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ULTRASONIC ATTENUATION OF HEAVILY
DOPED SEMICONDUCTORS IN THE WEAKLY
LOCALIZED REGIME
T. Sota, K. Suzuki
To cite this version:
JOURNAL D E P H Y S I Q U E
Colloque CIO, supplément au n012, Tome 46, décembre 1985 page C10-533
ULTRASONIC ATTENUATION OF HEAVILY DOPED SEMICONDUCTORS IN THE WEAKLY LOCALIZED REGIME
T . S O T A AND K. S U Z U K I
Department of Electrical Engineering, Waseda University, Shinjuku, Tokyo 160, Japan
Résumé
-
Nous avons étudié théoriquement l'atténuation ultrasono- re des semiconducteurs à la vallée-unique ou aux vallées-multi- ples fortement dopés dans des régimes faiblement localisés. Nous montrons que la modification quantique à la coefficient d'attén- uation, Au, décroit par l'augmentation de la témperature. Dans le cas des vallées-multiples, Aa décroit lorsque la diffusion inter-vallées devient forte. Le signe de Aa est le même que celui de Au, la modification quantique à la conductivité électrique dans le cas des vallées-multiples, mais opposé dans le cas de la vallée-unique.Abstract
-
We have theoretically studied the ultrasonic attenu- ation of heavily doped semiconductors with a single-valley or a many-valley in the weakly localized regime. The following has been found. A contribution from quantum correction terms to the attenuation coefficient Aa decreases with increasing temperature. In the many-valley case Aa decreases with increasing the strength of the intervalley scattering. The sign of Aa is the same as that of Au, the quantum correction to the electrical conductivity, in the many-valley case, but opposite in the single-valley case.1
-
INTRODUCTIONSince the proposal of the scaling theory by Abrahams et al. /1/, a number of experimental and theoretical studies of the electrical con- ductivity u of disordered systems have been performed in the weakly localized regime where, from a theoretical point of view, both the effects of randomness and mutual interaction between particles can be treated as perturbation and are expected to play a role in determining physical quantities.
Recently detailed measurements of the ultrasonic attenuation coeffi- cient a over a wide concentration region in S b doped Ge at T>2 K have been carried out / 2 - 4 / . The results show that the behaviour of a can be semiquantitatively explained on the basis of a simple free electron gas mode1 only in the concentration region much higher than the criti- cal concentration for the metal-non metal transition. Therefore it would be of interest to examine the behaviour of a in the weakly local- ized regime. Here we report a theoretical study of a of heavily doped semiconductors with a single-valley or a many-valley structure in this regime. In the many-valley case we consider any strength of the inter- valley scattering due to ionized impurities in contrast to previous work of a in which only the weak limit and the strong limit cases have been treated.
C10-534 J O U R N A L DE PHYSIQUE
THE ULTRASONIC ATTENUATION COEFFICIENT
When we assume that electrons responsible for a are the same as those contributing to O , an expression for a can be derived by using the field theoretical method developed to calculate O in disordered sys- tems 15-7/. We consider semiconductors with a single-valley or the many-valley structure. The electron-phonon interaction via the deforma- tion potential coupling is assumed. For simplicity, the following as- sumptions and approximations are made.
1.) The effective mass of electrons is isotropie.
2.) The frequency of the sound wave is low enough to satisfy the condi- tions ql<<l and w ~ < < l where q and w are the wavenumber and the aagular frequency of the sound wave and 1 and T are the mean free path and the total scattering time of electrons.
3.) Temperature is low enough to satisfy the condition k g T < < ~ F where E~ is the Fermi energy.
The following expression for a is obtained up to the leading terms of the quantum correction to the attenuation ~oefficient ( ~ e take unis of 6=kg=1). For the single-valley case,
a S = a S O + ~ a S = a S 0 ( 1 + 2 ~ ~ S R ~ 2 ~ ~ S I ) , (1) where 2 3 - 1 2
4
2 a S O = 2 ~ O I C D ] (pmvX) W Ts, Ts=1/(DqTF/q1,
bYSR=(3Q/4n) (~~~)-~(l-ni'i~i;/2), (2 ~Y,,=(3O/i6) (E,T)-~(~/~-F)+,,and for the many-valley case where we cqnsider n-type Ge as an example, a M = a M O t A a M = a M 0 ( 1 t 2 ~ ~ M R ~ 2 ~ ~ M I ) , (3) where 2 3 - 1 2 aM0'2~o~ICSil ( P v ) L! T M , T M = T r / 4 , 4 2 -bYMR=(3S7/4n) ( E ~ T ) (l-~d3T/T'tT/T~/2)
,
-~~,,=(3O/i6) (l/6-F/4)+1-3Fb2/4),
with @,=2nfiT$(l/%), +2=2nfiTi(1//wm+4/~'
),
wm=2nmT,-
1 F=x ln(ltn), (5Here k is the Fermi wavenumber and No, T, TI, T
,
and D represent, respecFiveiy, the density-of-states per spin, thfi total scattering time, the intervalley scattering time, the inelastic scattering time, and the diffusion coefficient at E.
F is the Fermi-surface average of the screened Coulomb interaction Eaicuiated using the Thomas-Fermi approx-imation, qTF is the Thomas-Fermi wavenumber, p is the crystal density,
v is the sound velocity, and C and C
.
are mthe dilational and the sBear components of the deformapion pgtential for eieetrons in the i-th valley. In Eq.(3) the term proportional to / C D ( is strongly screened by electrons and, consequently, the contribution to aM is neg- ligibly small. When we consider the electron-phonon interaction via the piezoelectric coupling instead of the deformation potential one for thesingle-valley case, a in Eq.(l) should be replaced by
2 3 2 9 2 2
a S P = 4 ~ e (pmvAzO) w T ~ , ~ ~ = 1 / D q ~ ~ ,
where ë is the effective piezoelectric coupling constant and is the
static dielectric constant.
A and AYMI only a contribution from the particle-hole channel is tagin into account according to Bhatt and Lee /6/. The factor of 2 in
front of AY (a=S, M, and b=R, 1) arises from the fact that they are
entered in %he expressions of a through both the density-of-states and the scattering time of electrons at E ~ . Here we would like to give a
few comments on AY
.
AY are the same as those obtained by Isawa etal. /7/ for o. exceck forSkhe sign of them and a lack of a contribution to AY from the particle-particle channel. AYMR is reduced to AYSR in the lsmit T I + m . bYMI becomes identical to the results obtained by Bhatt and Lee /6/ in the both limits T'+ and T I + O. Equations (1) and
(2) show that the sign of AYab in a is the same as that in o in the many-valley case but opposite in the single-valley case.
III
-
NUYERICAL CALCULATI0)JS AND DISCUSSIONSThe temperature dependence of AaM is shown in Fig. 1. In calculations we have used the following values jf the physica3 parameters: E T-2, F-0.8, r/r1=0.O~,0.l, p = .35(g/cm ) , v =5.57xlO (cm/sec), w=2xE0'
s e c ) i C
.
=(16/27yCi, and-
=16(e$). Furthe more, we have assumed T/TAT?'/^
aC&ording to ~ c h m i d - ~ / 8 / where A=lO-2 has been used so as to gatisfy the condition T<T'<<T€. Although we have calculated Aa it is extremely small because the electron-phonon interaction in 'the single-valley case is strongly screened by electrons. AaS is not shown here. The following has been found. Both AaS and Aa decreases with increasing temperature. In the many-valley case Aa Mdecreases and the temperature dependence becomes weak as the strengt# of the intervalley scattering increases.Since the behaviour of AY in the single-valley case has been already
discussed in literatures 79,9/, we discuss the many-valley case alone. The dependences of AY on T/T' and the temperaturehave been calculated, although it is not sgbwn here. The results are as follows. The l a r ~ e r
T/TI leads to the larger AYMR. The temperature dependence of AY 1s
weak when the relation T'<<T is satisfied. AYM4 decreases and MRthe temperature dependence becomgs weak with increa ing T/T'. Taking ac- count of the above-mentioned things and the fact that a M O is propor- tional to TI, the behaviour of AaM as shown in Fig. 1 can be understood.
-2
It can be seen from Eq.(4) that AaM are proportional to (E T)
.
As the carrier concentration decreases, E T becomes smaller and,Ftherefore,F
AaM becomes larger, see Fig. 2. Here it should be noted that the per-
turbation theory can be applied only in the region E ~ 1 1 . The magni- tude of AaM takes the maximum value at T=O K which c& be estimated to be at most ten percentsofaM. Therefore, when one investigates the weakly localized regime using the ultrasonic method, it would have to perform the measurement well below 1 K.
ACKNOWLEDGEMENT
We are grateful to Mr. M. Yawata for his help in numerical calculations. This work was supported in part by the Grant-in-Aid for Special Proj- ect; Research on Ultrasonic Spectroscopy and Its Application to Mate- rial Science from the Ministry of Education, Science, and Calture. REFERENCES
/1/ Abrahams, E., Anderson, P. B., Licciardello, D. C., and Ramakrishnan, T. V., Phys. Rev. Lett. @. (1979) 673
C10-536 JOURNAL DE PHYSIQUE
/5/ Fukuyama, H., in Anderson Localization edited by Y. Nagaoka and H. Fukuyama (Springer-Verlag, 1982) 89
/6/
Bhatt, R . N. and Lee, P. A., Solid State Commun. @ (1983) 755/7/ Isawa, I., Hoshino, K., and Fukuyama, H., J. Phys. Soc. Jpn.
2
(1982) 3262
/8/ Schmid, A., Z. Phys.
271
(1974) 251/9/ Lee, P. A. and Ramakrishnan, T. V., Rev. Mod. Phys.
3
(1985) 2870.006
-
t , , , ,
O 0.5 1.0
T i K 1
Fig. 1
