この論文は国立情報学研究所の電子図書館事業により電子化されました。The metallic impurity conduction is a typical example of the transport in the disordered metal. It shows anomalous transport at low temperature. After the developement of the perturbation theory of transport in "weakly localized region", these anomalies are interpreted to be the appearance of the effects of localization and electron-electron interaction in dirty metal. The theories explain the characteristic features of temperature and magnetic field dependence of conductivity, but quantitatively there remain some disagreements between theory and experiment. In this paper, we report the measurement of low temperature transport properties in the metallic impurity conduction of Ge:Sb, especially stressing the anisotropy of magnetoconductance

Katsumoto, Shingo

Ootuka, Youiti

Kyoto University Research Information Repository

Title8. Electronic Transport in Metallic Ge:Sb at LowTemperature(Experiments,I. Three Dimensional Systems)Author(s)Ootuka, Youiti; Katsumoto, ShingoCitation物性研究 (1983), 40(4): 26-31Issue Date1983-07-20URL http://hdl.handle.net/2433/91089RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversityElectronic Transport in Metallic Ge:Sb at Low TemperatureYouiti Ootuka and Shingo KatsumotoDepartment of Physics, University of TokyoBunkyo-ku, Tokyo 113, JapanThe metallic impurity conduction is a typical example of the transport in thedisordered metal. It shows anomalous transport at low temperature [1]. After thedevelopement of the perturbation theory of transport in "weakly localized region" [2] ,these anomalies are interpreted to be the appearance of the effects of localizationand electron-electron interaction in dirty metal. The theories explain thecharacteristic features of temperature and magnetic field dependence of conductivity,but quantitatively there remain some disagreements. between theory and experiment· [3].In this paper, we report the measurement of lo~ temperature transport propertiesin the metallic impurity conduction of Ge:Sb, especially stressing the anisotropy ofmagnetoconductance.I. 4-valley GeI-i) Temperature and magnetic field dependence of conductivityFig.l shows the conductivity of the sample with the donor concentration N IJf2.4xl017cm- 3 in various magnetic fields. The conductivity increases steeply astemperature is lowered below lK. In the magnetic field it increases first and c~t lowtemperature it begins to decrease, forming a hump in the temperature dependence.This hump of conductivity shifts to higher temperature as the magnetic filed isincreased. The magnetoconductance is positive at around lK and is negative at lowertemperature.Localization and electron-electron interaction are two important mechanisms inthe disordered metals. The former gives the tendency to localization remaining evenin the metallic phase and gives a positive magnetoconductance. The latter treatesthe Coulomb interaction which is modified by the diffusive character of electron indisordered material and gives a negative magnetoconductance.The expressin of conductivity which takes multivalleys and anisotropy effectsis [2,4]e& (~LOC + ~ :tNT )~(j = 2.TL&1\ (I){t <Xi (D"C )-IIL whe.~ H=o~LOC =. L", €.t 0.60) ci~ I .Q.~ eH Iwhel1 ~»DT~,,-.t Df' (-4- ~ Wl.seM H=o{,o,0.4-6 ,,' 3 - 3F) 10D~INT =v d'.1: R5 " FL tXi [ fCl(l)-tf¥ J .~ 0.4-6 -V (3 -F) 11D- l-' V i~when &faH, D~~ >,>Z,ItRI26(2)m, (S/cmK"')1&. ~.I G.Sb10Uotuk.\N.I US(~~;) .: II iO!·1001111101Onlukil~_ '0'N---..\ '\~".20!\:Iki,: ·110 1 ~ ',.,,uL.t,--,-N_L..- __ ._,,-..4lllU"cm J li\: .\: 1\10 :+\ t.: ,!.:! ',Fig.2 Donor concentration dependence of fittedvalues of mO' ml , J and B in Eqs.(2) and (3).The solid curves are the theoretical predictions.27....:..,Ge:Sb2.4 x 1017 ern- Jo20 ,..... J.. -.('...--..~30204036->;?"~ .~~----:,...I=-.~. _~_~0.5 1.0ff[J('2)Fig.l Conductivity of the samplewith N=2.4xl0 17cm- 3 as a functionof Tl / 2 •The parameters of D, 'tt, F, ml , mt , g, Y and g are the dif fusion constant, theinelastic scattering time, the screening parameter, the longitudinal and transverseeffective mass, the ~-factor of electron, number of conduction band valleys and theangle between magnetic field and the cylindrical axis of a conduction band valley.These theories are consistent with the features in Fig.l. The observed positivemagnetoconductance at high temperature can be attributed to localization and thenegative one at low temperature to interaction. In magnetic field, conductivityvaries as TI / 2 at low temperature with a coefficient almost independent of H. Themagnetic field dependence is approximately Hl / 2 . Therefore, the conductivity in theamgnetic field can be expressed asI~ ml Tl / 2 + J HI / 2In zero field we can fit the temperature dependence asSO- = m T l / 2 + B T (3)o 17 -3The similar T and H dependence is observed in other samples with N = 2 - 6 xlO em .Donor concentration dependence of coefficients mO' ml , J and B are given in Fig.2.The theoreticel values of mO and ml are also given in the figure. In the concen-tration range of N = 2 - 3 xlo17cm- 3 , there are discrepancy of the factor of 2 - 7between theory and experiment.The observed magnetoconductance at low temperature is negative, while in thetheory the negative magnetoconductance due to interaction is not large enough topredominate over the positive one due to localization. Here we should note that thevalues of moand ml are larger than the prediction of the interaction theory. On th0other hand, the positive magnetoconductance is smaller than the prediction of the~[mho/cml localization theory by 1/4 to 1/10. Therefore it----.. ------------------,- ----,--is conjectures that the present theory of locali-zation overestimates , while that of the interactionunderestimates the effects. If it is the case, low temperature magnetoconductanc:ecould be negative.I-ii) Anisotropy of magnetoconductanceAccording to the theories, the magnetoconductance has anisotropy reflecting theanisotropy of effective mass and g-factor. This will be a clue to understand theorigin of the magnetoconductance.Fig.3 shows the anisotropy of positive magnetoconductance measured at 4.2K.The donor concentration is 2.7xl017cm- 3 , the current direction is [110) and themagnetic field is rotated in the plane (001). The anisotropy in low magnetic fieldwas analyzed by Kawabata [4) and he found that it is explained by the localizationtheory very well, which is confirmed by our experiment also. At higher field theanisotropy changes. The theoretical expression of anisotropy given by Eq.(l) isshown in Fig.4, where we use the values of ml =1.59m and mt =0.082m. The character-istic "dip" in magnetoconductance at H /1 [110] is consistent with the experiment,although quantitative disagreements exist.The anisotropy of negative magnetoconductance at low temperature is shown inFig.5. Here the fitting parameter J in Eq. (2) is plotted as a function of magneticfield direction. The current is along [110) and the magnetic field is rotated inplanes of (110) in (a) and [001) in (b). Although the anisotropy of (b) isresembling that of the Zeeman energy term of the interaction theory. Theanisotropy of (a) seems not to consistent with it. The detailed analysis is beingdone now.(b)H...[Q01J3.23.63.03.4i.ll~cr;~) lieSb3.2"\\.\\~"(4,. , I'H···j·~· i9.0 ·60 -)0 0mOl e (degree) [1I0lFig.5 Anisotropy ofnegative magnetoconductanc('of the sample with N=3.5xlOl7cm-3 at 68mK. Thecurrent is along [110) andH is rotated in the plane(110) in (a), and (001) in(b) •Fig.4 Positive magneto-conductance predicted bythe localization theory.The current is along [110)and the magnetic field isrotated in the plane (001).The upper curve indicatesthe anisotropy at high H,and the lower one at low H.0.10.50.12.7X1017cni3S/cmr-----'4::.:.2'TK'---_....,...__-r~=:,.,0.6the plane (001).o t::::~~::::t=::::::S;1;(f;:::::::::p180".110 TIoFig.3 Anisotropy ofpositive magneto-conductance of thesample with N=2.7xI017crn-3• The conductivityis measured along[liD) and the magneticfield is rotated in28II . l-valley GeGe has equivalent four valleys of conduction band. Uniaxial compression alongthe [111] direction shifts the [1111 valley down and the remaining three valleys upin energy [5]. Hence, when the compression is strong enough, electrons distributeonly in the lowest valley. This is a I-valley metal with strong anisotropy. Thissystem is thought to give us clearer informations on low temperature conduction,because the anisotropy is large and the intervalley scattering which complicates thesituation is not need to be considered.Fig.6 shows the temperature dependence of conductivity of l-valley Ge below lK.They resemble that of 4-valley Ge and positive magnetoconductance at liquid He, 1/2 1/2temperature, negatlve magnetoconductance at lower temperature and T and Hdependence of conductivity in magnetic field are seen.II-i) positive magnetoconductanceMagnetic field dependence of positive magnetoconductance at 4.2K is shown inFig.7. Conductivity is measured along the direction of [1111. The magnetic fieldis rotated in the plane of (110). Anisotropy is very large. Little anisotropy isseen when the magnetic field is rotated in the plane of (Ill), which isconsistent with the symmetry of the system. Magnetoconductance varies as H2 atH~O.2T. At higher field magnetic field dependence is weaker than H2 .According to the localization theory, magnetoconductance reflects the anisotropyof cyclotron energy and is largest when the cyclotron mass is lightest, i.e. HII[lll].This is consitent with the experiment. Quantitatively, theory says the anisotropyequals to the effective mass ratio at low field and to the power of the one-forthof the effective mass ratio at high field. This means the magnetoconduceance ratio ofH II [Ill] to H.J. [Ill] is 1:0.05 at low field and 1:0.48 at high field, as shown in4.2KGe:Sb5,1x 1017cni 39klll Ao-I S/emJ\ 0"0-\ rl°'a,a I 0.3 I ",\ . r \. ,. I0\ (112) (1111l ')\ (112] ( [i~i]1 ,,1 "'. '\. I /Y-., 02\· j /.r\\ o. \. I'\\)1 \\0.1\ ,.j / }n-",.. ;\ ,.j I ,,,_"-u_..• '" Q / ... / I ..' I~\ \ .. II \ \.. I)n ,,"" / Ii .. ,/ /\ tl \ II11""11 n/ I" 1;/7kQwl111l7kllw1112JGe:Sb3,OxlO"crri3_c - Or.-If (K' "J........... .. ~.. . ~.-.- ...-180 ·-~aO·· -60 0 '·-·6~···~0--1~8-100f deorec)Fig.? Anisotropy of positive magnetoconductance of I-valley Gc with N~5.1xlOI7 cm- 3 at 4.2K. The conductivity is measured alonq 11111 ann thpmagnetic field is rotated in the plane of (110).29.,'.....;f'"!~ ...\Fig.6 Conductivity of I-valley Ge withN=3.0xl0 17cm- 3 as a function of Tl / 2 inthe magnetic fields of 0 and O.7T. Thecurrent is along [111].-0.48-0.67-1111+Fig.8 Theoretical prediction8of magnetoconductance anisotropyof I-valley Ge. The magneticfield is rotated in the plane(lIO). The meaning of eachcurves is as follow: (a) :local-ization theory in low II, (h):localization theory in high II,(c) :orbital term of interactiontheury in hiyh II and (d): upill-Zeeman term of interaction theo-ry in high H. Magnetoconductancpis isotropic in the plane (111).~c:o~ b----+-:----+-.,..----t---.~-0.05-g 0 0° 180°ouo~g~-As for the absolute value of positive magncto-conductance, the theoretical value is about 1.5times larger when H /I [Ill), and 0.6 times smallerwhen H~ [Ill) than the experiment. This discrepan-cy also is thought to be improved by consideringthe relaxation time anisotropy.In I-valley Ge, positive magnetoconductancecan be explained rather well by the localizationtheory, on the other hand in 4-valley Ge absolutevalue of positive magnetoconductance is smaller by1/4 to 1/10 than the theory as mentioned above.We conjecture here that the intervalley scatteringis important in 4-valley Ge. In theory theelectrons behave as if they moved in "one band"and positive magnctoconductance becomes small whenthere exist frequent intervalley scatterings. Thismakes the discrepancy small. But if it is thecase, it is puzzling that the anisotropy of positivemagnetoconductance does not vanish but shows theanisotropy which can be explained very Wl!ll hy Lhctheory ignoring the intervalley scattering.Fig.8. The experimental ratio is 1: 0.01 at lowfield and 1:0.37 at 0.9T. Concerning this dis-crepancy, we point out the anisotropy of the relax-ation time T. In the present theory, it is assumer1that T is isotropic, but it is not realistic. Forexample, the anisotropy of the electron mobility iRIJ.lIIJIl = 5'\, 10 [6) and is rather small compared witllthat of the effective mass. Hence, we have toconsider the anisotropy of relaxation time and theabove discrepancy is expected to be explained bythis.11-ii) Negative magnetoconductanceThe general trend of conduction of I-valley r.e shown in Fig. 6 is similar to thatof 4-valley Ge. Hence we analyze the conduction using the same formulae as Eqs. (2)and (3) and obtains the fitting parameters of mO= -1.53, ml = 0.4, J = -O.IO'\, -0.28and B =0.54 for the sample shown in Fig. 6 . The value of J dependes t:hedirection of the magnetic field as is discussed below. On the other hand, theinteraction theory qives the values of mo =-0.19 ann ml ·... O.04 for this sample.Discrepancy between theory and experiment which is same or larger than in the caseof 4-valley Ge exists. We can point out, however, that the theoretical valuesdepend strongly on the anisotropies of effective mass and relaxation time and t:hatthe agreement is improved if the anisotropy of the system is smaller than that ofmass as is the case with positive magnetoconductance.Next we examine the anisotropy.of negative magnetoconductance. 1"ig.9 shuws Llwmagnetic field direction dependence of J of the sample with N=3.0xl0 17cm- 3 at T =67.7mK. Magnetoconductance is largest when II~ [111] and smallest whenlJlI[lll].30This is contrary to the positive magnetoconductanceshown in Fig.7.According to the interaction theory, twoprocesses are important in magnetoconduction at lowtemperature: one is due to the spin Zeeman effect andthe other due to the interplay of the Zeeman and theorbital motion effects. The latter shows the aniso-tropy which varies depending on the ratio of Zeemanenergy to orbital ener.gy, A. In our sample, A isestimated small and this magnetoconductance showsthe anisotropy reflecting the mass anisotropy. Thisis shown in Fig.8(c). Magnetoconductance is largestwhen H II [Ill}. The origin of anisotropy of Zeemanterm is the g-value. The g-value of an eJectron inconduction band of Ge is 0.87 along [Ill} and 1.92along [110] and [112]. Therefore, the effect ofmagnetic field is largest when H.L. [Ill] as shown inFig. 8 (d) •90£111130 60e(degree)67.7mKGe:Sb3.0 X10'7cni3[1121o0.10.2the plane (110).netic field is rotated inBy using these theories, the experimental aniso-tropy is expressed as0.73x(negative M.C. with the anisotropy of Zeemanterm) + O.27x(positive M.e. with the ani!:iOtrupy uJorbital, or localization term). The experimentsuggests that the dominant contribution to low temperature maqneto-conductance is the Zeeman term of the interaction theory. This is consitent withthe theory, where the positive maqnetoconductance due to localization almost cancelsthe negative one due to the orbital term of the interaction theory and only the spinZeeman term remains.Fig.9 Anisotropy of neqa-tive magnetoconductance ofl-valley Ge with N=3.0xl017-3cm at 67.7mK. The currentis along [Ill] and the mag-By this experiment, it becomes clear that the positive magnetoconductance isoriginated by the localization effect and that the intervallev scattering and theanisotropy of the relaxation time need to be considered. The contribution of theZeeman term of the interaction theory is dominant in the negative magnetoconductanceof I-valley I.e at low temperature. But the theory cannot explain the absolute valueof temperature dependent conductivity even in I-valley case. This is afuture problem.References.[1] Y.Ootuka, S.Kobayashi, S.Ikehata, W.Sasaki and J.Kondo, Solid State Commun. 30(1979) 169.[2} "Anderson Localization" ed. Y.Nagaoka and H.Fukuyama (Springer Verlag) 1982.[3} T.F.Rosenbaum, R.F.Milligan, G.A.Thomas, P.A.Lee, T.V.Ramakrishnan, R.N.Bhatt,K.Deconde, H.Hess and T.Perry, Phys. Rev. Lett. ~ (1981) 1758, Y.Ootuka, S.Kobayashi and W.Sasaki, Proc. 16th Intern. Conf. on Physics of Semiconductors(1982) 700, S.Morita, Y.Isawa, T.Fukase, S.Ishida, Y.Koike, Y.Takeuchi and N.Mikoshiba, Phys. Rev. B~ (1982) 5570.[4] A.Kawabata, J. Phys. Soc. Jpn. ~ (1980) 628 and private communication.[5] H.Fritzsche, Phys. Rev. 125 (1962) 1552.[6] M.Cuevas and H.Fritzsche, Phys. Rev. 139 (1965) A1628.31

8. Electronic Transport in Metallic Ge:Sb at Low Temperature(Experiments,I. Three Dimensional Systems)

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8. Electronic Transport in Metallic Ge:Sb at Low Temperature(Experiments,I. Three Dimensional Systems)

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