Phonon Absorption by - Springer Link

Journal of Low Temperature Physics, Vol. 110, Nos. 5/6, 1998

Phonon Absorption by D- Band Tails in Ge:Sb B. Danilchenko, D. Poplavsky, and S. Roshko Institute of Physics of the Ukrainian Academy of Sciences, Prospect Nauki 46, 252650 Kier, Ukraine (Received July 31, 1997; revised October 9, 1997)

Uniaxial stress dependence of acoustical phonon absorption in intermediately doped Ge:Sb has been studied using heat pulse technique. Abruptly decreasing LA and FTA phonon scattering in stress interval of 3-7 . 108 dyn/cm 2 with further saturation up to 1.9 . 109 dyn/cm2 was observed. Results obtained from phonon measurements correlate with conductivity activation energy behavior on stress. Acoustical transparency stress dependence is assumed to be connected with phonon assisted electron transitions from impurity ground state D° to D- band. 1. INTRODUCTION Doped semiconductors have been studied both theoretically and experimentally last twenty years in order to understand the metal-insulator transition completely. It is supposed that both electron correlation effects and disorder in impurity distribution play an important role in this transition. The most detailed experimental results were obtained on Si:P.1-4 These experiments were carried out at millikelvin temperatures and represent conductivity, electrical susceptibility and magnetoresistance measurements for different impurity concentrations. Uniaxial stress was applied to samples in order to cross through the metal-insulator transition (it was possible due to the fact that uniaxial stress breakes down the symmetry of energy valleys and results in appearance of energy gap). According to the electron correlations representation metal-insulator transition can be treated as a "confluence" of the upper Hubbard band and the ground impurity state band (D° band). The upper Hubbard band is a result of the broadening of D- donor state, that represents a second electron attached to the neutral donor. Their reciprocal energy positions are schematically shown on the Fig. 1. The existence of D~ states in semiconductors was predicted by Lampert 5 in 1958. The theoretical value for 1029 0022-2291/98/0300-1029$15.00/0 © 1998 Plenum Publishing Corporation

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B. Danilchenko, D. Poplavsky, and S. Roshko

Fig. 1. Schematic diagram of energy bands in intermediately doped Ge:Sb.

binding energy of D- state is ~ 0.54 me V for n-type germanium. 5 Experiments gave 0.625 meV for Sb impurity and 0.75 meV for As impurity in Ge.6 Application of uniaxial stress along < 111 > direction resulted in a decrease of binding energies down to 0.55 meV and 0.57 meV for Sb- and As-doped germanium respectively, that is very close to theoretically predicted value. It should be noted, however, that almost all known experimental works concerning D- states in bulk semiconductors were carried out on samples with dopant concentrations far below metal-insulator transition (see, for example, Refs. 7-9) when D- state can be considered as an isolated energy level. Najda et al.10 carried out investigations of D- states in one-valley semiconductors (GaAs, InP, InSb) and revealed rather broad D~ spectral lines. Sakurai and Suzuki" measured stress dependence of ultrasonic waves attenuation in Sb doped Ge over a wide region of doping. Proposed simple model did not include D~ band formation in intermediately doped region and was not able to explain observed results completely. In the intermediate doping region the evidence for the existence of activation energy £2 that represents energy gap between D° and Dbands and its dependence on donor concentration in Ge was studied by Fritzsche.12 From Fig. 1 one can see that E2 represents the difference between energy position of mobility edge and impurity ground state. It was

Phonon Absorbtion in Ge:Sb

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shown that value of £2 is of order of few meV and it depends on the uniaxial pressure applied along directions corresponding to energy valleys. Such values of activation energy are comparable to the energy of ballistically propagating acoustical phonons in pure germanium at helium temperatures. Under this reason one can expect to obtain new information concerning energy structure of D~ band from the experiments on high frequency acoustical phonon beams scattering. This paper describes time-of-flight ballistic phonon spectroscopy investigations of n-germanium near metal-insulator transition as well as conductivity activation energy behavior with < 111 > uniaxial stress at helium temperatures. The treatment of experimental results is given in the frames of phenomenological model that deals with phonon assisted electron transitions from isolated state to D- band formed both due to electron correlation effects and disorder in space distribution of impurity atoms. Estimations of energy band parameters are performed through fitting procedure applied to experimental data in the frames of proposed model. As a result we obtain the value for D- band width and its stress behavior. The parameter of density of states (DOS) function formed by the dopant space distribution is also extracted. 2. EXPERIMENTAL DETAILS We investigated intermediately doped low compensated (K < 0.05) Ge:Sb sample with 5 - 1 0 1 6 c m ~ 3 donor concentration. Parallelepiped shaped sample had alloyed In:As ohmic contacts on < 111 > face. Uniaxial stress was applied along < 111 > direction. Electrical resistivity measurements were carried out with respect to the same direction. In order to get stress dependence of activation energy, resistivity measurements were performed at different temperatures from 2 K up to 4 K. Along with resistivity measurements the uniaxial stress dependence of acoustical phonon transparency was investigated. In order to get appropriate resolution of acoustical modes in time we used 4.4 mm thick sample. Phonon generator (2000 A Au film, 1 x 1 mm 2 ) was evaporated onto the < 1 1 0 > face of the sample. The film was heated by nitrogen laser pulses of 10 ns duration and repetition frequency of 100 Hz. Nonequilibrium phonons were detected by the bolometer deposited on the opposite side of the sample. In the chosen geometry of the experiment phonons propagating along direction were detected. As a phonon detector we used In film superconducting bolometer of 1 x 1 mm2 size. Bolometer superconducting transition point was shifted from 3.7 K to 2 K by weak external magnetic field (of the order of 0.01 T). As it was shown in Ref. 13, the magnetic freeze-out effect of A + and A° centers in Si:B is not important up to 12 Tesla. Because A + centers,

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formed in p-doping case, are completely analogous to D- centers in n-doping case one can neglect the influence of extremely weak magnetic field on donor states in our experiment. In order to get appropriate signal/noise relation phonon signals were accumulated up to 104 numbers of repetition. 3. EXPERIMENTAL RESULTS Time-of-flight phonon spectra at three typical values of uniaxial stress are represented on Fig. 2. These spectra are formed by phonon fluxes of different acoustical modes of vibrations. There are three acoustical modes in direction in Ge: longitudinal (LA), fast transverse (FTA) and slow transverse (STA). Time delays of observed peaks are in accordance with group velocities of corresponding ballistically propagating phonons. If no stress is applied only STA phonon flux reaches bolometer—LA and FTA phonons are strongly absorbed (Fig. 2, curve 1). Uniaxial stress causes noticeable decreasing of absorption of LA and FTA phonon modes. These phonons are well observed at the stress exceeding 1 . 10 9 dyn/cm 2 (Fig. 2, curve 3). The amplitude of STA mode does not change with

Fig. 2. Time-of-flight phonon spectra for different uniaxial stresses: 1—P = 0; 2-P = 3- 108 dyn/cm 2 ; 3-P = 15- 108 dyn/cm 2 .


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applied stress. Differences among observed signals of LA, FTA and STA modes at such high stresses are the result of different phonon emission rates from heater and additionally due to phonon focusing factors in Ge.14 Acoustical transparency dependencies on the applied uniaxial stress corresponding to LA and FTA phonon modes are represented on Fig. 3. One

Fig. 3. Uniaxial stress dependence of acoustical transparency: (a| LA phonons, (b) FTA phonons. Open circles correspond to experimental data; solid line corresponds to the result of calculations performed in the frames of proposed model; solid circles correspond to experimental data for weakly doped (6- 10 14 cm 3 ) Ge:Sb sample.

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can see that the sample is "opaque" for both modes approximately up to 3. 108 dyn/cm 2 then the transparency increases almost linearly up to 7.10 8 dyn/cm 2 and becomes constant for greater stresses. Here we also represent the results obtained for weakly doped (5 . 1014 c m - 3 ) sample for comparison. It is seen that phonon transparency behavior is completely different from that of intermediately doped one. This sample revealed rather high transparency at zero stress and resonant absorbtion at 1 • 108 dyn/cm 2 ; such resonance appeared due to the singlet-triplet splitting of D° state of donor. Stress dependence of electrical resistivity at different temperatures was investigated in order to obtain activation energy behavior with stress. In this experiment the direction of electric field was parallel to uniaxial stress applied in < 111 > direction. Observed resistivity stress behavior at temperatures from 4.2 K down to 2.1 K is represented on Fig. 4. It is noticeable that resistivity examines increase by 2-3 orders (depending on temperature) with increasing stress and saturates at approximately (7 -9). 108 dyn/cm 2 . Fig. 5 displays the dependence of activation energy on stress. As it was mentioned above activation energy K2 appears due to the energy gap between D- and D° states. We can distinguish two typical regions of dependence followed by the region of almost constant activation

Fig. 4. Longitudinal specific piezoresistivity at different temperatures.


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Fig. 5. Activation energy of conductivity as a function of uniaxial stress along < 1 1 1 > . Open circles correspond to experimental data; solid line corresponds to result of best fit obtained for acoustical transparency data; solid squares correspond to data deduced from Ref. 12.

energy. These measurements are in agreement with results obtained on Ge:Sb samples with the same concentration in Ref. 12 for zero and extremely high stresses (see Fig. 5). It must be noted that characteristic points (the beginning and the end of rising region) are the same as for the dependence of phonon transparency on stress (Fig. 3). 4. DISCUSSION 4.1. Phonon Scattering Phonon kinetics in crystals is controlled by the processes of scattering, decay, etc. The relevant scattering processes in doped germanium can be controlled by the following mechanisms: (i) (ii)

elastic scattering on isotopes and impurity atoms; resonant elastic and nonelastic phonon scattering on singlettriplet splitting of the ground state of donor impurity;

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(iii) phonon absorption in the processes of electron transitions from donor ground state to D~ band. The first mechanism does not depend on stress and determines only upper energy limit of ballistically propagating phonons. Its rate of scattering can be written as follows:

where w is phonon frequency, a is a constant (for undoped Ge a = 2.44. 10-44s3 (Ref. 15) and for Ge:Sb with Sb concentration of 5 . 10 16 cm -3 we used a = 4.44. 10 -44 s 3 (Ref. 16)). This scattering limits the acoustical transparency of our sample up to phonon energies hwmax= 1.8-2.0 meV. The second mechanism is due to phonon scattering on electrons populating impurity ground state. In germanium this state is splitted into a singlet and triplet states separated by valley-orbit splitting 4A0. Phonon scattering on these splitted states was well studied both theoretically 17,18 and experimentally 19,20 in weakly doped crystals. Strong elastic resonant scattering of LA and FTA phonons occurs at phonon frequencies w ~ 4A/h. Uniaxial stress applied in < 111 > direction increases the value of 4A (Ref. 21) respectively changing the frequency of phonons scattered in accordance with this mechanism. Maximum phonon flux attenuation for Planck-like phonon source energy spectrum will take place if splitting reaches 4A ~ hwmax. At stresses when 4A reaches hwmax the resonant frequency of phonon scattering exceeds the upper limit of acoustical transparency of the sample. Therefore phonon flux will increase and saturate with further increasing of stress. The results of experiments for slightly doped Ge:Sb represented on the insertion to Fig. 3 are in accordance with qualitatively described phonon flux behavior. Mechanism of phonon scattering mentioned above can not be applied to intermediately doped crystals where neighboring donor wave functions overlapping becomes significant and one cannot neglect other mechanisms affecting phonon absorption. The third possible mechanism of phonon absorption was not described yet in details. Its nature can be qualitatively explained using schematic diagram of energy bands, shown on Fig. 1. Being intermediately doped crystal is considered to have electronic structure with rather broad impurity levels that have to be treated as energy bands. At the temperature of experiment 2 K electrons populate impurity ground state. Phonons can cause electron transitions from this state to unpopulated states in D- band. If electron's final' state is higher in energy than mobility edge position then phonon assisted current change will occur. Such phonon induced current has been observed earlier22 in Ge:Sb sample with 2.10 1 6 c m - 3 donor


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concentration. Phonon assisted electron transitions to states below mobility edge do not change current but phonon is absorbed anyway. 4.2. Impurity Levels Broadening Broadening of impurity levels occurs due to two reasons: (i) quantum broadening that is a result of overlapping of neighboring impurity wave functions, ( i i ) statistical broadening that appears from disorder in impurity spatial distribution. Energy band width formed due to quantum broadening is proportional to overlapping integral of neighboring impurity wave functions. Germanium is multivalley semiconductor thus the total overlapping integral will be presented by the contribution from each valley of Ge. For weakly doped multivalley semiconductor Efros and Shklovskii23 give the following expression for overlapping integral:

where Iij-overlapping integral, n-number of equivalent energy valleys, £-dielectric constant,

where xij, yij and zij are projections of vector rij onto the axes of the coordinate system connected with energy minimum g, a and b-energy ellipsoid semiaxes (a = ( h / / 2 m t e 0 ) and b = (h//2m/£ 0 )). Here mt and ml-transverse and longitudinal effective masses, e0-donor ground state energy. As it was mentioned above, this formula can be applied only for weakly doped semiconductor. Nevertheless, for general case, obviously the sum over 4 valleys remains. After averaging in k-space this sum can be replaced by the factor 4. If the uniaxial stress along < 111 > direction is applied, energy minimum corresponding to < 111 > valley will shift to lower value while other 3 valleys—to higher energy values. At high values of stress only the lowest valley remains and semiconductor becomes onevalley. Taking into consideration small value of chemical shift in Ge:Sb, volume of state is almost constant,23 i.e., energy ellipsoid parameters a and b are independent on uniaxial < 111 > stress. Thus only one valley contributes to formula (2) at high stress therefore |Iij|2 becomes 4 times smaller. Respectively quantum broadening of state 2 times decreases. The second reason for broadening of energy bands is fluctuations in spatial distribution of impurities. It is important for our consideration to find the density of states (DOS) function for such broadening. Efros and

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Shklovskii23 considered acceptor-donor complexes in weakly doped low compensated semiconductors and taking into account large-scale Gaussian fluctuations they found DOS in the form of Gaussian function

where ND -donor concentration, e-energy, y-distribution parameter. However the assumption of Gaussian nature of fluctuations for weak doping is valid only in the case of fulfillment of the condition 5. 1 0 - 2 K - 1 / 2 > 1, where K is compensation level. It must be noted that this condition is not valid for our level of compensation but actually we have higher level of doping than considered in Ref. 23. From the other hand Halperin and Lax24 investigated impurity band tails for highly-doped semiconductors. They found that in the case of screened Coulomb potential with Gaussian statistics DOS has the following form

where B(e) varies from |e| 1/2 to e2, a(e) varies roughly as en (n varies from 5 to 2). The validity of application of Gaussian statistics in this case can be evaluated by the condition24

where Q-1 -screening length. For our case screening length is larger than average distance between impurity atoms so this condition fulfils. Under these reasons Gaussian nature of fluctuations can be assumed for the intermediately doped semiconductor. We took D~ band tail in the form of Gaussian function

where e1 is energy position of upper Hubbard band edge (see Fig. 1), C-constant. 4.3. Calculation of Acoustical Phonon Transparency

The phenomenological expression for phonon absorption rate in accordance with considered mechanism can be written in the following form


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where w-phonon frequency, F(e, e'; w)-matrix element of electron-phonon interaction, g D 0(e), gD- (e)-DOS functions of D° and D~ states respectively. Estimations show that for chosen impurity concentration ground state wave functions overlapping is significantly less than that of D~ state. It means that energy broadening of the ground state is negligibly small comparing with D- band width as well as its edge broadening as it is shown on Fig. 1. Under these reasons as a first approximation we set ground state DOS to Dirac d-function. Taking into account experimentally obtained activation energy values (see Fig. 5) it is reasonable to assume that electrons populate only ground state at temperature 2 K. After these simplifications expression (6) can be rewritten as follows

where i denotes phonon mode (LA or FTA). In order to get an expression for matrix element V ( w ) we considered phonon assisted electron transitions between two neighboring donors as it was done by Miller and Abrahams. 25 They considered D° —> D° transitions but we deal with D° -> D - transitions. This approach can be exploited in our case because the highest phonon energy in our sample is almost equal to the activation energy therefore the overwhelming part of phonons will cause transitions from isolated ground state to isolated D- state because electrons do not reach mobility edge (see Fig. 1). Under this reason we supposed that the expression for matrix element can be taken in the form analogous to that obtained in Ref. 25

where v i —group velocity of corresponding phonon mode, A—constant, a is defined as a-1 =a-1 + b-1 (here a and b—Bohr radii of D° and Dstates which wave functions were taken in hydrogenlike forms). As it was mentioned above application of stress decreases D~ band quantum width that is determined by overlapping integral (2). At the same time uniaxial stress does not affect the spatial distribution of impurity atoms. Under this reason band tail DOS function does not change. It only moves together with band edge position in accordance with quantum width change with stress. As a first approximation we suppose that quantum band width decreases linearly with stress, i.e.,

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where p-uniaxial stress, W0—initial value of band half-width (see Fig. 1), psat—stress of saturation (taken from experiment). Here the stress of saturation corresponds to the value of stress when 3 valleys have "died off." It is close to the value of activation energy saturation (p sat ~ 8.10 8 dyn/cm2, see Fig. 3 and Fig. 5). Band edge position e1 directly depends on quantum broadening of D- state

where eD- -activation energy of isolated D- state. Here energy s1 is counted from the bottom of conduction band. Such approximation may seem to be rather rough but it will enable us to examine main peculiarities of the system. Total energy detected by bolometer at the distance L from phonon source with a spectral density Pi(w, T0 , Th ) is

where i = LA, FTA-considered phonon modes, e i -phonon transmission coefficient from metal heater to the crystal, T0, Th -crystal and heater temperatures, respectively, ri1 =risotopic + r e l - p h , i — t h e total rate of phonon scattering,

density of nonequilibrium phonon energy injected to the crystal, where c i -phase velocity of i-th phonon mode in heater. In the frames of described model we have calculated phonon signals for different values of uniaxial (111) stress according to the formula ( 1 1 ) and formulae (1), (7). Results of calculation of LA and FTA phonon flux behavior with stress are shown on Fig. 3 (solid line). Saturation of acoustical transparency at large stresses appears due to two reasons: (i) D band edge shifts to considerably higher energies at high stresses and phonons (with their highest energies ~ 2 meV) cannot reach it; ( i i ) at high values of stress semiconductor becomes one-valley and according to formula (2) no change in band width and therefore in acoustical transparency is observed with increasing stress. The second also explains saturation in activation energy dependence (see Fig. 5) that appears at the same value of stress. Best fit was reached at the following values of unknown parameters: D- band initial half-width W0 = 7.5 meV, band tail FWHM (full width at


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half maximum)—3.3 meV. Such values of these parameters yield activation energy behavior close to that observed in experiment (see Fig. 5). In general we had three parameters to determine—two mentioned and a dimension constant in the final expression for r - 1 . But we must have the same parameter values both for LA and FTA phonon signal therefore only two independent parameters remain. From the other hand we have activation energy dependence on stress that enables us to exclude the second parameter. Thus we have fitting procedure with only one parameter. Fitting result did not depend on heater temperature that agree with experimental data. Obtained values of band parameters show that phonon absorption mainly causes transition of electrons to isolated states that do not induce the current. 5. SUMMARY High frequency acoustical phonon absorption in intermediately doped Ge:Sb was studied by means of heat pulse technique. Acoustical transparency behavior of different phonon modes with respect to uniaxial stress applied in < 111 > direction was qualitatively different. Phonon fluxes of LA and FTA modes increase and saturate while STA phonon mode signal remains constant with stress. Stress dependence of LA and FTA phonons transparency was strictly different from that of weakly doped Ge:Sb sample. Also kinetic measurements were performed and conductivity activation energy behavior with stress was obtained. Main peculiarities in stress dependence of phonon transparency and activation energy agreed with each other. D~ band structure formation including localization effects connected with impurity disorder was employed in order to treat experiments on phonon transparency. Proposed model includes broadened band with TABLE 1 Parameters of D

Band and Activation Energies for Sb Concentration ND = 5. 10 16 cm Different Values of Uniaxial < 111 > Stresses

Parameter D band half-width ( m e V ) Band tail FWHM (meV) Activation energy £ 2 ( m e V ) Activation energy e 2 from 12 ( m e V )

0

7.5 3.3 2.15

2.2

P (108dyn/cm2) 9.3

18

3.75

3.75

3.3 3.8

3.3 3.9 4

3

at

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gaussian tails, d-like impurity ground state and phonon assisted electron transitions occuring between these two states. Introduced model made it possible to explain results obtained both from acoustical transparency and activation energy measurements. Also D - band parameters were extracted from experimental data according to employed model. Their values are listed in Table 1 and also activation energy values are compared to that obtained by Fritzshe.12 ACKNOWLEDGMENTS This work is supported by Ukrainian State Foundation for Basic Research (Grant 2.4/93). D.P. also acknowledges the support of ISSEP Foundation through Grant GSU062085. REFERENCES 1. M. A. Paalanen, T. F. Rosenbaum, G. A. Thomas, and R. N. Bhatt, Phys. Rev. Lett. 48, 1284(1982). 2. T. F. Rosenbaum, R. F. Milligan, M. A. Paalanen, G. A. Thomas. R. N. Bhatt, and W. Lin, Phys. Rev. B 27, 7509 (1983). 3. G. A. Thomas, M. A. Paalanen, and T. F. Rosenbaum, Phys. Rev. B 27, 3897 (1983). 4. M. A. Paalanen, T. F. Rosenbaum, G. A. Thomas, and R. N. Bhatt, Phys. Rev. Lett. 5, 1896 (1983). 5. M. A. Lampert, Phys. Rev. Lett. 1, 450 (1958). 6. M. Taniguchi and S. Narita, J. Phys. Soc. Jpn. 43, 1262 (1977). 7. S. Narita, T. Shinbashi, and M. Kobayashi, J. Phys. Soc. Jpn. 51, 2186 (1982). 8. M. Taniguchi and S. Narita, J. Phys. Soc. Jpn. 47, 1503 (1979). 9. M. Taniguchi and S. Narita, Solid Slate Commun. 20, 131 (1976). 10. S. P. Najda, C. J. Armistead, C. Trager, and R. A. Stradling, Semicond. Sci. Technol. 4, 439 (1989). 11. H. Sakurai and K. Suzuki, J. Phys. Soc. Japan 52, 4192 (1983); J. Phys. Soc. Japan 52, 4199 (1983). 12. H. Fritzsche, Phys. Rev. 125, 1552 (1962). 13. S. Roshko and W. Dietsche, Solid Stale Commun. 98, 453 (1996). 14. G. A. Northrop and J. P. Wolfe, Phys. Rev. Lett. 43, 1424 (1979). 15. M. G. Holland, Phys. Rev. 132, 2461 (1963). 16. V. Radhakrishnan and P. C. Sharma, Can. J. Phys. 58, 1268 (1980). 17. R. W. Keyes, Phys. Rev. 122, 1171 (1961). 18. T. Miyasato and M. Tokumura, J. Phys. Soc. Japan 53, 210 (1984). 19. R. C. Dynes and V. Narayanamurti, Phys. Rev. B, 6, 143 (1972). 20. M. Gicnger, P. Gross, and K. Lassmann, Phys. Rev. Lett. 64, 1138 (1990). 21. P. Price, Phys. Rev. 104, 1223 (19561. 22. B. A. Danil'chenko and S. Kh. Rozko, Fiz. Tverd. Tela (Leningrad) [Sov. Phys.-Solid State] 32, 579 (1990). 23. B. I. Shklovskii and A. L. Efros, Electronic Properties of Doped Semiconductors, SpringerVerlag, Berlin (1984). 24. B. I. Halperin and M. Lax, Phys. Rev. B 148, 722 (1966). 25. A. Miller and E. Abrahams, Phys. Rev. 120, 745 (1960).