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Screening effect on piezoelectric phonon-electron scattering and the mobility
Solid State Communications, Vol. 5, pp.62l-623, 1967. Pergainon Press Ltd. Printed in Great Britain
SCREENING EFFECT ON PIEZOELECTRIC PHONON-ELECTRON SCATTERING AND THE MOBILiTY Y. Murayama Hitachi Central Research Laboratory, Kokubunji, Tokyo, Japan (Received 15 May 1967 by T. Muto)
Starting from a phonon Boltzmann equation and a momentum conservation equation for the coupled electron-piezophonon system the linear electron mobility was calculated with the Debye-Hueckel screening included (classical RPA electron gas).
IN ORDER to calculate the relaxation time due to any relaxation mechanisms, e. g. electronphonon scattering, most theories start from the electron distribution function or the Boltzmann equation for electrons. However, so far as the linear theory with respect to the applied field is concerned, the other approach should give the same result; this approach is connected with
where 1 / ~
the phonon Boltzmann equation, in which the time varying phonon Boltzmann equation) flux (the is made solution use of of the Instead phonon of the electron distribution function. This type of procedure was taken by Yamashita and Nakamüra~ when they treated the so-called “hot” phonon problem, where the amplified phonon causing
2 w~, ~ K2 = (averaged electro-mechwith ci~ anical coupling K constant)2 and w~= q2 ~2 (s: unrenormalized sound velocity). All~thescreening effect comes from the (RPA) polarizability of nondegenerate electron gas ~ iT a (9, w) and thus obtained w1 = qv 8 (q) gives the renormalized sound velocity, v5 flj).
=
c(q,w) =1+4
2~ (~) is the solution
iT a(q,w)
-
4
=
___________ -
(3)
+ ~
—
(U 1
electron current to saturate was of main concern. We calculate the linear mobility due to the piezoelectric phonon-electron interaction including the screening effect. Let the electron system be drifting by v6, which is proportional to the applied 2 field E. Now the phonon Boltzmann equation N (t~= N (t~ N (0’ = Ne (quilibrium) 9 ‘ ~q q’” q’ ‘ q
0
+
—
(
Of 2
U)2
On the other hand, the momentum conservatlon equation for the coupled system is obviously given by v~
=
v0 v~j
______ -
-
1
—~-
<~qcos(q. !d) Nq(t)>~(4)
-
where v 0 is an Ohmic drift velocity determined by any other relaxation mechanisms than the electron-plezophonon Interaction and T0 is its relaxation time, and the bracket means the summation over q. n is the carrier density and m the effective thiss of the carrier. Equation (2) and the explicit expression of used In equation (4) give the last term
‘1’ /
~ k~T/ is lved ~ 50 b ng Nq(t)
=
N exp
Pzw~
-
J
<....>
~•
(1/T) dt,
(2) 621
622
PIEZOELECTRIC PHONON-ELECTRON SCATTERING
Vol. 5, No. 8
4n~nm~dq5thq2~~qwlx(vdx~vs) Cq2q~ / ~2q2 ~ C=K2J~m/2kBT). exp((q2~qb)2 \ ~‘~B~”
~.-~--—-.—
1’
~“ .2
~
I+F(Q)
‘N.,
—F(0)
In this procedure N integrand was taken to be linear approximation.
9(t) involved In the N~ following the
I
.05.01
For simplicity, v~(q)is assumed to be constant, and q~ = 4itne2TcokBT stands for the Debye-Hueckel screening constant. It is an easy task to carry out the integration of equation (5) and the result leads to 2 ~ ~ + F’~’~ Vd — Vd 4 v[2 K2/’~mkBT’/ e ~2 “~‘~‘ rr ~0 ‘6’ —
—
‘
F”~’-—
~‘ ‘~
~1+ 12+ r%~.nQEi/ ‘ ~ ‘~‘~‘
—
~
‘~
/
~2q2D 8mk~T
The assumption underlying this treatment reads implicitly q . > 1(q: the wave number of a typical piezophonon, 2: the electron’s mean free path). However, thus calculated result coincides well with what Hutson4 reached on the classical basis (ordinarily considered to be q .t <1) without numerical factor. Equation (6) reduces equation (4) to (vi, Vd)/T with r = 1 + and = (r/r~)v 0. if we use thus defined ~r, equation (4) is restated as Vd
vd
=
-
1
V~,-V~ =
______
-
<~qcos (9• ~)
1 —
9. (4a) ‘N (t’ 9. /
Ne9’
‘
which is the same as the equation deduced by virtue of the projection procedure in Ref. 1. Our main interests reside on F(Q), the screening effect. When Q -. 0, the effect vanishes completely. For Q << 1 (for m = 0. 1m 3, Q is nearly 0, T 10-s), —~ 300° Ei(-Q) K, e~ =—. 10, 1~dte_t/t~0.5772 n -~ 1015 cm +lnQ. Hence, F(Q) 2Q1nQ. F(Q) is sketched in Fig. 1. -
~
FIG. I and 1 (for + F(Q) vs. Q. When QF(Q) increases larger n or lower T) the mobility ri,, = e’r~/mIs no longer proportional to T-°~5.The deviation increases along with Q as 1/(1 + F(Q)). In case Q> 1, the screening effect appear (forcalculation Q — =, 1+isF(Q) ~ 1/i — may 0), but then drastic the above Invalidated; neither RPA nor the approximation taken in order to obtain w~and ~2 is not a good approximation for ~2 q~/m> kBT (Q> 1). This theory is limited to the scope ~tth Q < 1. Experimentally, the screening effect will appear more easily on higher n samples than in lower temperature cases. The observation of M of photoconducting CdS by Pujita. ~ shows a good agreement with the T °~-Iaw down to some degrees Kelvin. The effect, therefore, should be testified by performing experiments using samples with more carriers. In conclusion, our new approach to calculate mobility, which uses the phonon Boltzmann equation, has shown to give the same resuit as what the ordinary theory gives to lowest order in the applied field, pointing out some aspects concerning the screening effect. The T and n dependence of the mobility was calculated for the piezophonon scattering in case 2q~ / mkBT < 1. ~ Acknowledgements
-
The author wishes to ac-
who gave him knowledge t)r. valuable S. J. Miyake discussions and Mr.and E. encouYamada, raged him throughout the course of the work.
Vol. 5, No. 8
PIEZOELECTRIC PHONON-ELECTRON SCATTERING References
1.
YAMASHITA J. and NAKAMT.JRA K., Progr. Theor. Phys. 33, 1022 (1965).
2.
PINES D., Elementary Excitations in Solids, Chap. V, Benjamin, New York (1963).
3.
MELJER H. J. and POWER D.,
4.
HUTSON A.R., J. Appi. Phys. 32, 2287 (1961).
5.
FUJ1TA H., KOBAYASHI K. and KAWAI K., J. Phys. Soc. Japan 20,
Physica 19, 255 (1953).
109 (1965).
Ausgehend von einer Phonon Boltzmann Gleichung, sowie der Impuls-Erhaltungs-Gleichung fUr eta gekuppeltes Elektronen-Piezo Phononsystem wurde die lineare Elektronenbeweglichkeit errechnet einschiiesslich des DebyeHückelschen Siebes (mit klassischem RPA Elektronen-Gas).
623
SCREENING EFFECT ON PIEZOELECTRIC PHONON-ELECTRON SCATTERING AND THE MOBILiTY Y. Murayama Hitachi Central Research Laboratory, Kokubunji, Tokyo, Japan (Received 15 May 1967 by T. Muto)
Starting from a phonon Boltzmann equation and a momentum conservation equation for the coupled electron-piezophonon system the linear electron mobility was calculated with the Debye-Hueckel screening included (classical RPA electron gas).
IN ORDER to calculate the relaxation time due to any relaxation mechanisms, e. g. electronphonon scattering, most theories start from the electron distribution function or the Boltzmann equation for electrons. However, so far as the linear theory with respect to the applied field is concerned, the other approach should give the same result; this approach is connected with
where 1 / ~
the phonon Boltzmann equation, in which the time varying phonon Boltzmann equation) flux (the is made solution use of of the Instead phonon of the electron distribution function. This type of procedure was taken by Yamashita and Nakamüra~ when they treated the so-called “hot” phonon problem, where the amplified phonon causing
2 w~, ~ K2 = (averaged electro-mechwith ci~ anical coupling K constant)2 and w~= q2 ~2 (s: unrenormalized sound velocity). All~thescreening effect comes from the (RPA) polarizability of nondegenerate electron gas ~ iT a (9, w) and thus obtained w1 = qv 8 (q) gives the renormalized sound velocity, v5 flj).
=
c(q,w) =1+4
2~ (~) is the solution
iT a(q,w)
-
4
=
___________ -
(3)
+ ~
—
(U 1
electron current to saturate was of main concern. We calculate the linear mobility due to the piezoelectric phonon-electron interaction including the screening effect. Let the electron system be drifting by v6, which is proportional to the applied 2 field E. Now the phonon Boltzmann equation N (t~= N (t~ N (0’ = Ne (quilibrium) 9 ‘ ~q q’” q’ ‘ q
0
+
—
(
Of 2
U)2
On the other hand, the momentum conservatlon equation for the coupled system is obviously given by v~
=
v0 v~j
______ -
-
1
—~-
<~qcos(q. !d) Nq(t)>~(4)
-
where v 0 is an Ohmic drift velocity determined by any other relaxation mechanisms than the electron-plezophonon Interaction and T0 is its relaxation time, and the bracket means the summation over q. n is the carrier density and m the effective thiss of the carrier. Equation (2) and the explicit expression of used In equation (4) give the last term
‘1’ /
~ k~T/ is lved ~ 50 b ng Nq(t)
=
N exp
Pzw~
-
J
<....>
~•
(1/T) dt,
(2) 621
622
PIEZOELECTRIC PHONON-ELECTRON SCATTERING
Vol. 5, No. 8
4n~nm~dq5thq2~~qwlx(vdx~vs) Cq2q~ / ~2q2 ~ C=K2J~m/2kBT). exp((q2~qb)2 \ ~‘~B~”
~.-~--—-.—
1’
~“ .2
~
I+F(Q)
‘N.,
—F(0)
In this procedure N integrand was taken to be linear approximation.
9(t) involved In the N~ following the
I
.05.01
For simplicity, v~(q)is assumed to be constant, and q~ = 4itne2TcokBT stands for the Debye-Hueckel screening constant. It is an easy task to carry out the integration of equation (5) and the result leads to 2 ~ ~ + F’~’~ Vd — Vd 4 v[2 K2/’~mkBT’/ e ~2 “~‘~‘ rr ~0 ‘6’ —
—
‘
F”~’-—
~‘ ‘~
~1+ 12+ r%~.nQEi/ ‘ ~ ‘~‘~‘
—
~
‘~
/
~2q2D 8mk~T
The assumption underlying this treatment reads implicitly q . > 1(q: the wave number of a typical piezophonon, 2: the electron’s mean free path). However, thus calculated result coincides well with what Hutson4 reached on the classical basis (ordinarily considered to be q .t <1) without numerical factor. Equation (6) reduces equation (4) to (vi, Vd)/T with r = 1 + and = (r/r~)v 0. if we use thus defined ~r, equation (4) is restated as Vd
vd
=
-
1
V~,-V~ =
______
-
<~qcos (9• ~)
1 —
9. (4a) ‘N (t’ 9. /
Ne9’
‘
which is the same as the equation deduced by virtue of the projection procedure in Ref. 1. Our main interests reside on F(Q), the screening effect. When Q -. 0, the effect vanishes completely. For Q << 1 (for m = 0. 1m 3, Q is nearly 0, T 10-s), —~ 300° Ei(-Q) K, e~ =—. 10, 1~dte_t/t~0.5772 n -~ 1015 cm +lnQ. Hence, F(Q) 2Q1nQ. F(Q) is sketched in Fig. 1. -
~
FIG. I and 1 (for + F(Q) vs. Q. When QF(Q) increases larger n or lower T) the mobility ri,, = e’r~/mIs no longer proportional to T-°~5.The deviation increases along with Q as 1/(1 + F(Q)). In case Q> 1, the screening effect appear (forcalculation Q — =, 1+isF(Q) ~ 1/i — may 0), but then drastic the above Invalidated; neither RPA nor the approximation taken in order to obtain w~and ~2 is not a good approximation for ~2 q~/m> kBT (Q> 1). This theory is limited to the scope ~tth Q < 1. Experimentally, the screening effect will appear more easily on higher n samples than in lower temperature cases. The observation of M of photoconducting CdS by Pujita. ~ shows a good agreement with the T °~-Iaw down to some degrees Kelvin. The effect, therefore, should be testified by performing experiments using samples with more carriers. In conclusion, our new approach to calculate mobility, which uses the phonon Boltzmann equation, has shown to give the same resuit as what the ordinary theory gives to lowest order in the applied field, pointing out some aspects concerning the screening effect. The T and n dependence of the mobility was calculated for the piezophonon scattering in case 2q~ / mkBT < 1. ~ Acknowledgements
-
The author wishes to ac-
who gave him knowledge t)r. valuable S. J. Miyake discussions and Mr.and E. encouYamada, raged him throughout the course of the work.
Vol. 5, No. 8
PIEZOELECTRIC PHONON-ELECTRON SCATTERING References
1.
YAMASHITA J. and NAKAMT.JRA K., Progr. Theor. Phys. 33, 1022 (1965).
2.
PINES D., Elementary Excitations in Solids, Chap. V, Benjamin, New York (1963).
3.
MELJER H. J. and POWER D.,
4.
HUTSON A.R., J. Appi. Phys. 32, 2287 (1961).
5.
FUJ1TA H., KOBAYASHI K. and KAWAI K., J. Phys. Soc. Japan 20,
Physica 19, 255 (1953).
109 (1965).
Ausgehend von einer Phonon Boltzmann Gleichung, sowie der Impuls-Erhaltungs-Gleichung fUr eta gekuppeltes Elektronen-Piezo Phononsystem wurde die lineare Elektronenbeweglichkeit errechnet einschiiesslich des DebyeHückelschen Siebes (mit klassischem RPA Elektronen-Gas).
623