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OBSERVATION OF KOHN-TYPE ANOMALIES IN THE ELECTRON-PHONON INTERACTION ON THE
FERMI SURFACE OF DEGENERATE SEMICONDUCTORS
D. Carlson, A. Segmüller
To cite this version:
D. Carlson, A. Segmüller. OBSERVATION OF KOHN-TYPE ANOMALIES IN THE ELECTRON- PHONON INTERACTION ON THE FERMI SURFACE OF DEGENERATE SEMICONDUCTORS.
Journal de Physique Colloques, 1972, 33 (C4), pp.C4-81-C4-83. �10.1051/jphyscol:1972417�. �jpa-
00215094�
JOURNAL DE PHYSIQUE Colloque C4, supplkment au no 10, Octobre 1972, page C4-81
OBSERVATION OF KBHN-TYPE ANOMALIES IN THE ELECTRON-PHONON INTERACTION ON THE FERMI SURFACE
OF DEGENERATE SEMICONDUCTORS
D. G . CARLSON and A. SEGMULLER
IBM Thomas J. Watson Research Center Yorktown Heights, New York 10598
Rhurnk.
-
L'effet acousto6lectrique a CtC observe pour des vecteurs d'onde acoustique compa- rable au vecteur d'onde de Fermi. Pour des 6chantillons de GaAs avec une distribution d'klectrons d6gkn&rk, on trouve une discontinuit6 dans l'amplification piCzoelectrique acoustique quand I q I = 2 1 k~ I, oh q est le vecteur d'onde acoustique et k~ est le vecteur d'onde de Fermi. La rai- son en est que, dans un gaz d'tlectrons d&gknkr&s, I'emission d'un phonon avec conservation de 1'6nergie et de l'impulsion est impossible si I q I 3 2 k ~ . Ces effets ont Cte observ6s avec des fre- quences acoustiques de 1-3 x 101 1 Hz a I'aide de la diffusion inklastique de rayons X.Abstract. - Acoustoelectric effects for acoustic wavevectors comparable to the electron Fermi wavevector have been observed. For GaAs samples with degenerate electron distributions, we find a discontinuity in the piezoelectric acoustic amplification at 1 q I = 2 1 k~ I, where q is the acous- tic wavevector and k~ is the Fermi wavevector. This occurs because, in an electron gas obeying degenerate statistics, phonon emission with simultaneous conservation of energy and momentum between initial and final electron states is not possible for 1 q 1 3 2 k ~ . These effects were observed at acoustic frequencies of 1-3 x 1011 Hz using inelastic x-ray scattering.
Using x-ray scattering, acoustoelectric amplifica- tion as a function of acoustic wavevector has been observed in GaAs in the quantum limit, i. e., where the electron system obeys degenerate statistics and the phonon wavevector ( q ( = q is comparable t o the electron Fermi wavevector,
k,
[I].In samples with degenerate carrier distributions, the amplification decreases abruptly for q 2 2 k,.
This amplification discontinuity is a consequence of the sharpness of the Fermi sphere and the require- ments of energy and momentum conservation in the phonon emission process. This phenomena has the same physical origin as the Kohn velocity anomaly [2].
However, the amplification decreases by several orders of magnitude a t q = 2 kF, whereas the velocity change observed at this point is less than one percent.
Thermal shear waves were piezoeIectrically ampli- fied [3] by applying pulsed dc electric fields of approxi- mately 150 V/cm along the [Ol 11 direction of GaAs.
Typical samples consisted of a 10 p n-type epitaxial layer of GaAs grown on a semi-insulating GaAs substrate ; free electron concentrations were in the range loi5
-
5 x 10" c m P 3 and sample tempe- ratures were held close to 20 OK. The pulsed dc field was normally increased until a weak current satu- ration occured due to the acoustoelectric interaction with the intense acoustic flux. Under these conditions, the thermal shear waves were amplified by about lo4.The acoustic intensity as a function of q was obtain- ed by measuring the scattered x-ray intensity as a function of sample orientation [4] ; this measurement
involved scanning over the (400) GaAs reciprocal lattice point, along the [Ol I] direction, by rotating the crystal through the Bragg angle 8 and keeping the detector fixed at 2 8. CuK,, x-rays were used, and were incident on the sample 1 mm from the positive contact.
Figure 1 shows acoustic intensity vs q for a sample
I I I I
'0k5 , , , , , , lo6 lo7
ACOUSTIC WAVEVECTOR IN cm-'
FIG. 1. - Experimentally observcd acoustic intensity versus acoustic wave number for fast transverse waves in the [ O l l ]
direction. The symbol A indicates the point where q = 2 k ~ . with degenerate electron statistics (carrier concen- tration = 1.35 x 1017 ~ r n - ~ , temperature = 20 OK).
The acoustic intensity has a maximum at
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1972417
C4-82 D. G . CARLSON AND A. SEGMULLER
This maximum occurs where the amplification is greatest, namely at q = q,, where q, is the inverse of the electron screening length. The acoustic intensity decreases abruptly a t q = 2 kF = 3.2 x lo6 cm- l, reflecting the discontinuity in acoustic gain at this point. The secondary maxima at q = 3 x lo6 cm-"
represents acoustic sum frequency generation due to the piezoelectric electron-phonon coupling. This cou- pling, which has a singularity a t the Fermi surface, has been discussed elsewhere [5].
A physical argument is now presented to explain the gain anomaly at q = 2 k,. First, we observe that the acoustoelectric gain represents phonon emission by the free electrons of the sample ; this is a real process, i. e., one in which energy and momentum are conserved. (This is in contrast to those electron-phonon scattering events which result in a change in acoustic velocity [2].) The initial and final states of the electron will be labelled by k, and k,, and we will discuss only the case for emission of a phonon with wavevector in the x direction. If we assume a free electron disper- sion relation for the electrons, then for phonon emis- sion, the conservation conditions require
4 mv,
k,,
=5 + h
where m is the electron effective mass, V, is the acoustic phase velocity, /z is Plancks' constant and k,, is the x component of k ,
.
Figure 2a shows one process for the case of q
<
2k,.
FIG, 2. - a) Momentum relations for phonon en~ission when q < (2 k~
+
2 (ka - mValh)). Electron initial statc is kl, final state is kz. The Fermi surface is displaced by thc dc electricfield. 6 ) Case for q > (2 kp
+
2 (ka --mva/h)).The circle represents the Fermi surface, which for GaAs is a sphere ; the electron distribution is assumed to be degenerate. Due to the dc electric field, E =
~ 2 ,
the sphere is displaced from the origin of momentum space by kd = ezE/h, where z is the electron collision time and e the electronic charge. To achieve acoustic amplification, the electron distribution must be dis- placed enough to make phonon emission more pro- bable than phonon absorption, i. e., k , > mV,/h.
The phonon wavevector is nearly (but not exactly) bisected by the k, axis, and
I
k,I
wI
k,I
; both of these conditions occur because the phonon energy is much smaller than the electron energies involved.The process shown in 2a is allowed, since it repre- sents an electron transition from a filled initial state (k,) to an empty final state (k,). Figure 2b shows the results of the conservation conditions for a case where q > 2 k,
+
2 (k,+
mV,/h). Here k, corres- ponds to an empty initial state. Thus, this process will not occur, so that phonon emission (and acoustic gain) is absent. The gain discontinuity is expected at q = 2 kF+
2 (k,-
mV,/A), but kd<
kF and mVJh<
k, so the anomaly occurs at q w 2 kF.The linear acoustoelectric gain, a(q, w), is related to the frequency-dependent electrical conductivity, o(q, a ) , by [6] a(q, o ) =
-
q ~ ' Im(1-
i 4 7c o(q, o ) /cueo)-' where K is the electromechanical coupling constant, e0 is the real dielectric constant of the medium, and w is the acoustic angular frequency.The conductivity o(q, w) is found from a single par- ticle density matrix calculation. We will neglect elec- tron diffusion currents by assuming o z $ 1. When plane wave electron eigenfunctions are assumed, o(q, o ) is given by [I]
-
in, me2 L14%
w) =mq where
A ( X ) = 1 ( k l
-
x 2 ) log(x - k,) +
Xli, q m o imq m + - f k
2 hq Aqz d
q mu, im q = - - - + - - + k
2 hq hqz d '
In the above, no is the free electron concentration.
Figure 3 shows the calculated acoustoelectric gain vs q/k, for GaAs with no = 1.35 x 1017 cm-3 and k, = 2 molhq. When the nonelectronic acoustic losses and the finite collision time are included, the maxi- mum gain occurs at a lower value of qlk,.
The effect of electron collisions is to relax the conservation requirements, which in turn causes the
OBSERVATION OF KOHN-TYPE ANOMALIES I N THE ELECTRON-PHONON (24-83
FIG. 3. - Calculated acoustoelectric grain for no = 1.35 x 1017 cm-1 in GaAs. Zero temperature and w7 1 was assumcd.
rate of change of a(q, o ) with respect to q at q = 2
k,
to be finite. Non-zero temperatures reduce the sharp-ness of the Fermi sphere and will also result in round- ing off the gain discontinuity. When these real crystal effects are included the acoustic gain is still found to have a dramatic drop at q = 2 k,.
The decrease in acoustic gain for q
>
2k,
is related to the acoustic velocity anomaly seen in metals by Brockhouse [7] et al, and predicted by Kohn [2].However, the gain anomaly is much stronger than the velocity anomaly. The velocity and gain depend on Re ~ ( q , 0) and Irn ~ ( q , o), respectively, where Re ~ ( q , 0 ) is the real part of the dielectric response function, ~ ( q , CO) ; ~ ( q , o ) is related to the conductivity by ~ ( q , o ) = 1
+
o(q, 0 ) / ( 4 nio). The velocity ano- maly appears (in the absence of collisions and a t zero temperature) as a logarithmic singularity in d[Re ~ ( q , w ) ] / d q at q = 2k,
; the gain anomaly appears as a discontinuity in Im ~ ( q , a ) a t q = 2k,
and hence the singularity in 2[Im e]/dq is stronger than that in ?[Re e]/dq.It is a pleasure to acknowledge the important contri- butions of D. R. Vigliotti to our experimental tech- niques.
References
[I] CARLSON (D. G.) and SEGMVLLER (A.), Phys. Rev. [S] CARLSON (D. G.) and SECMULLER (A.), Phys. Rev.
Lett., 1971, 27, 195. Len.. 1972. 28. 175. , ,
[2] KOHN (W.), Phys. Rev. Lett., 1959, 2, 393.
[3] WHITE (D. L.), J. AppZ. Phys., 1962, 33, 2547. [6] SPECTOR (H. N.), Solid State Physics, edited by Fre- [4] CARLSON (D. G.), SECMULLER (A.), MOSEKILDE (E.), derick Seitz and David Turnbull, Vol. 19, p. 291.
COLE (H.) and ARMSTRONG (J. A.), AmZ. - - Phys. 171 BROCKHOUSE (B. N.), RAO (K. R.) and WOODS (A. D. B.),
Lett., 1971, 18, 330. Phys. Rev. Lett., 1961, 7, 93.
DISCUSSION
K. WIGMORE. - A matter of trivial experimental D. G. CARLSON.
-
Yes. The scattered X-ray inten- detail -you are monitoring only the first few microns sity is proportional to the cosine squared of the angle of the GaAs. Presumably therefore you are very sensi- between the X-ray scattering vector and the phonon tive to any layer of surface damage, such as has been polarization.mentioned several times at this meeting ?
K. RENK.
-
DO YOU find also phonon up-conversion D. G. CARLSON. - Our sensitivity to surface condi- by direct phonon-phonon interaction via anharmo- tions depends on the sample and on the X-ray wave- nicity ?length used. Using heavy element samples and long wavelength X-rays, the technique does, as you say, probe only the first few microns of the sample. Using light element samples and short wavelength X-rays, you can probe several thousand microns in depth.
D. G. CARLSON.
