 if 5AA > 1 and s1 = -1 if if 5AA < 1, and so on for other Bi' There are total 32 distinct porosity sets for some of which there is no conduction and for some of which the intergallery space is completely permeable for all x. They are given in table V.l. As one can see, many of them are not physically significant. Those that are significant can be selected by a judicious choice of the species A, B, and C. The choice of the porosity parameters is limited by the available pillaring agents. Table v.2 lists dimension of some of the agents one can insert in to the gallery of a clay host. In order to perform the mass conduction measurement, the diffusing spices should be such that they do not have any chemical reactions with the host or with intercalants. Inert gases are thus an ideal choice for the diffusing species. We have chosen [C03+(en) [Cr3+(en)31x-Fluorohectorite (where en 3]l-x = ethylenediamine, and fluorohectorite is a layered silicate denoted as FHT) as the ternary mixed ion system and Argon as species C under the following considerations. First, pillaring agents such as Co(en)';+ and Cr(en):+ have very large dimensions as can be seen from table v.2. They prop up the gallery to a height of 5.4 A which is suitable for Argon atoms (dynamical size = 4 A). Secondly, they are trivalent ion complexes which are far separated in the gallery so that the lateral dimensions between them are large enough for Argon to pass through (the average distance between two adjacent intercalants is about 10.22 0). 3+ 3+ Thirdly, there are no water molecules in the [Co(en)3 ]x[Cr(en)3 ll-x- + + FHT because three en ligands fully coordinate the Cr3 and C03 ions. These structural characteristics indicate that this system should be 70 Porosity 81 S2 S3 S4 S5 Percolation set S1 + + + + + Open S11 - + + + + L-S S111 - - + + + L-B SIV + + + - + V-S Sv - + + - + L-S/V-S SVI - - + - + L-B/V-S SVII + + + + - V-S SVIII - + + + - L-S/V-S Slx - - + + - L-B/V-S Sx All remaining 23 combinations Closed Table V.1 Porosity sets and their corresponding behavior. L=laterally controlled, V=vertically controlled, S=site, B=bond. Openznon-percolative systems. 71 Ion Diameter Height Me4N+ 4.8 4.2 MegNH’r 4.0 3.2 MeNH+3 3.2 2.8 NH+4 2.8 1.5 Co(en)33 + 5.4 5.4 Cr(en)33+ 5.4 5.4 Table V.2 Dimensions of pillaring agents. Diameters and heights are in unit of Angstrom. 72 open to Argon atoms for all concentration x. The most interesting and important characteristics of these two pillaring agents are as follows. Coy-(en)3 and Cr3+(en)3 have different responses to heat. When the sample is heated at between 100C and 150C, a C03+(en)3 complex will be break into (C03+(en)2+en]. What happens is that an en ligand is dissociated from its parent complex and this ligand remains in the vicinity of the original complex cation, while Cr3+(en)3 cations remain unchanged. It turns out that [C03+(en)2+en] has smaller vertical dimension ( about 3.8 A) and larger lateral dimension ( more than 6 A) than the parent cation. Argon atoms can not pass through the lateral space between a pair of [Coa+(en)2+en] and between [C03+(en)2+en] and Cr3+(en)3 (see figure v.1 for schematic view), while they still can pass between two Cr3+(en)3 cations. Thus, the annealed system ( [Coy-(en)2 + en ]J‘[Cr3+(en)3 ]1_x-FHT ) has porosity parameter set { -, -, +, -, +}, which is identified as SVI in table V.1. As we can see, this system should be percolative. Figure v.1 shows the inplane structure of FHT with CoEH'(en)3 and Cr3+(en)3 as intercalants before and after annealing. The actual distance between two adjacent intercalants is larger than illustrated because the charge/unit cell for this particular FHT is 1.6e- instead of 2.0e-. We now present our experimental results for mass absorbtion of Argon in [Co(en):+]x[Cr(en)g -FHT bulk. It has been found that the .1. ll-x -FHT shows mass uptake of Argon in the annealed [Co(en)§+]x[Cr(en):+]1_x an interesting behavior which indicates a percolative process.54 In order to obtain direct evidence of the dynamical movement of Argon atoms in the gallery of this mixed ion clay system, we have also performed studies of mass conduction of Argon in the unannealed and 73 annealed [Co(en)§+]-FHT films. Although Argon atoms can move in the gallery space of this particular sample before annealing, they are blocked after annealing since the gallery collapses and the gallery space is blocked by the dissociated ligands. Besides diffusing Argon, we have used Helium to prob whether the sample is complete blocked by undesired reasons such as the seal off by glue. Since the diffusion rate of Argon is expected to be very small, we have perform the diffusion studies of Argon in a thin [Co(en):+]-FHT membrane. A special experimental set up with high sensitivity and a unique technique to diffuse gases through thin membranes has been developed. We have successfully observed the complete blockage of Argon in the annealed film which gives a strong evidence for the dynamical movement of guest species in the gallery spaces between the cations. V.2 Experiment and results The starting material is Li+-Fluorohecteriteso which is a trioctahedral 2:1 layered silicates with following unit cell stoichiometry (here Li+ is the gallery cation): L11.6 (“94.4Lil.6) $13 020 F4 The main structure is essentially the same as vermiculite. Samples for mass absorption measurement were made through an ion exchange reaction by putting the parent compound to solutions with 3+ 3+ r desired concentrations of Co (en)3 and Cr (en)3. Water molecules swell the gallery so that the ion exchange can take place. Once these 74 large complex particles enter the gallery, the compound stops swelling and reaches a state in which no water molecules are present inside the gallery. Co3+(en)3-FHT films for mass conduction experiment were made by immersing a self supporting L12+-FHT film into a dilute Coa+(en)3 solution. As the ion exchange process terminates, the film stops swelling and do not ingest water even when immersed in the solution. The film thickness is about 15 um. Annealing has been done by slowly heating samples in vacuum ( starting from room temperature to 120 C with AT/At a 15 C/hour, and remaining at 1200 for another 6 hours). The in plane structure of [Co(en)3+]1_X[Cr(en):+]x-FHT before and after heating is schematically shown in figure V.l which includes all structural aspects of the system as stated in introduction. Figure V.2 shows the x-ray diffraction pattern of Coa+(en)3-FHT before and after it has been annealed. The c-axis basal spacing decreases from 14.45 A to 12.82 A. Other features in this pattern are broadening of (001) Bragg peak and the disappearance of higher Bragg peaks after the film has been annealed indicating shorter coherence length along c-axis and basal spacing fluctuation. Figure v.3 presents the surface morphology of an oriented film sample and a bulk sample from a scanning electron microscope. As can be seen, the clay particles orient very well in the film while the bulk sample exhibits considerable amount of large voids ( with dimension of order of micrometer). This is the other reason that we have chosen film sample. Even in the oriented film, there are small defects such as micro-cracks and voids which are unavoidable resulting in difficulties in the experiment. The diffusing gas is applied perpendicularly to the surface of the 75 0.0.0.010. , ”is "e" .0 r\v A 1A v ’r‘" x v - Intercalants 4.. (d .V 65355-133 Ce Argon atoms \ 1” A \ VJ’A’ A ,_. gee V :10 (Joe 0 00 Figure V.1 The in plane structure of [Co3+(en)3]1-x[Cr3+(en)3]x-FHT. Single cricles represent Cr3+(en)3. Concentric circles present Co3+(en)3, in which smaller circles are for before annealing and larger circles are for after annealing. a=5.28 angstrom, b=9.15 angstrom. Actual cation density is less than shown because unit cell charge is 1.6e' instead of 2.0e'. 76 d = 14.45 A g r c: D E ’ d = 12.82 A :3 .o a h :3 a i at 3 J a J Al A I 0 10 20 30 4O 50 20 Fig. v.2 The x-ray diffraction patterns of C03+(en)3-FHT before and after annealing (the c-axis basal spacings are respectively 14.45A and 12.82A ). These patterns were acquired at the room temperature using Cu Ka radiation. 77 (a) (b) ..J‘*. 19.9u £099 Fig. v.3 The surface morphology of 003+(en)3-FHT. Picture (a) is the surface SEM image of an oriented film and (b) is the image of a bulk sample. 78 oriented film. There are three possible channels for Argon to pass through in this particular geometry (illustrated in figure v.4). Path 1 can be a hole on the film or a channel of connected voids. This type of defect results in a huge background signal instantly. Path 2 is a so called void-grain boundary-void path. Diffusion through these two channels will not depend on whether or not the film is annealed. Path 3 is the void-gallery-void path. Diffusion through this channel will be different before and after the gallery is blocked as a result of annealing. The last situation is desirable and depends on the quality of the membranes. Mass conduction measurement was carried out on an apparatus which is equipped with a residual gas analyzer and is automated by a personal computer (see figure v.5). A constant pressure of gas C is maintained in the reservoir and applied to the membrane, which is mounted on a sample holder to avoid leakage from other sources (also shown in figure v.5). The sampling chamber is always pumped to maintain a vacuum of normally 10-7 torr. The partial pressure of the interested gas is measured by a quadrapole residual gas analyzer and signals are sent to a IBM PC. Valves are controlled by the computer to accomplish certain specific functions at desired times. The mass absorption of Argon has also been conducted by using conventional gravimetric method.54 Figure v.6 shows the amount of Argon absorbed for samples with different x ( x is the concentration of Cr3+(en)3) and for samples which are the direct physical mixture of two pure compounds.54 They have been heated to the correct temperature range beforehand. As one can see, the amount of Ar absorbed in the physical mixtures shows a linear dependence 4. on 3: because the amount of available gallery spaces are from Cr3 (en)3- 79 l I Figure v.4 Path 2 V H H .__Jl Path 3 Pathl 1 Three possible path' ways. Each clay particle is presented as a block. 80 Roughing Pump a -2/3 Dq h <:1| vo |:1> = 8/3 09 h <0| VOnIO> . -4 09 <12| vbh|12> . <12| voh|tz> . 1 10/3 lg— 09 (AI.7) 101 where Dq is normally defined as D9 = constant———;——— . 7l4u Evaluation for the energy levels for 002+ and Ni2+can be carried out by using (AI.2), (AI.5), (AI.6) and (AI.7). The procedure is lengthy but straight forward and the results are as follows 2+ 3 3 3 Ni . E( A29) - -12 D9; E( T29) = -2 D9, 3( T19) - +6 Dq 2+ Co : E( 4rlg) = -6 Dq; E( 4T = +2 Dq; E( 4A29) . +12 Dq. 29) As we can see the ground state for Ni2+ in an 0h field is an orbital singlet ( A29) and the ground state for 002+ is an orbital triplet (T19). And the energy spectra for 002+ and Ni2+ are off by a sign. When the trigonal distortion along the C axis (z-axis) is turned on, the crystal field becomes V = V + V where V = a r2 Y0 to first c 0h 1 T 2 3 order. Two triplets both break into a doublet and a singlet. To first order, we only need the matrix elements as follows = 5Dt 3 t 3 <232| v |112> = 0 "12+. 3 1 3 . <2§1| vtlyt1> = -30t 8 0 <23| VTIY3> = -4Dt (AI.8) =-SDt 3 1 3 . 0 002+. 3 T 3 <1§1| VTIY31> a 3Dt 0 (Y3l v1|13> a 4Dt which can be obtained by using (AI.5), (AI.6), and the proper C-G coefficients. Dt is defined.as Dt a constant . The energy 102 splitting can be evaluated by calculating the expectation value of arng and results are as follows: 2+ 6Dq + Dt (1) Ni : T ”299 + 599 ‘1)- A -12Dq (1) lg 6Dq - 1/2 Dt (2); 29 I -2Dq - 5/2Dt (2)' 29 002+: A 12Dq (1); T ZDq + 5/2 Dt (2). T 6Dq + 1/2 Dt (2) 29 2g l 2Dq - 5 Dt (1)' lg 60q - Dt (1) where the numbers in parenthesis indicate the degeneracy. The energy spectrum is shown in Fig. AI.1. The corresponding wave functions are still the functions in (AI.1). 01 and 82 are for the doublet of T19 and 83 is the singlet. 01 and 02 are for the doublet of T and 03 is the 29 singlet. As can be seen, the ground state of a 002+ ion is an orbital singlet (03) when the trigonal distortion is turned on. 3F (7) 103 Octahedral field Tng‘mil perturbatlon 311g (3) (1) 6Dq+Dt 6Dq —: Free N i2+ ion (2) 6Dq-1/2 Dt (1) -3Dq+5Dt 3123 (3) -2Dq — (2) -3Dq-5/2 Dt 3 A (1) 1 2“ -12Dq U -12Dq Figure AI.1 Energy splitting of a N i2+ ion in an octahedral field with a trigonal perturbation. Numbers in parenthesis present orbital degeneracy. For a C02+ ion, the spectrum is off by a sign and the spin degeneracy becomes 4. 104 Appendix II Divalent Cu in an octahedral field with trigonal distortion A free Cu2+ ion has electronic structure of 3d9 and a ground state of 2D which has an orbital degeneracy of S. It behaves as a d hole and the five degenerate wave functions are Y2 , m=-2,...,+2 (normally denoted as dm ) . In an 0h field these wave functions form a reducible representation 1‘ consisting several irreducible representations of Oh. The energy splitting for a d hole in an 0h field is the most common example given in crystal field theory literatures.44 I' splits into a . Here we use lower case doublet called eg and a triplet called t29 since we deal with a single hole. The energies for each levels are: E( eg ) = -6Dq; E(tzg) - 409, here D9 is defined exactly the same as in Appendix I. And the basis wave functions are as follows ( assigning the C3 axis as the z-axis) tgg = J27; d(x2-y2) - J17; d(xz) tzg: tzg = /2/3 d(xy) + l1/3 d(yz) ; 0 2 t29 d(z ) (AII.1) e; = /173 d(xz—yz) + {273 d(xz) e 3 9 e; a /2/3 d(xy) + /1/3 d(yz) 2 2 - 2 . - where d(x -y )=l//2 (d2+d_2), d(z )=do, d(xy)=1/(1/2 )(dZ-d_2), = Dt 0 o . -2Dt t i = 0 (A11.2) t t — = /2 Dt As we can see, the trigonal perturbation mixes tig and e:. By solving the 5x5 secular equation, one can obtain the new energy eigenvalues and eigenstates (in parenthesis) 4Dq + Dt + 2Dt2/(5Dq); (A t1 + B ei) 29 0 4Dq -2Dt; (tZg) (A11.3) E(e ) - -6D - 20t2/(5Dq)° (-B t: + A at) 9 q ' 29 9 where A ~ 1 - th/(IOOqu), B 1' 1/2 Dt/(lODq). The perturbed levels are plotted in Fig. AII.l. The most important feature is that the ground state energy is down shifted but its orbital degeneracy is not removed. Spin degeneracy can be added to the wave function sets by multiplying the spin eigen functions a and B, where a is for spin up and B is for spin down. So the total degeneracy of the ground state is four and we can rewrite it as c1 d(xz-y2)a + c2 d(xz)a 106 c1 d(xz-y2)B + c2 d(xz)B (AII.4) c1 d(xy)a - c2 d(yz)a c1 d(xy)B - 02 d(yz)B where c1=1/1/3 A - /2/3 8, and c a /2/3 A + 1/1/3 B. By evaluating the 2 matrix elements for 11.3, one can easily show that all matrix elements are equal to zero ( see appendix II in chapter 6 of ref.44). Therefore, the spin-orbit coupling can not lift the degeneracy of the ground state. It is straight forward to evaluate to expectation value of the 9 factor operator (1 + 2;) and obtain the 9 factor to be exactly 2 for the degenerate ground state. 107 Octahedral field Trigonal perturbation (2) Z0(5) 2+ . Cu free 10n 2 2tzg (3) 4Dq+Dt+2Dt /(5Dq) 4Dq (l) 4Dq-2Dt 2e, (2) Figure AII.1 '6Dq —_|——(2)— -6Dq-2Dt2/(5Dq) Energy splitting of a Cu2+ ion in an octahedral field with trigonal perturbation. Numers in parenthesis present the orbital degerenacy of each state. 108 REFERENCE 1. 10s 11. 12. 13. 14. 15. 16. 17. 18. 19. M.S.Whittingham, Intercalation Chemistry (Academic Press, NY 1982), p1. K.Watanake, et.al., Proc.R.Soc.Lond.,A;§;, 51 See Solid State Batteries, ed. by A.C.Sequeire and A. Hooper (Marinas Nighoff Publishes, Dordrecht, 1985) P.B. Weisz, Science, ;_2, 433 (1973) n. Gelbin, AIChe J, 2;, 358 (1979) S.Lee, x-ray studies of layered rigidity and c-axis expension in clay intercalated layered compounds, Ph.D thesis, p2 K.Norrish, Trans. Faraday Soc. 19, 120 (1954) S.A. Solin, Adv.Chem.Phys.,g§§, (1982) M. Suzuki, H.Ikeda, J.Phys.C, Solid State Phys.,11, L923 (1981) J.T.Nicholls, G.Dresslhaus, Synthetic Metals.,;g, 519 (1989) D.G. Wiesler, H.2abel, Synthetic Metals, 93, 505 (1989) J. 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Grim (McGraw-Hill Book, 1968) S.W. Bailey, Am. Miner. fig, 175 (1975) See Crystal structures of clay minerals and their x-ray identification by G.W. Brindley and G. Brown (Mineralogical Society, Lodon 1980), p98 M. Suzuki, et.al., Phys. Rev. B 49, 11229, (1989). Note: The ac susceptibility of Ni-ver also shows a shoulder at 2.21! and a sharp peak at 2.9K indicating possible magnetic phase transitions at these temperatures. See N. Wada et al., J. Mater. Science, 9, 864, (1987). J.A. Rausell-Colom, et.al., Clay Miner., 1;, 37 (1980) N.W. Ashcroft, N.D. Mermin, Solid states Phys. (Saunders College, Philadelphia 1976), p56 W. Low, Paramagnetic rasonance in solids (Academic Press Inc., NY 1960), p113 C.K. Jorgensen, Inst. Intern. Chim, Solvay, Conseil Chim., Brussels, 1956 B. Bleaney, K.D. Bowers, Proc. Phys. Soc. (London), A§§, 667 (1952) 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. S4 55 56 110 J.H. Van Vleck, J.Chem, Phys,;, 807 (1935) 8.8. Hendricks, M.E. Jefferson, Am. Miner. 2;, 851 (1938) A.M. Mathieson, G.F. Walker, Am. Miner., _§2, 231 (1954) A. Grudemo, Handl. Svensk. Forskn. Inst. Cement., 2;, (1954) A.M. Mathieson, Am. Miner., __§, 216 (1958) H. Shirozu, S.W. Bailey, Am. Miner., 5;, 1124 (1966) de la Calle, et.al., Clays and Clay Miner., 3 , 481 (1988) W.F. Bleam, Clays and Clay Miner., 99, 527 (1990) Ballhausen, Introduction to ligand field theory (McGraw-Hill Inc., NY 1960), Chapter 4 J. Yamashita, J. Phys. Soc. Japan, 2, 339 (1954) J. Yamashita, Progr. Theort. Phys. (Kyoto), 12, 808 (1954) N.D. Mermin, H. Wagner, Phys. Rev. Lett.,11, 1133 (1966) L. Onsager, Phys. Rev. §§, 117 (1944) B. Kanfmann, Phys. Rev. 1Q, 1132 (1949) R.M. Barrer, D.L. Jones, J. Chem. Soc. (A), Inorg. therot, 1531 (1970) 2.x. Cai, S.D. Mahanti (unpbulished). J.A. Cowen, G.T. Johnson, J. Chem Phys., 51, 2217 (1966) H.A. Bethe, z. Physik, 1;, 205 (1931) R.Kim, T.J.Pinnavia, to be published. Bivins, et.al., The 3-j and 6-j symbols, MIT Press. P.G. de Gennes, La Recherche, 7, 919, (1976). 111 LIST OF PUBLICATIONS PHYSICAL RENEW 3 VOLUME 4|. NUMBER 18 112 —.. 15 JUNE [9904! Dynamics of exciton transfer between the bound and the continuum sates in GaAs-Al. Ga. ..,As multiple quantum wells H. X. liang. E. X. Ping. P. Zhou.‘ and J. Y. Lin Department of Physics. Cnrdwell Hell. Kansas State University. Man/tetra. Kansas 66506-260! (Received 2 January l990: revised manuscript received 21 May 1990) Experimentally observed two-exponential decay o! excitonic transitions in GaAs-Al.Gai-.As multiple quantum wells has been successfully interpreted in terms of the exciton transfer between the continuum (free camera) and the bound state. The calculation results obtained from this excitonotransl'er model are in excellent agreement with expenmental observations. The rates of the exciton transfer and the free-carrier mombination have been obtained. We have demoutrat- dthutheemnsmnqnergydependenadthedecayumecoauantolthemdeaycom- ponntucaumdbythennauonmexumnbindingeurgymduadbyintmteamughnumthe quantamwells. Optical properti- of semiconductor quantum wells (QW‘s) and snperlattias (SL's) have been intensively in- vutigatadduringthepastfewyeanbeceuseottheirnovel ' “’ ThedecayotexcitonictransitionsinQW‘s was hand to show exactly two-exponential behavior.“‘ W the physical origin of this two-exponential decayandthedynamicprocessesolexcitonsinQW‘sare notyetwellunderstood. lnthispeper.theoriginofthe two-exponentialdecayol'excitonictransitioninaQWhas been studied by using time~reeolved photolumineecence. We demonstrate that the origin of the two-exponential de- ‘cay of eacitonic transitions in QW‘s is caused by exciton transferbetweentheboundandthecontinuumstatm (free carriers) via acoustic phonon-exciton (phonon-free carriers) interaction. ThesampleusedforthisstudywasaGaAs- AlesGaasAs MQW which was grown by molecular-beam epitaxy on a GaAsilOO) substrate without growth inter. ruptinn. it consists of alternate ISO-A well layers and 278-AAhsGauAsbat-rierlayerswithatotalol‘ten period‘s. Experimenul details have been described previ- ously. . Figurelshowsasemilogarithmicplotoltemporal responses of photolumincecence of a GaAs-AthauAs MQW measured at three representative emission energies aroundthebeavy-boleexcitnntransitionpeak. Theinset shows the time-iotegrated emission spectrum with heavy- hole and light-hole exciton luminescence peaks at 1.5253 and l.5297 eV. respectively. These values are consistent with those calculated by using the transfer-matrix method with the conduction-band ofl’set parameter being 0.65 and the binding energies of heavy- and light-hole excitons be- ing 5.8 and 6.2 meV. respectively. ’ The shoulder at about 1 meV below the heavy-hole exciton peak is due to either impurity-bound exciton or bicxciton transitions.” It is known that the exciton emission linewidth is predominant- ly caused by interface roughnessce. ‘° Therefore. lumines- cence signals measured in the Vicinity of the exciton trano sition peak correspond to exciton recombination occurring in difl'erent QW domains.“° In Fig. l. the nonexponen- tial decay of photolumineecence is clearly observed with $1 dilueatdeceyratmoecurringatdfl‘ersntem'usionener- gin. thoanlotstheluminmanceasafunctionoldelay timetameuuredattheheavy-holsexcitontransition peak. Therisepartotthelumineeceneeisnotshownbere andu-Ohasbeenchmeaatthepeakpoeitioninthe luminescenceumporalruponsmdthl. Crossesshow dttingsusingone-andtwo-eaponsntisldecay.reepective- ly. methLitisclsarthatthedecayot‘thebeavy- ”Wannabedeecribedbyasingle-exponential 5 1° ; V” l' fir 'r r: : 300 i 5 £200 § 4, l' ' 3 1.00 e‘ m r /-" 3 1 a : o , a > : : L53 1.535 < ’5; . . x ‘ Enertytm ; S ' ~ \ C s 3 _ ,t “ .E ‘0 E -z-tszs«v t d f -- EILSZGSeV :’ . z-tszsscv d 1 5 ‘ ‘ ~We i 102 A: tL .. la ALLA t.; l- 0 2 4 6 8 :3 Time (ns) FIG. I. Semtlogarithmic plot of photoluminescencs intensity vs delay time for three representative emission energies around the beatmhole exciton transition peak at a GaAs-GsesAlesAs multiple quantum well. Well and barrier thicknesses are 250 and 278 A. respectively. The excitation energy was 2.125 W with an average power density of 0.2 W/cm‘. The inset shows the low-temperature (8.5 K) time-integrated photolumtnescence emission spectrum showing the heavy- and light-hold exciton emission bands. )- p b \, um ctmmamrtmm A. 12950 F ‘ ti ‘ ' ._A46 d (rm) 'U/‘u A A #Aul.___-.h Intanslly lath. unite) is (no) FIG). Mashactioaddelaytimeumea- “mmmmmpeak. Cmmmshow themaedvalnmwbilathedashedandsolidlinmaredttinp “mdmuddeeayn-peeuvely. u-Oh. benchm-etthepeahpu'tieaiathelumineseeneetemporal mdml. mummmn-m ”yummde’ummmmm madman. e,+e.bmbesannrmalisedtoenity 80 form However, the two-exponential decay fits perfectly withtheeaperimentaldata. Mommgenmlthedr caydtheheavy-holeexciton wansitioninQWcanbe writtenas ((r)-Aexp(-t/r.)+8exp(-t/r2). (l) wheres. andrzarethetwodeceyttmeconstantsrepre- senting faster and slower decay components and are about 0.9 and 4.2 as measured at the heavy-hole exciton transi- tion peak. respectively. Furthermore. decays of the exci- ton recombination in entire heavy- and light-bole exciton emission bands are also exactly two exponential. We have obtainedthetwodecaytimeconstantsasfunctionsol emission energies in the heavy-hole exciton emission band shownin Fig. 3. Weseethat r. tobeshownmainlydueto theradiativedeceyotexcitonsdependsonlyweaklyonthe emission“ Incontrast. rgdependsstronglyonthe emissionenergissanddecreasesmomtonicallywithin- mdanimionenergyfromeensat l.5232tho about 2.1nsatanenergyo! 1.5268eV. We wanttoindi- cateherethatthetimeconstantsol'theslowdecaycom- ponentatthelower-energysideareevenlargerthanthe exciton lifetime in the bulk Om (3.3 ns). ” [t is known that at room temperatures. optical non- linearities in absorption spectra of GaAs~Al.Gai -.As multiple and single quantum well structure under high laser excitation intensity is due to the ionization of exci- tons after their creation. ‘1'” and the transfer between the bound and the continuum states becomes important. " The excited states of excitons (including the continuum states corresponding to the ionization threshold of exci- tons) in QW‘s have also been observed at low tempera- tures under low laser intensity excitation. ""‘ in this Rapid Communication. we interpret that even at low tern- 113 H. X. HANG. B. X. PING. P. ZHOU. AND J. Y. LIN 0| ’3 2 = a h s. O ‘5’ I Q 2 b .' — _ ‘r _ ‘ ‘ : v . o J— 1 L l 1323 ”2‘ 1523 '52! 1527 Energy (mev) .FIOJ. Demytlmecouaatsdphotolamineecencevsemh. annuagimintheemiminnbaaddtheheavy-boleescttn. lentil-mom. The asterisks remnant the experimental mea- suredn (lowervalumlaedtnghervalu-l. Thesolidlined niaaleast-snaarmdttingheetextl.Theselidlinernrnisthe calalatioarults. peremrmacitensarenotoalyoecupyingtheboundstatm alloweuenergies. betalsohigherd'neociatedstates (ion- ized excitons or free carriers) due to thermal ionization. Here. we concentrate on the behavior of the free exm'toa decay after (4 '0. Which is about 500 p aher laserexcita- tioa. ltisltnown thatbotcarriersinitiallygeneratedby above band-gap excitation relax to the bottom of the sub- band and thermally distributed to generate exciton within about zoom: [.8 K.”andthusoursystemisin thermal equilibrium at :4 -0. Under the above considera- tion. the rate equations for the exciton populations in the bound and the continuum states at delay time t, > 0 can be written as dn. 4: m. UN. +an. dm 7‘— - - ylrflf-DRI+UR. , where n. and m denote. respectively. the exciton popula- tion in the bound state (13 ground state) and the free- carrier population in the conduction band. 7, and r; are the recombination rates of excitons and free carriers. U (D) is the rate of exciton transfer from the bound (con- tinuum) to the continuum (bound) states. in writing Eq. (2). we have included all other excited states into continu- um statee since experimentally observed transition from excited ZS exciton state is corresponding to the continuum edge and the transfer time of the continuum to excited states is expected to be negligible. Efl'ect due to impurity-bound excitons or biexcitons have been neglected since their populations are very small in our case. The solution of Eq. (2) is an exact two-exponential form of Eq. (2) . fl DYNAMICS OF EXCITON TRANSFER BETWEEN THE BOUND . . . (”with FIT}!‘1'{7e+7f+D+U:[(70+U-7/-D)z Ham“). (3) In order to compare with experimental results. we need to know two more quantities among four unknowns. 7,. r]. U. and D. Exciton transfer rates U and D can be calculat- ed by using quantum theory and consider that the exciton transfer process is assisted by emrssion or absorption of acousticphonons. DandUthuscanbewrittenas . o-(wn)§§£ lMI’f(ki)In(q)+ I] Q 3‘80. firemen ‘Ez‘ha) . (4) U‘(2flh)§§§ lul’ftk;)e(q) 8801-1. +q)3(£z‘£i+ho) . I-Ierell'uthematrixelementdependingontheinterac- dutch)-Ieap0te(q)/kT-lll" istheprobabilityof dndinganacousticphonoeofwavevectorqattempera- tureT.sndf(ht)theprobabilityoftheinitialesciton statebeingoocupiedhaabeenamumedtobeafloltzmann dhrihutioa. EiandEzaretheexcitonenergyandkiaud kzarethetwo-dimensionalwavevectorsofexcitonsinthe planeper'pendiculartothegrowthsax'nintheinitialand haslstatmuupectinly. Hesetheboundsatepresents initial (dual) state for MB). The detailed calculation procedure will be published in a forthcoming paper." However.animportantresultobtainedisthatDandUat-e relatedbythefollowingexprusion: D-UexpuE/kT) . (5) whereAEistheenergydil’erencebetwsenthecontinuum andtheboundstates. whichisthebindingenergyofexci- tonta. Withanapproximationofthematrixelementll being a constant and independent of k. “-1.”. from Eqs. (3)-(5). we can calculate r. and rs. The key factorwhichcauses rztobestronglydependent onemiasionenergyistheemission-energydependenceof the exciton binding energy A5( -5.) as a result of inter- face roughnesses. The observed exciton emission linewidthcorrespondstoawellwidth fluctuationofabout zolttzsoknwmuutmximtionora linesrrelationfortsasafunctionofwellthicknessl. Esq.) -ELILn) 'CQ'LQ). Wm Lo-Zso A ll 1h. average well thickness and e is about 0.0I2 meV/A in this regioaofL." Wealsouseanexpression. ri-te -fi(L -Lo). to account for the changing of the radiative recombination rate with L. where s-somo" its/A and to -0.85 as. which are consistent with those deduced from the exciton transition peak position shift with delay time.“ However. taking r. as a constant will not alter the behaviorof r1. Strongdependenceof rzonernissionener- gies is a direct consequence that the transfer rates 0 and 114 1295! U are functions of exciton binding energies. E,. which de- pend on emrsston energies as a result of interface rough- nets. The recombination rate of free carriers 'CXClton in the continuumstates) 7/ is assumed to be independent of well thickness in this region. r: as a funcnon of emission energy has been calculated and the result is piotted as a solid line in Fig. 3. which is in good agreement with up". mental data. The best fitting between experimental data and calculation Yields 7/" -20 as consistent With the value obtained preViously” and the matrix element M ‘lf39‘ 10 " meV. Based on these results. the physio cal origin of the two exponential decays have been com- pletely resolved. The fast decay rate is mainly due to the radiative recombination of excitons. The slow decay coni- ponentisdeterminedbytherecombinamnrateoffree carrisssandtheratesofexcitoatransferbetweenthe bound and the continuum states. AtL-250 A. instruct rathandDare 7.52Xl0‘ and 2.0st0‘ s". respectively. Although the transfer rateofDisJordsrsofmaintudelargsrthan thatofU. theabeolutenumbsrsoftxm‘toastramferred betweenthe bound (continua) and the continuum (bound) states are onmpetibhsineethepopulationofexcitonainthebound state(n.)isabnutJordsrsofmagnitudelargerthanthees inthecontinuumstatehflinourcanashowninthe in-tdFigJ. IntbimmdFiglweplotthevariatinns ofe,andn,/e. asl'unctiomofdelaytimeu. Hm~+flf hmbeennormaliaedtounityattg-O. Weseethattha relativepopulationofexcitonsinthecontinuumstate (freecarriers) increaamwiththeincreaseofu. Thisis causedbythe fact thatthetranaferrathandD.aswell as the free-carrier recombination rate. are muchsmaller than the exciton radiative recombination rate in the bound state. which also leav- rt being hardly aflected by the ex- citon transfer between the bound and the continuum states. Another point we want to indicate is that the pop- ulation of excitons in the bound and the continuum states are in thermal equilibrium only at f4 '0 due to the transfer rate being smaller than the radiative decay rate in the bound state. The discrepancy between the calculated and experimental results of r: in the high-energy side of Fig. 3 is caused by the luminescence intensity overlap be. tween the heavy- and light-hole exciton transitions. The absence of the transition line from the free carriers. espe- cially at high temperatures. may be due to the free carrier recombination rate being much smaller than the transfer rate 0. yet remains to be investigated. In conclusion. the origin of two-exponential decay of ex- citonic transition in Gus-Alfie. -.As MQW‘s has been investigated. Our results demonstrate that the two- exponential decay of exciton transition is a direct conse- quence of the exciton transfer between the bound and con- tinuum states (free carriers) via acoustic phonon-exciton (phonon-free carriers) interaction. The recombination rate of free carriers as well as exciton transfer rates are obtained. 11952 'Presentaddr‘fl DepartmentofPhysiaandAstronomyand Center for Fundamental Materials Research. Michigan State University. East Lansing. MI 488244 1 l6. 'L. Esth. J. Phys. (Paris) Colloq. 4|. CS-I (I987). zILCI..\»lillet'.D.AKleinman.W.A.lilortllalid.andA.C.Gus» turd. Phys. Rev. D as. “I (I9IO). J.l.l-‘eldman.(i.l’ster.E.O.(‘iolisl.P.Dawsou.K.Moors.C. Foul. and R. I. Enha- Phyt Ru. Lett. 99. 2337 (I987). ‘l.Chr&n.D.Bhberg.A.Steekenboru.andG.Weimann. Aw. PhytLetLKuU9M). ’MKthD.Heitmann.S.Taruchs.K.Lee.andK.Ploog. Phys. Rev. I 39. 7736 (I939). "P.Zhee.fl.x.llang.ll.lanewart.8.A.Solia.endG.ld. MM.IC.II“Z (I939). 7H.X..IiangadJ.Y.LILLAflPhyaflJflflfl‘l). 'lCMillm.D.A.Klsiamaa.A.C.Gnmard.sndO.Mua— “PM".IEWS (I992). EMTSteiamhtLWfihsweltltm LY.G.I‘IMMM.I‘JSI3(I9IS). "CVMlMACMaflW.W~I-. “newsrooms”. 115 III. X. HANG. E. X. PING. P. 2801!. AND J. Y. LIN 11 ”G. W. Hooft. w. A. J. A. van 4.: Feel. L. w. , and c. r. Foxon. Phys. Rev. 3 35. mi (19m I-w. l-l. Knox. R. L Fork. M. c. poem. 0. A. a. wine. 3 Chemla. c. v. Shank. a c. Gossard. and w. an.“ 3 ”Sb: 23152"? s'éfl’” a. an" . . . . la.D..I. 'IenW.P.Ws A.C.Gaaard.andW.T.Tsan A M” “(Hui " ”L PM I“ ‘I‘ "9 W.PickinandJ.P.ILDa' A Ph' . (1990). M "L n Lm 5‘ '6' "I. C..Mlller. D. A. Woman. W. T. Tsang. and A. C. cm. «Phys. Rev. are use (19"). "P.Dswsee.LJ.htoore.G.Deggaa.l-l.l.lalph.andc.r ‘ Fosoml’hyalevJJdmilO“). ' '7J.Kumno.Y.Segawe.Y.Aoyagi.ead3.Namba.Phya|t. seamstress). ' "LXPlagadI-LXIiangkewblhhd). "I'JJhagJeIISussCmQJflflml Nammnacs-uuomuchhyu. asassot (tats). ' PHYSICAL REVIEW 8 VOLUME 40. NUMBER 17 116 1: DECEMBER 339.; Excitonic tt’ansitions in GaAs-Al, (2321,.x As multiple quantum wells aflected by interface roughness P. Zhou Center for Fundamental Materials Research and Department o/‘thrtcs and Astronomy. Michigan State Critter-rm. East Lanstng, Michigan 48824-l116 H. X. liang Deparatment afPhystcs. Candice“ Hall. Kansas State University. Manhattan. Kansas 66506 and Canterfor Fundamental .Ilatertals Research. Michigan State University, East Lansing. Michigan 68824-1116 R. Bannwart‘ and S. A. Solin Center for Fundamental Materials Research and Department of Physics and Astronomy, Michigan State University. East Lansing, .lft'clngan daze-UM G. Bai Dles'ss‘anoftngt‘neert‘ngand AppliedSct‘ence. Callfimu'a fudtuteofl'ecltnolno, Pasadena. Callfisrnt‘a 91125 (Received l9April l9b9tteviaedmanimcriptrecmvedllluly I989) nwmphaoluminmbmbemmedmuudythedecmdintmfacemughn-oau- citonsc mooninGaAs-Al.6ai-.As multiplequantumwella. Insdditiontotheluminmcence wmmhgandSmkmmdeeinurfacemughn-ahomudyammedynamm pmcemofopucalaanunomiothuthaexcimnien-anmoonpakshihswithddaydmc However. thehnvy-holeexct'toumononuredshtltsetshortdelayummandexhibitsamnovcatlonger delaytimm. AmammumshiftofaboutMmeVatadelsyumeofenswmobtained. Wehave mmmmmnmwmmmummw Further- monthedecayoftheexmmntctrannuonufoundtodtatwo-eaponennslform. Busdonamodel mwlnngmurfacemughnmsmdmnxponmufldemy,nalcuaudmepoduoadtheeammnic transition peak a a function of delay time. Our calculaums are consistent with experimental re- sults. I. INTIODUCI'ION Recently. quantumowell (QW) and superlattice (SL) structures have attracted a great deal of attention because of their novel properties.‘ " Since the proposal of these exciting new structures," ‘they have been studied exten- sively. and thus many important features have been discovered. For fundam-tal physics. quantum-well and superlattice structures have been used to explore the physical properties of a whole new field of low- phenomena in the quantum regime have been discovered. such as resonant tunneling of double-barrier quantum ' wells vn'tli negative dilerential resistance. 'By separat- ing the. impurities with charge carriers by modulation doping, a significant mobility enhancement in GaAso AI Ge(-,As quantum wells has been achieved.’ ‘° Ap- plications of these quantum wells and superlatticcs in- clude high-speed electronics. optoelectronics and photon- ic devices. such as quantum-well lasers.” ‘2 modulation- doped deld-ed’ect transistors” moor-11'). photodetec- tors. etc. Although much work has been done in this field. there are only a few investigations that concentrate on the dy- $0 namic processes of excitonic transitions in QW and SI. The lifetimes of excitons in GaAs-Al Ga,-,As quantum wells were first reported by Christen er al. " An increase of almost one order of magnitude in the transition rate of excitonic recombination in GaAs quantum wells with a well width of 52 A compared with bulk GaAs has been observed. They also found that the decay of the excitonic recombination is nonexponeritial. Molecular beam epitaxy (MBE) techniques can be used to grow QW and SI. with very high quality but roughness at the interfaces of two materials of QW and SI. can still not be completely eliminated It is of interest and impor- tance to know how this roughness affects the optical pro- ceases in QW‘s. The interface roughness (or interface de- fects) in QW's has been studied previously by low- temperature continuous-wave (cw) photoluminescence and by photolumincscerice excitation spectroscopy. “‘7 The main effects so far observed due to the interface roughness were exciton linewidth broadening and the red shift (Stokes shift) of the emission: the emission of the lowest heavy-hole exciton generally is slightly shifted to low energy (typically a few meV) with respect to the ab- sorption or excitation spectrum maximum. This was at- tributed to localization of excitons within potential fluc- II 862 ©1989 The American Physical Society i0. EXCITONIC TRANSITIONS {N GaAs-Al.Gai-.As . . . tuations due to interface roughness. The cw lumines- cence and phOtoluminescence excitation spectroscopy used to date to study the interface roughness of the QW yields very little understanding about the role of the in- teri‘ace roughness in the dynamic process of Optical tran- swam. in this paper we investigate the etl’ects of interface roughness on the dynamic process of optical transitions. Low-temperature time-resolved photolumrnescence of GaAs-Al. Ga.-.» multiple QW‘s has been studied. The peak positions of heavy-hole exciton luminescence shift- ing with delay time has been observed. The shift is ac- counted for by interface roughness. A calculation based on interface roughnms and a two-exponential decay forms fits the experimental results very well. 1mm ThessmpleusadforthisstudywasaGaAs- AluGauAs multiple quantum well (MQW) which was grownbymolecularbeamepitaxyonaGaAsllmlsub- stratawithoutgrowthinterruption. ltconsistsofalter- nate 275-A AluGauAs barrier layers and 2:0-A well layerswithatotaloftenperiods. Thex-raydidraction detaindicateaperiodlengthof52SA.whichisingood agreement with the design parameters. Escitationpulsesofabout7psindurationatarepeti- tion rate all MHz were provided by a cavity-dumped ul- tral’ast dye laser (Coherent 70229) with an average power of lo mW. which was pumped by a yttrium alumi- num garnet (YAG) laser (Quantronix 416) with a frequen- cy doubler. The pulse duration was continually moni- tored by using a rapid-scan autocorreletor." The Lens- ing photon energy was 2.125 W with a spectral width of 2 meV. A time-correlated single-photon counting system with a double monochromator (Jarell Ash 25400) and a computer were used for the measurements. The ed’ective time resolution of the system is about 0.2 as. The sample was mounted strain free inside a closed-cycle Ha refri- geratorandmaintainedatatemperatureofSJ K. III. RESULTS AND DISCUSSION Experimental results of low-temperature (SJ-K) time- resolved photoluminescence at three difcrent delay times for a GaAs-AluGaQw MQW are plotted in Fig. l. The exciting photon energy was 2.125 eV at an average power density of ~50 mW/cm’. The luminescence at dilercnt delay times has been rescaled for presentation. The peaks at [.3252 and l.5295 eV are ascribed. respec- tively. to transitions of is heavy-hole (n- l. e-HH) and light-hole (n- l. e-LH) excitons. which are composed of an electron and a heavy (light) hole belonging to the lowest state (n-l) in the QW. These values are con- sistent with those calculated by using the transfer-matrix method" so with the conduction-band od'set parameter being 0. 63 and the binding energies of heavy- and light- hole excitons being 6.0 and 6.2 meV. respectively. The observed spectral width (about 1.78 meVl is attributed to the interface roughness of the QW. '°'” In Fig. I there is a shoulder at about 1 meV below the heavy-hole exciton 117 ll 363 . 7-6 5 K a 1'00 3.1 1'22 3: ° 1-55 :is Lumines. liilciisily (nrb iiiiils) fi'r‘r'rfl ‘1 l I I‘ I s l I s I " {’7'} __e.._s _ I h ' l I .2 l 5225 1.525l. 5275 1.53 1.5325 E (all) FIG. l. W (8.3 K) time-resolved photo- luminmesncaatthrasdidhrentdelaytimmforGaAs- AluGauAs mulnplaquantnmwellswiththicknesamofthe wmmmmmmnmniy. Theescita- nonenergywuLlZSveidiaaavetagspowes-densityabout SOmVl/cm‘. 3F peakwhichhassisobesnreportedbyMillerandco- workers." 3 Based on the excitation internity. tempera- ture. and polarization dependencies of this low-energy peak (shoulder here). they concluded that it was due to a biexciton transition with a binding energy of about 1 meV. By using excitation-intensity-depardent lumines- cence and time-resolved spectroscopy. Charbonnesu and coo-workers” recently showed that the lower-energy com- ponents of heavy-hole excitonic transitions have diluent origins in dill’erent samples and can be attributed either to biexcitons or to impurity-bound excitons. One important feature depicted in Fig. l is that the ex- citon transition pealt shifts toward lower energy as the delay time increases. This is manifested as the intensity ratio of two data points at the exciton luminescence max. imum changing with increasing delay time. Here. delay times were measured from the end of the excitation pulses. The origin of this shift cannot be correlated with the filling state phenomena“'” because of the temporal behavior as will be discussed later. In order to fully explore the fact that the exciton tran- sition peak shifts with delay times. we have performed a high spectral resolution experiment around the peak posi- tion of the heavy-hole exciton transition. The results are shown in Fig. 2. where all the parameters and experimen— tal conditions are the same as those in Fig. l. The solid lines in Figs. 1 and 2 are a guide to the eye. Figure 2 clearly demonstrates that the peak position of the heavy- hole exciton transition shifts toward lower energy with increasing delay times. We have used a least-squares lit for the experimental data to find the peak positions at different delay times. The effect of the lower energy 118 1136‘ -- p q...- ..- 1: 1‘ 1 ‘~. ‘. - Z 'A I E .. .t"’ ‘V\ \\‘~ ‘ 3 - I. e \ . . _. gt’ \ ‘e _. a . it / a-\ \.. \‘. 1 h u ,7 ~‘ ‘-. . e I b // n 0 “\ . V V” 1, \\ ‘0 ‘ . Q t“ ,1” °\. \. 1. so ‘. a’ . c t’ 1 \. 0 l- ‘i. ' d \‘ .5 t‘ . t-0.0n.s '\ "‘_ g l: o t-2.2na \3 .. 3 .5 0 £36..” —-a E l7 4. 3. .. i F_-. (.m-mLa.._‘l l 5245 1.525 1.5255 1.525 3 (all) FIG. 2. l-llfl recitation time-reolved photoluminecenca thpehpomtionofthehevy-holeecimatran- sideaatthrsedilerentdalaytimel'he slitwidthand ”WWW.OAIMOJA. Otherparams andsapmimntslcondidonsanthesameasthoseinfigl. shoulder on the peak position of the heavy-hole exciton transition have been eliminated by deconvolution. This elect is shown to be negligible. Only a few data points around the maximum intensity of the excitonic transition were suficient to obtain the peak position at diferens de lay time. The reults for the heavy-hole exciton peak position at dilernt delay time are shown in Fig. 3 (cir- cles). FromFig.3 weseethatthemaximumshihisonly about 0.1 mall at about 4 as. Another feature is that the 1525.30 - , , I. | L J : r-e.ss ‘ _ 0 Data 1 .. 1525.25 +- "'"‘“"'7 q % r . E " 1 is - i I E h o 4 “3 1525.20 - - i- C 1 1525.15 ' 1 ' ' -2 O 2 4 5 5 Td (ris) FIG. 3. Energy position of the emission intensity maximum of heavy-hole exciton as a function of delay time for GaAs- Alg,Ga..,As multiple quantum wells. The quantum well pa- rameters are the same as those in Fig. l. ZHOU. HANG. BANNWART. SOLIN. AND 8A1 :3 peak positions shows .i red shift at the short deiav time 1.0—; n31. and then turns over oeyond $ ris. One comes that the shift at short :leiay times 15 almost linear. The amount of shift is smail: neverthezess. it contains very .m- portant information. The experimental results shown :n Fig. 3 can be interpreted in terms of the interface rougn- ness in QW's and .i :wo-exponential decay of the excrton. to transition. Figure 4 presents _a schematic diagram of the :ntert‘ace roughness in a QW along the growth axis '3). and :lie concomitant erfect on the photoluminescencc linewidth broadening lb) and exciton lifetime (c). The fluctuation in well (barrier) thicknes can only be integral multiple of one monolayer. 1n the quantum-well layer plane. there are domains formed by the diflerent well thicknesses with size varying from a few hundred angstroms to a few mi. crometers. Difl‘erent emission energie around the pane- ple excitonic peak con-epond to excitons recombined in dilerent spatial domains. As we have shown in Fig. s, line A repreents the lower photon energy which corre sponds to excitons mombined at the location of a wider quantum well while line B repreents the higher photon energy which corresponds to the excitons recombined at thelocationofanarrowerquantumwell. lnFig.4lb) Windicatethefulllinewidthatthehalfmaximum. There was no investigation carried out previously ad- (6) l/ W "§\\' (In) _ me | I ' I (a) 1 . ; re TI FIG. 4. A schematic diagram of the interface roughness in J quantum well shown along the growth axis (a). lb) and :ci shou effects of roughness on the linewidth broadening and excitor lifetime. £9. EXCX'TONIC TRANSITIONS [N GaAs-Al,Ga.-.As . . . dressing the fact that excitons recombined at line .4 will also have a larger lifetime than those recombined .it '.ine B. Thus near the excztonic transition intensity max- imum. the lifetime of the exciton. r. Will have a depeno dence on emusion energy E. which can be written :is 715). Because the well fluctuation normally is within a few monolayers the change in lifetime of excitons around the perk position is small. This is probably the reason that this erl’ect has been preViously negieCted; However. the change of exciton lifetime with respect to the emis- sion energy E. dr/dE. can provide very important physi- cal information. since the recombination rate of the exci- ton depends on the QW thickness. “'37 The larger the quantum well width. the longer the exciton lifetime. as indicated in Fig. Me). First. let us prove that energy-dependent exciton life- times can cause the transition peak to shin with the delay time. Becausethetotalamountofpcakshifiisverysmall (0.1 itself) we can write r as a function of E. near the en- ergy ofmasimum intensity Ea. as flE)-flzo)+(E-Eo)ai (I) here 11qu is the lifetime of excitons measured at the en- ergy of maximum intensity of delay time i, =0 and a is the lifetime change rate with repent to energy. dr/dEI‘G. Assuming the line-shape intensity distribu- tion around the peak is a Gaussian with a single exponen- tial decay. the luminecence of the excitonic transition . can be written as a function of delay time t, and energy E as (E-Eo)z t‘ IlEng'Ioexp -T-m . (2) Here to and £0 are the maximum intensity of the exci- tonic transition and the energy position of the intensity peak at delay time t. II0. repectively. while 0 define the ‘ linewidth which correlate the QW thicknes fluctuation parameter.“ In Eq. (2). foexp(-(E-Eo)‘/a"'] repre sents the Gaussian intensity distribution at t, II0. and exp( ‘f‘ /r(£)] is the decay factor which depends on en- ergy. The peak positions at did'erent delay time can be obtained by setting dIlE.t,) -0 (3) 45 ’ which give 2 dr E (i )-E +——‘1——i . (4) mas ‘ 0 “115)? d5 1 Here we have assumed that the linewidth of the excitonic transition is independent of the delay time. which is con- sistent with the experimental results shown in Fig. 1. Be- cause the second term in Eq. (1) is much smaller than the first. we obtain from Eq. (4) _J 119 ll 3.65 7' it- -; '1‘ Eli Enos (1:35,) _ at, . 5, Equation '5‘; indicates that the peak posmon of the exci- tonic transition will shift linearly to lower energies as the delay time increases. since aaarxdE =* d.‘ 'a'L./ ltrE/dl.) <0. i.e.. dr/dL>O and dE/df. <0. Here L is the average thickness of the QW‘s. The same calculation shows that the peak position of the excitonic transition will shift as the delay time in- crease because of the preence of interface roughnes in the QW's. However. Eq. (5) only give a linear red shift ot peak position with the delay time. which is incon- sistent with experimental observation depicted in Fig. 3. The experimental reults are more complicated and can- not be fully described by Eq. (3). As we will see. by using the above treatment with the assumption of a two- exponential decay for luminescence. the experimental re sults can be well accounted for. This is obtained by rewriting the luminescence intensity as a function of emusionenergyEanddelsytimestguroundthepeak position. as it£.:,i-ap(-i£-Eoi‘/a*] x l .4 exp( -t‘/r‘(E)] +BCXPI-54/ffls)" s (6) where AandBareconstants. Karenandrgaretime constants which are both functions of energy E. From Eq. ('3) we get ,1 flEJ‘) ‘ Ew(:d)=Eg‘—2-m, (7) with 2 Ct dr, t f(E.t‘)- I?! #7 -d—E'- lexp [- 13:5) I (7a) and glE.t‘)- é Csexp [- 1 1, (7b) .M l fllE) where l (isli q- B/A (i=2). ‘7" One notice that C , IB/A is the ratio of two lumine- ccnce components at t, '0. For energies near that of the excitonic transition peak. we can write r,(El'r,(Eo}-lE -Eo)a, (t' ‘1.2). (8) By using Eqs. (7). (71). (7b). and (8) we obtain 2 , 2 [C,a,t, /r,z(E°)]cxp[ -t‘ /r,(Eo)] Emiidiaso-i-Zzl-m z 2 C,exp[-t‘/r,lE.,)] i9) 120 11366 Parameters a. a. .and C : can be deduced from ex~ perimental measurements. One sees that a is related to the FWHM by FWHM- -(lnl:” 5.7. The value of FWHM obtained from Fig.1 is about 1.78 meV, which gives ail-1.07 meV. miEq) (~03 nsi is the time con- stant of the fast decay component of exciton lumine- cence measured at the energy of the intensity maximum occurring at t, '1). which correponds to the exciton life- time of radiative recombination. rzl Ex) is the time con- stant ot' the slow decay component. which is measured to be 4.2 as. One notice that this time constant doe not repreent the lifetime of exciton recombination since it is even larger than the exciton lifetime in bulk GaAs (3. 3 its). (The physical origin for this two-exponential decay is under invengation.) Now C3 is measured to be 0.07. We insert thee value into Eq. (9) and adjust a, and a, to obtain the least-square ht with experimental data. We find that. with a, - -0.06 ns/meV and a:--0.42 na/meV. the plot of Eq. (9) shown as the solid line in Fig. 3 is in excellent agreement with experimental reults (cir- cle). Here n, represents the change rate of exciton life- timewithrespecttoemissionenergy. and thus thedtting value ( -0.06 us/meV) give a total lifetime change of the fast decay component of about 0.06X 1.8-0.1 (as) within the FWHM. However. as we will show later. one does not need to measure the exciton lifetime of QW‘s with dilerent thicknese in order to obtain the change rate of exciton lifetime with respect to well thickness. dr/dL. A much easier way to obtain dr/dL is to measure the peak position shift as a function of delay time. We studied the temporal response of the luminecence attheexcitonictransitionpeahasshowninFig.5. The circles are the experimental values which have been deconvoluted to account for the temporal reponse of the Wfivf ee.. v -ee Ft “000' T-OS‘ 1 40000? otso OIL: d - - — One-exponential — Two-exponential . Luminea. Intensity 3000 '- 2000 ‘ * 0 Td (ns) FIG. 5. Luminescence of heavy-hole exciton as a function of delay time. The circle show the measured value while the dashed and solid lines are fit using one- and two-exponential de~ cay models. respectively (see text). ZHOL'. HANG. BANNWART. SOLXN. AND BAl fl detection system. The rising part of the luminecence :3 net shown here. The dashed and sciid line are the theoretical fit Obtained using one- and two-exponential decay models. respectively. From Fig.‘ . it is :. ear that the decay ot heavy-hole excitons cannOt be described by 1 single exponential f.orm The two-exponential decay tits the experimental results very well. The decay time con- stants ootained from Fig. 5 are 0.9 and 4. 2 as. repective- ly. for the fast and slow components. This two- exponential decay behaVior has been observed recently by Other groups.‘0 Following the discussions presented above. 4 ME. the time-constant change rate with repect to emission ener- gie E obtained from the peak shift. can be used to deduce the lifetime of the exciton in QW‘s of diflerent well thickness. since we knew from previous work that thelifeumeoftheexcttonincrsasealmostlinearlywith well thicknes.” From a-dr/dE-ldr/dLl/ (dE/dL). wehave d_f- a“ 4" Oz. (10) Here E -E,(GaAs)+E.+E. -E.. . (11) withE,lGaAs)beingtheenergygapofGaAsmaterial. E, (E.) isthecondnementenergyoftheelscn'onlholelin theQW which ismeasuredfromthe bottom ofthecon- ductionltopofthevalencelband. E... isthebindingen- ergy of the exciton. By noting that the change in the binding energy of the exciton with respect to well thick- nee L. dEm/dL. is much smaller than the change in confinementenergyoftheelectronorhohwecan neglect 4E,“ /dL. in addition. the confinement energies of the ground-state electron and hole (of the order of 10 meV for our sample: are much les than those of the con- duction and valence posential barriers (a few hundred meV here). Thus we can etimate E, and E. by treating electrons and hole as particle bound inside an infinitely deep QW. Therefore we have «114 #373 iii RUE, Tin-2F m- 271-1. (12) Here I. is the average well thicknes and u'Imfni; (m,‘+m.,') is the etl’ective reduced mass of the exciton. From Eqs. (10)-(12) we obtain dr 51.13 2 -- " - ——-.- -n . dL ’ .p'L - L We used in ' '0. 067m, and m '.-0 45m for the eleco tron and hole ed‘ective masse inside the QW. ’3 3‘ where m is the chain mass of electron in the free space. With well thicknes parameters of L I230 A and - -0. 06 ns/meV obtained in the above we find iii-2.6x 10" its/A. (13) This value is in good agreement with those extrapolat- ed from the experimental reults of Ref. 26. i. e.. ~ 2 .5 x 10"1 ns/A. Here we want to indicate that in Ref. 121 4_0 EXCITONIC TRANSITIONS 1N GaAs-Al.Ga. -.As . . :6 the authors defined the time at which the intensity has dropped to in of its maximum value. r.,,. as the lifetime of the exciton. The real lifetime of the exciton radiative recombination will be did'erent from this value. Never- theles. dr/dL deduced from their experimental results will be very close to the rel value. Since exciton lifetime is approximately proportional to well thicknes. one can deduce the exciton lifetime for difiererit well thicknescs by mesunng :- and dr/dL for one quantum-well thick- nes. IV. CONCLUSIONS We have studied the elect of interface roughnes on the time-resolved photoluminecence of the excitonic transition of GaAs-Al,Ga,-,As multiple QW’s. We found that the energy position of intensity maximum of the excitonic transition shifted as the delay time in- creased. Based on a model involved interface roughnes in the QW‘s and the form of a two-expoa-nal decay for luminescencewecalcnlatedthepeakpositionasafunc- tionofdelaytimes. Thecalculatedreultsareingood agreemsm with experimental observations. From thee measurements. we obtained the change rate in exciton lifetime with respect to the well thicknes L. We showed that this is a very etTective method to measure dr/dL compared with the usual procedure of measuring exciton 11367 lifetime at difl'erent well thickneses direCtly. From the above discussions we see that besides :he Linewiam broadening and Stones :ed Shut. the :ntert‘ace roughness also strongly atl'ects the dynamic orocess of the optical transition. The reults obtained here are very :moortant tor tundamentai reearch and also very useful z’orlaracti- cal applications. sucn as narrowing the spectral and time pulse widths of QW lasers. ObViously. the etl’ect of QW thickness fluctuations on the dynamic proces of optical transitions is very impor- tant. There are still many important quetions to be answered. One of the most important quetions here is why the decay of the exciton transition is composed of a twocexponential form. In future work. we will study the dynamic proces of the optical transition by using QW‘s with difl‘erent well thicknese and at diflerent tempera- ture. We hope that the studie will provide more un- derstanding of the complicated elects of interface rough- nes. MW Thereearchreportedherewsssupportedbythe Michigan State University Center for Fundamental Ma- terials Reearch. H.J(J. would like to acknowledge use ful discussions with I. Y. Lin. 'Presnt sddrem: Solid State Laboratorim. 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Tarucha. and H. 1. Ohm Phys. Soc. Jpn. ss. 37 (19361. 2’14. WW F. Henneberg. V. Women. 1. N. Uraltsev. and H. G. Bruhl. Phys. Scr. 35. 76 (1907). 2‘R. Hoses. E. O. Gobel. I. Kuhl. K. Ploog. and G. Weimann. in Proceedings of the Seventeenth International Conference on the Physie ofSernicanducror-s. San Francisco. 1984. edited by I. D. Chadi and W. A. Harrison (Spnnger-Verlag. New York. i9ssi. p. 573. ”I. Feldmann. G. Peter. E. O. Gobel. P. Dawson. K. Moore. C. Foxon. and R. 1. Elliott. Phys. Rev. Lett. 59. 2337 (1987). ”H. X. Jiang and I. Y. Lin. 1. Appl. Phys. 63. 1984 (1988). 2’G. W. Hooft. W. A. I. A. van dcr Poel. L. W. Molenkamp. and C. T. Foxon. Phys. Rev. B 35. 8231 (1987). 3°]. Christen. D. Bimberg. A. Steckenborn. and G. Weimann. Appl. Phys. Lett. 44. 34 (1986). 3‘1’. Dawson. G. Duggan. 1'1. 1. Ralph. and K. Woodbridge. in Proceedings of the Seventeenth International Conference on Semiconductors. San Francisco. 1984. edited by I. D. Chadi and W. A. Harrison Springer. New York. 1985). p. 551. ”M. Kohl. D. Heitmann. S. Tarucha. K. Leo. and K. Ploog. Phys. Rev. B 39. 7736 (19391. 3’c. Mailhiot. Y. c. Chang. and 1'. c. 51:01". Phys. Rev. 3 as. 4449 (1982). 1‘C. Priestcr. G. Allan. and M. Lannoo. Phys. Rev. B 30. 7302 (19841. Q 122 PHYSICAL REVIEW 3 VOLUME 39. NUMBER 1 '. JANUARY L9,” lnterfacial effects and superconductivity in high- T. materials S. Vieira.' P. Zhou. Center for Fundamental Materials Researcn. and S. A. Soiin Department of Phi-sic: :no Astronomy. .llicriignn State University. East Lansing, .Wicnigan 488:: N. Garcia. M. Hortal. and A. Aguiio Deportmento Fistco .llnterto Condensoda. (.‘nit-erriaad .sutonorno llodrtd. Madrid .‘3049 Spain (Received 2 March (.988; revised manuscript received 20 July (988) A series of quasihomogeneous superconducting powders of YBaiCui01-s have been prepared by sievmg and characterized by electron microscopy and x-ray dirTraction. Quantitative sizeo dependent and temperature-dependent magnetic levitation and susceptibility measurements of random powders and of field-oriented (at T < T. -92 K) small grains ( < 20 um) confirm previ- ously observed symmetry-breaking effects associated with the twin boundary planes and suggest that the bulk anisotropic superconductivity is measurably perturbed by interi‘acial etl'ects. Current models of intert‘acial superconductivity in high-T. materials are evaluated in light of these and other results and are found to be lacking. It is commonly assumed that YbazCu;Os-, is a pure built anisotropic superconductor": (BAS) whose proper- ties are influenced by the (110) twin boundary planes (TP's) which are a natural consequence of its orthorhom- bic structure’ and act as “weak links" between BAS re- gions. But the only theoretical model to date which quan- titatively relates the bulk superconductivity to TP's is the superconducting glass model.z This model is controversial and the subject of serious challenge. ‘ Recently. Garcia et al.’ suggested that the supercon- ductivity in YbazCuJOi -. is a nonouilt TP eti’ect. On the basis of scanning tunneling microscopy (STM). ac suscep- tibility, and qualitative levitation measurements they identified the TP‘s as "strong links" and attributed the su- perconductivity to a purely inter/betel Allender-Bray- Bardeen (ABB) excitonic mechanism’ which would func- tion for all temperatures below 7'... While x-ray studies of oriented small-grained powders’ showed symmetry- brealting supercurrent paths parallel to the TP’s in addis tion to those parallel to the basal (do CuO) planes. these studies were compatible with either a wea'xolinlt or strong-link origin. Subsequently. TP supercurrents have been associated with the hybrid phenomenon of twin- plane superconductin’ty’ (TPS) and with oxygen vacancy ordering‘ (0V0) both of which are activated at a temper- ature—T; < Tc below which the BAS mechanism obtains. (Nae-tut BAS is the underpinning of TPS and 0V0 neither of which can exist in pure form and both of which are strong-link models.) To date the TPS model has been employed to quantitatively account for the temperature dependence of the upper critical fields’ and of the specific heatm of YbazCu301-.. While the evidence for interfa- cial effects in the superconductivity of YbazCuiOs-. is suggestive it is not conclusive. Accordingly, we present in this article the tirst quantitative studies of the magnetic levitation and susceptibility of oriented powders of Yba1Cu301-,. These studies are apparently incompati- ble with the assumption of pure BAS but support a hybrid mechanism which partially localizes the supercurrents to g the neighborhood of the TP's. . Quasihomogeneous powders of YbazCuiO~-.. cw). were synthesized by the usual method” and Sieved into the following size distributions: d<20 um; 6-e 20 am also levitate in bath LN and LH at these same heights. All of the loose and compact random powders are superconducting with typical Mcissner fractions (mea- sured according to methods established by Krusin- Elbaum. Malozemorf. and Yeshurun") in excess of 50% as shown by the data of Fig. 4. The random powder data of Fig. 4 are typical of pure phase 1:2:3 material and show none of the features. e.g.. kinlts at T < T. which are nor- mally mated with impurity phases and/or reduced ox- ygenation. " Assume for the sake of analytic solutions that the YbazCu301-. particles are magnetically anisotropic pure BAS ellipsoids whose susceptibility and demagnetization tensors have coincident principal axes. A particle of volume V in an applied magnetic field Ho will adopt a cOnfiguration which minimizes U (Ref. 16) where . . . z. , L ‘fva'Hodu--yfv1;'mflif,. (I) Here. M is the magnetization of the particle: .V,. 1,. and ”0, are the demagnetizing factor. susceptibility. and pro- jection of the tield along the yth prinCipal .ixis. respective- IY: and SN, the The torque and force on the particle can be obtained from U. e.g.. .- -&L'.’30. F: Will, 8:. etc. Equation (1) can be used to establish the : dependence of the .evitation force when the free particle has oriented ~Ifl 31.".' ‘e a. (a 01. 3\\ (l1 l‘ .‘ _._, f N _ U \ J "1 , /"5‘\ (F (L- FlG. 2. A schematic diagram of the levitation experiment with axes and aegis labeled for reference in the text. The prin- cipal axes of the demagnetization tensor are labeled 1. Z. 3 ind are parallel. respectively. to the crystallagraphic axes c. a (or o). b (or a). itself to minimize U. Suppose a particle has descended into a region where its local field is less than the lowest critical field H.307 K) (II— Holldo planes). Then I, -- l/4n. j-l.2.3. and the term in Eq. (1) with minimum N, minimizes U. The particle will align with its longest principal axis (axis 2 in Fig. Z) ((110. If .V;<<4n or Ho, 2’ HA. the levitation condition is no) - .- V.)!l.(aH,(:)/dz) -(p-pLg~i)gV, (2) where M, -H./4n. p '6.38 glcmJ is the mass density of YbazCuyos-.. p... -0.6 g/cm’ is the densuy of liquid acceleration. nitrogen. and g is the gravitational s . ‘ao we J - 1 2 e H. C p i a“: e u- ‘i f ‘2. .35 S I \ d - (- x -e . = l 3 l- l ‘. ‘ e E J «.20 z ‘ a?- .' "i ‘. . S 3' Z 3 ‘3 .i ... i :5 . a I . i . e g a ‘ :- : C ' 3 ‘ .‘ - "' -'5 3 ~ I I J \ ~ . ‘ P Q 9 E‘Ih.’ IQ. 3 x : 1. :l \ 1 g tio :.4 .; _ _ = .: ‘1 .5...‘ .3 1‘ ‘5 x i ‘~ 3 z = L- “““ o .0 - 00 S to is 20 llfl'll'fll FIG. 3. The measured transverse field (circles) and a seven- parameter polynomiai tit (solid line) (see text). The gradient (dashed line) was obtained analytically from the polynomial it. The additional ordinate scale on the far right renormalizes the gradient to on'...(:)(.‘.‘ K) the maximum value of which .s indi- cated along with D'JLN. the buoyancy-corrected grayitational mass density of YBa2CaiOi-.. 124 336 VIEIRA. ZHOU. SOLIN. GARCIA. HORTAL. AND AGL’ILO 3 3 5 . 3 - ' . I 5 -. l i l O ' 2 - 9 g . Oh—T ] J 200 m 0- HlGeusei tn ; .. :2: . i _ _ : _ l -o.as-:1 C 2 - ~ g l g L o 1 a -o.e' ' ¢ 3 L . I 3 3")". e e a I I I ' o o 3 e l l o 0 ° ‘: e," -o.sl- l 2 l ° l -i.oi-e e co e ' g l ‘1.’ . l i I . ' ' L ~ ‘ 0 io . so so so so so 7o so so we run FIG. 4. The temperature dependence of the mass susceptibili- ty of YBasCusOs-g powder (d<20 am) in a random con- duration (circles) and held-oriented (see text) configuration (squares). Open (solid) symbols represent held cooling at l! G (zero-held cooling followed by held heating at 25 G). Inset. the held dependence of the S-K susceptibility of random powder (a) showing in. mm critical field HM: K) -300:25 6 (vertical arrow). Equivalently. if W define Pmsg(Z)-F: (2)/V: to be the magnetic mass density. the particle will levitate if When the local field excads the lower critical field H51. along one or more of the principal axes of the particle. it enters the vortex state. For YBa1Cu301-g. Hell -He’l -H:1>H¢'h Hell -Het )He‘l and 17:. < H}: < Hg: < H21. (Magnetic anisotropy in the ab plane. although expected.J has been observed' but not yet measured.) In the range 29 \ H1. < H). «HQ. M is approximately linear in H in sq.“ case D x, - -"1/4,r)(H_11:’H,). a, :4) "5) hen the levitation condition is given by Eq. (2) but “H the term H.(:I replaced by Héi. For Ho, >> H!” «1m: ', netizing erfects are negligible and the particle will align m a direction which minimizes flux expulsion. :.e.. along :3, axis with the minimum value of H5. in the language .3.- anisotropic Ginzburg-Landau theory ‘7 this is the direction corresponding to the smallest erTective mass and thus the highest ”5;. But the product of HA and H5; is roughly constant. i.e., (Hilffézfl’zaflc where H. is the thermal dynamic critical field. So the direction with maximal Hg, also has minimal Hf. as asserted above. We have used Eqs. (1)-(5) to determine the levitation force on and preferred orientation of a 3A5 particle. Die results are summarized in Table I. The values of (D.L.).m cited in that table were deduced from Fig. 2 us- ing H507 K) '90:! G. The latter was computed" from in: lower critical field. mils K) -3oo:25 o. a which leJ' '5 K) of a random powder linearly departs from constancy (see inset. Fig. 4). This departure com- mences when Hail”. for particles in the powder which happen to be aligned with their ob planes ilHo. Although the departure is in principle quadratic in applied field for single-crystal specimens or fully aligned widely spaced grains. its analytic :‘orm. which depends strongly on uncal- culable local-field corrections. is not known for powder specimens. The assumption of a linear departure which we used to compute H2. (5 K) from the inset data of Fig. 4 is thus reasonable and uncertainties which this assumption introduce are reflected in the large error bars on our mea- sured value of ”2.. Moreover. the above specified value of HA (5 K) is consistent with our measurements of oriented powders and lies at the low end of the range of reported values"" for that parameter. NOtice from Table i that a pure BAS analysis predicts that the particle will not levitate in the Meissner state be cause field limitations constrain p}... to values 4: ,2. How- H.’-H0j*(4v1/I4R)Hé| . 1.1/(1+.V,r,)-(-within/UH”) TABLE I. Orientation and levitation of a bulk anisotropic superconducting particle in a transverse field. H,(:) at 77 K. The symbols as. o. and n denote the Meissner. vortex. and normal states. respec- tively. and S - ‘1- .ll, V(dH,(z)/dr ). where V is the particle volume and M, .s the induced moment. Axis labels refer to Fig. 2. Applied Axis (93...)... State field orientation F. (dyn) y'cm‘ i‘ v‘ M H' H}! (l) UH, 0 0 n H. > H}; None 0 0 39 NTEi aracw. EFFECTS AND SUPERCONDUCTIVITY IN . . . «- ever. 'the particle should levitate in the vortex state for which (9.}...)..m> lip-pm). If flux is more ed'ectiveiy ”gelled from the TP's (of width t with spacing f.) than from the intervening regions (of width L - t) as would be the case in a strong-link hybrid model. (94...)... will be reduced by approximately t/ L. Then under the condition i/‘L < (p‘nL~)/(p.:...)m '022 small particles will not levitate at 77 K. as is observed. Nonlevitation is thus like- 1’ since the spread in reported values of t (Refs. 4 and 20) and L (Ref. 21) limit their ratio to gig d.. The height of a cell within a healing length i. of the A ion is also increased to 4.. A second A ion in this region does not afi'ect already expanded cells but expands unexpand- ed cells within 1 of its location. The process of random replacement of the 8 ions continues to saturation. II we define d(x) as the fraction of cells with height 4... then d.(.r) -o(x). The simulation results for d.(x) are shown in Fig. 3 for several difi’erent healing lengths. Clearly. in the floppy-layer limit h-O. a Vegard's-law behavior obtains whereas the initial slope Id.(.r)l'.-o -- c- as x-- a. As can be seen from Fig. 3. there is no percolation threshold even for finite x because d.(x) de- pends upon all oi‘ the large ions. n0t only on those be- longing to the infinite percolation cluster. Note that the nonlinearity in d.(x) for A. > O is a collective efi'ect asso- ciated with the individual interaction between the larger ions through their distortion fields. The sublinear r dependence and the rapid rise in d..(x) near .r, is outside the monolayer model. Several mechanisms including the relative magnitudes of host- guest and host-host interactions. interlayer correlations. and the presence of defect (d) sites can produce sub- linear behavior in d.(x) but only the latter two can gen- erate threshold efi'ects. Since the ions of interest here are relatively incompressible we treat the guest species as hard spheres as noted above. The interlayer correlation mechanism is one in which large guest ions locally puclt- 2l69 129 PHYSICAL REVIEW LETTERS VOLUME 60. NUMBER ll .0 . I 23 MAY 1988 ’ mechanics calculation gives to - E .fg'l/(:+I). {1) on} x-[l/(l+f)lx, 5 o.s- +[f/(l+f)lll/I:exp(-a/kT)+-lll. (2) . D ‘0 Z 0 v v rv FIG. 3. Moanlnyes triangular lattice computer simulations (dostdlinm)ofthecompnsitiondependenceofthenormaliaed bunlspacingofaternaryinterealationcompoundforseveral velumofthe healing length.Landrigidity parameter. p. The solidlinoasefrnmflq.(elofthetextwith(l)p-l.L-0:(2) p-7.L-“(J)p-l3.1-fi¢elm (4)p--.L--. inset; ThepectaredregioeofatriangelarlatticewithL-ee. Here the nembcofexpanddsitesisp-Zs-t-7 whereZisthe aumberofnur-tneighbors. ertheboundinglayerssothatatlothheyadoptstar gered lateral positions. in. no line joining the centers of any pair of large ions in adjacent galleries is perpendicuo lar to the silicate layer. This mechanism is relevant to host materials with low transverse layer rigidity such as graphite while the d-site mechanism is more appropriate to the more rigid layers of clays. Sourcesofdsitesinourspecimensareshownina scanning tunneling electron micrograph of (Cl-lith-iT- Vin (inset. Fig. 2). The region imaged consists of homo- geneously intercalated areas (g sites) which are bounded laterally by layer edge dislocations and are capped by free surfaces (4 sites) that bind guest species without in- ducing c-axis expansion. Since the clay grains have typi- cal basal dimensions of a few micrometers. it is clear from the scale of the micrograph that these free surfaces can represent a significant fraction of the total surface availabletoguestspeciss. Additional minorsourcesofd sites are the microcracks and folds that are visible in the micrograph. We have explored the d-site mechanism by construct- ing a two-site model in which the basal spacing is as- sumed to depend upon the gallery A-ion concentration x, which itself is a function of the total A-ion concentration x. The functional dependence of x, on x is determined by two parameters. 1' and A/ltT. where I'M/N, is the fraction of ions in 4 sites relative to those in g sites and A is the effective binding-energy difl’erence between these sites. the 4 sites having a lower binding energy. For sim- plicity we assume only one type of 4 site. A statistical 2170 where z-expl(e,-u)/le is related to the t'ugacuv and the binding energy s, of the g sites. Equations (l) and (2) can be solved to obtain x,-e(x.f..i/kT) for different values of f and a/kT. Physically then for x < in. the A ions first preferentially displace 8 ions from the 4 sites. This reduces the gallery A-ion concen- tration for a given it and yields a sublinear increase in d.(x). For x > is. additionally ingested A ions enter the galleries. Theresultisarapidinaeaseinddx). Using methods developed by Xia and Thorpem one canobtainthefollowinganalyticsolutioa forourmono- layer simulation: d.(.s,)-l -(i -x.)'. can: I. . (Jl wherepisalayerrigidityparamsasr. Th'nequatioa fits thesimulatioadataextremelywellushownbythedot- tedlinesinFig. 3. Forourlattics-gassimulatioa. p-Z+l where Z is the number of neighboring sites thatarepuckeredbytheinssrtioaofanisolatedAion (see inset Fig. 3). in the continuum limit (lunch/2). p~(21/44)1. Using Eq. (3) and x, -e(x.f.a/kT). we obtain d.(x) -l - {l -o(x.f.A/kT)l'. (4) Note that the slope of d.(x) at 32:. is governed by a combination of p and A/kT while 3. is determined prio marily by f for large a/kT. We have used Eq. (4) to obtain a nonlinear least- squaresfittothe dataofFig. 2. Theparametervalues which give very good fits (solid lines in Fig. 2) for the two CIC systems [(CH3)4N’I.I(CHJ)3NH’II-.-Vm and Cs.Rbi -.-Vm are lp-SD. f-0.5. AMT-4.3) and {p-7.0. f '12. A/kT'4J). respectively. The smaller value of the rigidity parameter in the Cs-Rb system is consistent with the fact that alkali ions in CIC’s can par- tially penetrate the bounding silicate layers. The mecha- nism which gives rise to this penetration is a torsional in-plane distortion” of the tetrahedral sheets which ex- pands the hexagonal pockets that contain the guest species. [on penetration of the clay layers causes a reduCtion of the apparent healing length. But the (Cl-marl" and (CHJth-l" ions are much too large to penetrate the clay layer significantly. even in the pres- ence of torsional distortions. Thus one expects the Cs- Rb-Vm system to exhibit a lower value of the rigidity parameter p. The / values deduced for the two systems also reveal interesting properties of the clay structure. For singly ionized guest species N, "GA, where a is the layer charge density and .«l, is the surface area asso- VOLUME 60. NUMBER 21 ciated with j Sites. j-d.g. [f A -.-td +24, then .44 'If/(l 4'1.) IA and (Ad kph/(.44 )ICH,i,-cu,i, 3 2. Thus. of the surface which provides 4 Sites for small Cs- Rb ions only about half (the portion not adjacent to edge dislocations. or derived from some microcraclts or folds) can also accommodate the robust (Cl-iniN '- (ct-1,) ,NH' ions without inducing basal expansion. Fi- nally. the difference in the Mid values for the two pairs indicates that the d sites are more attractive for the larger ions. This makes physical sense because the more spatially demanding ions prefer the less constrained de~ feet environment to the more restrictive gallery. The layer rigidity model which we have developed here should be directly applicable to other lamellar solids such as zirconium phosphates and layered niobates which have relatively rigid layers. It can also give in- sight into the behavior of intercalation compounds whose has layers have low or moderate rigidity. For example Li.C. (Ref. 4) and Li.TiS:” exhibit no threshold in 4.6). and therefore contain few if any d sites. Also. there are conflicting reports of a Vegard's-law d..(x) for Rbg‘l-hc. prepared from single-crystal graphite” and s ' sublinear behavior for the same com- pound prepared single-crystal graphite. ” highly on'ented pyrolysis graphite.“ or powder." For clay hosts sub- linear threshold behavior can be reasonably associated with 4 sites But for do ~layer hosts such as graphite there is much evidence‘ that interlayer correlations and their associated strain fields dominate the behavior. Therefore. even though the non-Vegard’s-law behavior of Rb.Ki -.C. (Ref. 12) has been attributed to 4 sites. we do not believe that the model addressed here is applic- able to that compound. Finally. we have assumed that the site binding ener- gies in our model are independent of concentration. This assumption might be relaxed if the binding energy of the g sites drops once the galleries are initially expanded. PHYSICAL REVIEW LETTERS 23 MAY 13%? The resultant transfer of ions from d to g sites wouzc then contribute to the rapid increase In J.(.r) for .t 21.. We thank M. F. Thorpe for the derivation of Eq. (3). Useful discussions with H. X. liang and Y. B. Fin are acknowledged. This work was supported by the National Science Foundation under Materials Research Center Grant No. DMR 854-1154 and in partby the Center for Fundamental Materials Research of Michigan State L'niversuy. ‘0Department of Chemistry. (”Department of Physics and Astronomy. lS. A. Solin. in Intercalation in Layered Materials. edited by M. S. Dresselhaus (Plenum. New York. I986). p. I“. 2T. l. Pennavaia. Science 220. 365 (1983). 1Until» graphite which is mimic clays have a and lay- er charge. Intercalation in thme materials is thus an intragal- lery ion-exchange precu. ‘There is one exception. K(NHi).Cis. whose superlinear :- depeadeat basal spacing is qualitively distinct from that of claysbutiswellaccouniedforbyarigsd-layermodelwhichino cludssboslielasticandelectreaieea'ects. SeelRYorhaad S. A. Solis. Phys. Rev. I 31. 8206 (I905). 5:. a rm and it. 1. itta. Phys. Rev. 3 35. ms (I987). and referenc- therein. °B. R. Yorker al.. SolidStaeeCommea. 94. 415mm. 7H. Kim er al.. to be '5. Hendricks and E. Teller. 1. Chem. Phys. IO. la? (19“). ’D. M. c. MJCEWIII. Nature (London) m. sis (I953). low. Xia and M. F. Thorpe. Phys. Rev. A (to be published). "I. R. Dahn. D. C. Dahn. and R. R. I-Iaering. Solid State Commun. 42. H9 (1982). '10. Medjahed. R. Merlin. and R. Claret. Phys. Rev. B 36. 9345 ( I987). IJl’. Chow and H. label. to be published. “P. Chow and H. Zabei. Synth. Met. 7. 243 (I983). 1’S. A. Solin and H. Zabel. Adv. Phys. (to be published). illlllllllllllllllllllll||||||lllllllllllllllllllllllIII|| 312930088203