Quantum corrections to conductivity and energy relaxation in InP and InSb

Phys~a 117B & 118B (1983) 75-77 Nort~Holhnd Publal~n8 Company

75

QUANTUM CORRECTIONS TO CONDUCTIVITY AND ENERGY RELAXATION IN InP AND InSb

A.P. LONG a) and M. PEPPER b)

a) Cavendlsh Laboratory, Unzversity of Cambridge, England. b) Cavendish Laboratory, University of Cambridge and General Electric Company, P.L.C., Hirst Research Centre, Wembley, England. We present results on conduction in the metalllc xmpurity band of InP at low temperatures. Below 4 K, in zero magnetic field, we observe a T correction to the conductivity consxstent with the Coulomb interaction model, although the sign of the change is not in agreement with theory. The observation of a negative magnetoresistance also indicates the rSle of localization in the conduction process and we have attempted to separate the two mechanzsms by measuring for both mcT I. We have measured the energy relaxation time in InSb for various electron and lattice temperatures. The results are dzscussed in terms of energy loss vxa acoustic piezoelectric, deformatlon potentzal and polar optzcal scattering. We have measured the electronic specific heat on the insulating side of transition and find it is conxstent with a small density of states, but not a gap. Two corrections to the low temperature conductivity of a disordered 3D Fermi system have been proposed. The first is a localization theory zn 2D, which in 3D gzves a small correctzon arzsing from quantum interference during elastic scatteringl,2, 3. This results in a correction to the conductivity -e 2 i (I) 6°LOt = ~-'~ " L ~ where Lxn is the inelastzc diffuszon length. This correction is lost above %10 K where Lln~,~, the mean free path, and is not szgnificant at low temperatures when Lin is large. The second correction arises from electron-electron interactxonsb, 5. This Coulomb interaction model predicts a correction to the conductzvity ~o c = mT ½

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where Qo(O) ffi .O25[TF(m*D/~)3]-~ with T F the Fermi temperature, m* the effective mass, D the diffusion constant and o(0) the zero temperature conductivity. The 4/3 term comas from exchange interaction and the 2F from Hartree interaction. The parameter F depends on the Fermi wavevector k F and the Thomas Fermi screening length ~F i.e. F = I/x(l+x) where x~(2kFAF~ For InP wlth an impurlty concentration 7 x 1022 m -3 we find x = 1.41, F = .624 and (4/3-2F) = +.0853. Therefore, the theory predicts a positive contribution to the conductivity. Even allowlng for the possibllzty of 235

(2)

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The coefficlent m includes contributions from both exchange and Hartree interactions and the sign of the coefficient will depend on their relative magnztudes.

n-type ~= ~fOmcm"=

230 We have made 4-termlnal reslstance measurements of a metallzc sample of InP as a function of temperature to investigate the relative import~nce of the correction terms discussed previously. The sample is n-type with an impurity concentration of 7 x 1022 m-3(kF~%2) and is nominally uncompensated. Temperatures down to 40 m K w e r e reached by use of a dilution refrigerator. In Fig. I we plot the conductxvlty as a function of temperature In the high temperature regzon 1.5 K • T < 4 K. The solid llne represents a least squares fit to an equation of the form ~(T) ffi o(O) + mT~. The close agreement between our results and this equation suggests a correction of the form mT½ with m negative: This is consistent with the Coulomb xnteraction model. The same temperature dependence is observed at lower temperatures, at least down to IOO mK. Below 50 mK a sllghtly stronger dependence is observed.

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A P Long, M Pepper / Quantum correcttons to conducttvtty

76

localized states leading to a reduction in the free electron concentration zt is not posslble to produce, with free electron formulae, a negative coefficient for impurity concentrations above that required for a metal insulator zn the impurity band, i.e. the Mott criterion. Several possibilities exist which may explain the above discrepancy. First the simple Thomas Fermi screening may be inadequate. Second, in the presence of spin-orbit coupling the interactions will be modified. Third, spin-fllp scattering can suppress the particle-partlcle interaction wlthout affecting the Hartree term. We note that if the Hartree term is unaffected by the spinorbit interaction or spin-flip scatterlng but the partzcle-partlcle Interaction Is modzfied~ then the slgn and magnitude of the conductivity change could be brought into agreement with experiment. To galn further inslght into the nature of the quantum correction to the conductivity we have measured the magnetoreslstance at 700 mK and 50 mK. At both temperatures the low field (~c T < I) magnetoreslstance ms negative: For 700 m K the change in conductlvlty is proportlonal to B½ for B greater than %300 Gauss and saturates when B = 4T and 0~cT~,l. This is the expected behavxour when quantum interference is suppressed and the cyclotron length L c << Lin. We have not observed the B 2 behavzour expected when L c ~ Lln and presumably only occurs at fields less than 300 Gauss. However, at 50 mK the extent of the negative magnetoreslstance was decreased for the same field range. We attribute this to the addltlon of a positive magnetoreslstance arising from the electron-electron interaction. Whereas the negative magnetoreslstance zn the B½ regime 24 InSb 22

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Large changes in magnetoresistance are found for ~c r just greater than one. We feel that this arises from two sources. The first being the effect of Landau level formation on the density of states, and the second being a change zn the magnitude of the znteractlon correction. In thls context it has been shown 6 that for ~c r ~ l the interaction contribution is reduced in 2D but as ~x increases the interaction contribution increases. Accordingly we have measured the temperature dependence of reslstance for ~c ~ = i and ~c Y = 2.2• We find that the resistance increases with decreasing temperature, consistent with a change in the Hartree contribution. For ~c r = i the resistance increases by 0.35% zn the temperature range 500 - 50 mK, whereas for ~c T = 2.2 the equivalent change zs 1•5%. It appears that the 3D behavxour is similar to that predicted xn 2D In thls section we outline results concerned with energy relaxation in InSb. Conservation of energy requires Ce(Te-T L)

~E2T E

(4)

with C e the electronic heat capacity, Te-T L the electron temperature rise produced by electric fleld E and r E the energy relaxation time which depends on details of the electron-phonon coupling. C e is important through its dependence on N(EF) , the density of states at the Fermi surface. The inset of Fig. 3 shows Te(E) for an n-type sample ND-NA=7.7xlOI9m-3. ~(T) confirms this sample is on the insulating side of the transitlon. We determine r(Te,TL) by a pulse technique (7) In Fig. 2 we plot Ce(Te). Below T F ~ 6K we observe a linear temperature dependence. The coefficient of Ce(Te) is y=2.6xlO-5j .kg-iK-2 a factor 34 less than that predicted from free electron theory For n~l.4xlO20m-3 we expect electrons in localized states. Therefore~ the linear specific heat indicates N(E F) is finlte and there is no gap at E F. The steep rise in C e above 6 K, we suggest, represents impact zonzzatlon to the conduction band 8

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should be independent of temperature, the interaction contrlbutlon is more effective at lower temperatures becoming significant when gSH~kBT. At 700 m K the condition gBH~kBT is close to ~ T ~ I and there is little detraction from the B~ negative magnetoreslstance. However, at 50 m K the interaction contribution will become significant at %2kG.

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Finally we consider {he dependence of T E on Te, T.. Fig. 3 shows r~(T ) for T L = 4.2,1.2 K. • ~ e • T~ree scattering mechanisms are considered acoustic plezoelectric potential, deformation potential, and polar optical phonons. Kogan 9 has derived expressions for the flrst two terms for degenerate and non-degenerate statistics. For a degenerate semiconductor the piezoelectric and deformation terms are both approxmmately linear in T e.

A P Long, M Pepper / Quantum correctsons to conductiwty

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Te(K) Fig.3: Energy relaxation time, TE, as a function of the electron temperature, T e. Open circles correspond to a lattlce temperature, TL, of 1.2 K, close circles TL = 4.2 K. The inset shows the electron temperature, for Che two lattice temperatures, as a function of the electric field E. A -I ~2 k B Te F~(n) ACKNOWLEDGEMENTS - We thank Prof. Sir Nevlll T(Plezoelectrlc) = p -2-- T;i TL Te3/2 F-~-~ (5) Mot£ for useful discussions. The low tempera2 2 ture measurements were performed at the 8ERC 8e~4 e 2 m,3/2 ck Rutherford Laboratory and we are most grateful where A = o for the help and advice of Dr S.F.J. Read and P 2 ~ ~2 5r ~o p with el4 the Mr. G. Regan. This work was supported by the SERC• piezoelectric stress constant, p the density and c ~ .4. Fa(~) Is the Ferml integral of order a REFERENCES and argument ~ = EF/kBT e. I. L.P. Gor'kov, A.8. Larkln and D.E. ~2 kB Te --F~) Khmel'nltkii, JETP Lett. 30, 228, 1980 r (Deformat lon Potent ial) =AD I 2. M. Kaveh and N.F. Mot-~, J. Phys C. I_~4,L177. 2 !TLTT-I FI 2 e L e 1981 3. K.F. Berggren, J. Phys C. I_~5,L45, 1982 8m,3/2E12kB3/2 2 4. B.L. Altshuler and A.G. Aronov, Soy. Phys where A ffi with E] the deformation JETP 5_OO, 968, 1979 potential. U n ~ n t y in ej~ and EI 10, the need 5. T• Rosenhaum, R•F. Milllgan, G•A. Thomas, for exact statistics and the inclusion of screenR•N. Bhatt, K. DeConde, H• Hess and T. Perry, ing makes quantitative agreement between theory Phys Rev. Lett. 4~7, 1758, 1981 and experiment difficult. However, using the 6. A• Houghton, J . R . Senna and S.C. Ying, reduced value of Ce and el4,E I from Ref.(IO) we Phys'Rev. B. 2_55, 2196, 1982 have agreement (within 20%) for TeIOK may be attrlbuted to Phys Rev. L e t t . 23, 848, 1969 scattering by polar optical phonons II for which 8. R. Mansfield and I . Ahmad, J . Phys C. 3 , T~T~ exp(e/Te). Good agreement is o b t a i n e d at 423, 1970 the highest temperatures. The relaxation times 9. M. Kogan, Soy. Phys S o l i d S t a t e , 4 , 1813, we flnd are less ~han those reported for metallic 1963 samples 7 consistent with the decreased importance 10. M.A. Kinch, Proc. Phys Soc. 90, 819, 1967 w . of selection rules on the localized side of the 11. H. F r o h h c h and B.V. P a r a n j a ~ , P r o c . Phys transition. Soc. B6__99,21, 1956