Some properties of an electron gas in a quantizing magnetic field

Physica 85A (1976) 575-588 © North-Holland Publishing Co.

SOME PROPERTIES OF AN E L E C T R O N GAS IN A Q U A N T I Z I N G MAGNETIC FIELD Amal K. DAS DOpartement de Physique ThOorique, Universit~ de Genbve, CH-1211 Genbve 4, Switzerland

Received 11 May 1976

The electron gas neutralized by a rigid positive background and in a quantizing magnetic field, is studied in a 'quasi-classical' model previously proposed by the author and collaborators. The plasma oscillation of the electron gas shows an acoustic-type dispersion for small wave vector. The potential behind an ion moving along the direction of the magnetic field is calculated and is found to have a sinusoidal behaviour. Some consequences of this potential are pointed out. The energy loss by a moving charge in the electron gas is shown to exhibit some interesting properties. Other quantities such as light scattering and magneto-acoustic oscillations are also discussed. A derivation is given for a 'quasi-classical' linear response in the finite relaxation time approximation.

I. Introduction T h e e l e c t r o n gas in a q u a n t i z i n g m a g n e t i c field has b e e n the s u b j e c t of n u m e r o u s studies. T h e d y n a m i c a l p r o p e r t i e s of the s y s t e m s u c h as the p l a s m a o s c i l l a t i o n s h a v e b e e n s t u d i e d b y a n u m b e r of authors'-3). H o r i n g ' ) has a p p l i e d field t h e o r e t i c m e t h o d s while o t h e r s e.g. B e n f o r d a n d R o s t o k e r 2) h a v e c o n sidered the e q u a t i o n of m o t i o n of a single particle d i s t r i b u t i o n f u n c t i o n in a m a g n e t i c field. M e r m i n a n d C a n e l 3) have c o n s i d e r e d the e q u a t i o n of m o t i o n of a single particle d e n s i t y m a t r i x in the r a n d o m p h a s e a p p r o x i m a t i o n . It is k n o w n that these m e t h o d s are o f t e n p e r m e a t e d by i n v o l v e d a l g e b r a that t e n d s to o b s c u r e the p h y s i c s of the s y s t e m b e i n g studied. In r e c e n t y e a r s the a u t h o r a n d c o l l a b o r a t o r s ° - ' ) h a v e p r o p o s e d a simple q u a s i - c l a s s i c a l m o d e l to s t u d y the p r o p e r t i e s of a n e l e c t r o n gas in a q u a n t i z i n g m a g n e t i c field. This m o d e l has m o s t of the p h y s i c s of the r a n d o m p h a s e a p p r o x i m a t i o n or e q u i v a l e n t l y the s e l f - c o n s i s t e n t field a p p r o x i m a t i o n , while r e n d e r i n g the form a l i s m i n v o l v e d r e l a t i v e l y t r a n s p a r e n t a n d simple. T h e static s c r e e n i n g of a c h a r g e d i m p u r i t y in an e l e c t r o n gas in a m a g n e t i c field was s t u d i e d by Das a n d de A l b a 4) a n d b y Das a n d H e b b o r n ' " ) . T h e m e t h o d has b e e n a p p l i e d b y the a u t h o r ' " ) to a c h a r g e d Bose gas. W e refer to t h e s e p a p e r s for the details a b o u t the m e t h o d a n d the a s s u m p t i o n s of the q u a s i - c l a s s i c a l model. It is k n o w n that the e l e c t r o n gas in a m a g n e t i c field can s u s t a i n o s c i l l a t i o n s 575

576

A.K. DAS

transverse to the direction of the applied magnetic field, at frequencies equal to multiples of cyclotron frequencies. These modes are k n o w n as the Bernstein modes. There is another type of collective mode that propagates along the direction of the applied field. This mode, for low wave vector, has an acoustic dispersion i.e. the f r e q u e n c y to o~ q, the wave vector, in contrast to the usual plasma dispersion (to ~ q2) in the absence of the magnetic field. This is a special feature of the electron gas in a quantizing magnetic field. One of the objectives of this paper is to study this mode. The longitudinal linear response of the electron gas in a quantizing magnetic field to a test charge is the basis of the physical results derived in section 3. This response function has been calculated in the next section. The particular structure of this response function is responsible for the special features of the physical quantities presented in section 3, namely the acoustic plasma resonance, the potential due to a moving ion and the energy loss by a charged particle. One can think of other physical quantities that should exhibit similar behaviour. Light scattering in a quantizing magnetic field f r o m a degenerate semi-conductor or a semi-metal is an example. We have made some brief c o m m e n t s on the quasi-classical model and on the physical results derived thereof. These are in section 4. The response function has been calculated in section 2 on the assumption that the electrons (or the quasi-particles) have an infinite life time. There is certain unsatisfactory limitation associated with this approximation. In the appendix we have presented a semi-phenomenological formalism to calculate the linear response function, assuming a finite relaxation time for the electrons.

2. Longitudinal response in the quasi-classical model We consider an electron gas, a single c o m p o n e n t plasma, in a quantizing magnetic field. Considering the system first entirely classically, we can write the following set of B o l t z m a n n - P o i s s o n equations:

Of, ~_ p__.Of,+Oeh. O f o + e ( v × H ) . O f , at m Or 9r Op c Op

f, ~"

V%b(r, t) = -41re 2 f f,(r, p, t) d3p + 4~rZe26(r - vt).

(2.1)

(2.2)

4) is the self-consistent potential of the charge Ze moving with the velocity v. fl is the first order (proportional to ~/,) deviation f r o m the equilibrium distribution function fo for the electrons. H is the magnetic field. The a b o v e equations are classical in that they do not take into account, the Landau quantization of the electron energy levels in a magnetic field. As discussed in the introduction, in order to study the quantum problem a n u m b e r of methods have been applied. We shall replace the a b o v e set of equations by a quasiclassical model in the following manner: the effect of the quantizing magnetic

AN ELECTRON GAS IN A QUANTIZING MAGNETIC FIELD

577

field, a s s u m e d to be in the z-direction, is included in the zeroeth order in &, i.e. in the distribution function fo.

fo = fo(E,.,,~) = [exp (E .... - tx)/kBT + 1]-',

(2.3)

where

E,.p~ = (n +~)h~oc+p~/2m '

¢oc = eH/mc

and

are the L a n d a u levels and the cyclotron frequency, respectively. We then have the following quasi-classical equations:

Of~ + p_~ Of, + Odp Ofo _ Ot m Oz 0z Op.~ d2 dz2&(z,t):-

4

f, -c'

2[ eH \ 1re k4~--~--~c) ~

(2.4)

f

f , ( z , p ~ , n ) d p : + 4~rZe26(z - vt).

(2.5)

We have assumed that the test charge is moving parallel (or anti-parallel) to the external magnetic field. The integral f d3p has been replaced by

eH \4~2h2c] ~ f doZ"

f d3p__> [

This has been discussed in Das and de Alba.') The interesting point to note is that we now have a s y s t e m which b e h a v e s effectively as one dimensional. This is a special feature of our model in a magnetic field. It will be seen later that this particular feature has some interesting consequences. We shall solve eqs. (2.4) and (2.5) by linearising the latter in ~b; without linearization the solution looks very difficult, though it deserves some serious consideration. We note that the spatial b e h a v i o u r of q~ can be determined only for the z-direction. H o w e v e r , this limitation is not serious since we are interested here mainly in the longitudinal response of the electron plasma. It should be pointed out that the formalism set up in eqs. (2.4) and (2.5) would need modification if it is to be extended to study the transverse response e.g. the Hall effect in a quantizing magnetic field. The quasi-classical model itself is quite capable of being applied to investigate such effects as the Bernstein modes. The interested reader is referred to Horing and Danz6). There is one more limitation e m b e d d e d in eqs. (2.4) and (2.5). This is the use of the semi-classical Boltzmann equation itself. It is well k n o w n that such an equation describes a system in the region of long wavelength. Since we shall consider a degenerate electron gas characterized by a Fermi distribution, we should expect the familiar F r i e d e l - K o h n oscillations. But working with eqs. (2.4) and (2.5), we shall miss these oscillations. We can h o w e v e r r e m e d y this deficiency in a s o m e w h a t ad hoc, though not in an arbitrary manner. L e t us write eqs. (2.4) and (2.5) in Fourier space. .

G

- iwf, + iqp~m -' f + lq ~

f ,

ck - - --r'

(2.6)

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A.K. DAS

-qZ
f f,(q,p~,co) dp=+4rrZe26(co-qv). (2.7)

In passing, let us r e m a r k that by noting the following q

Op~

= lim [f0(E ...... ) -f,,(E..,,~)]

,,~,

and

(2.8)

qpz _ lira [ E ...... - E,,p~] m

q~O

we can arrive at an i m p r o v e d quasi-classical model. In fact this c o r r e s p o n d s to the usual R P A model. Eq. (2.6) is replaced as follows: - icof, + i[E,.,,z~, - E,,p~]f, + i[f,,(E ...... )-fo(E,.,,z)]cb-

f' .

(2.9)

T

If eq. (2.9) is used to calculate the dielectric r e s p o n s e , it will have the F r i e d e l - K o h n singularity in it. H o w e v e r , in this paper we shall c o n s i d e r the 'quasi-classical' model as r e p r e s e n t e d by eq. (2.6). F r o m eqs. (2.6) and (2.7), ~bo(q, co)

c~(q, co) - 1 + (4~re2/qZ)x(q, co)' w h e r e ¢~o(q, co) -~ - ( 4 ~ Z e 2 / q 2 ) 6 ( c o - q v ) m o v i n g charge, and

(

x(q, co)-- ~

eH

)~

,~,

f

(2.10) is the u n s c r e e n e d

potential of the

iq(afo/Op~) d P : ( _ i c o + l / r ) + ( i q p ~ / m )"

(2.11)

T h e dielectric f u n c t i o n is then given by

e(q, c o ) = l +

4.n-e2 q2 x t q , co)-

(2.12)

If the e l e c t r o n plasma is a s s u m e d to be d e g e n e r a t e and at T -~ 0 K. f,, is a unit step f u n c t i o n and the s u m m a t i o n o v e r n is up to nv given by (2.13)

Ev = (nv + ")hcoc + p~/2m,

E(q, w) can then be easily calculated. , ( q . co)

1-

4

"{

7re-~)

eH

I

1

[EF - (n + 9hcoA: ~ 2 ~ / 2 m -F - - - m - - .,Z,(o + i/r) 2 - (2q2/m)[Ev -- (n + ')hw~]" (2.14)

In the static limit i.e. w h e n w--+0, eq. (2.14) r e d u c e s to the result of Das and

AN ELECTRON GAS IN A QUANTIZING MAGNETIC FIELD

579

de Alba 4) •

2[

eH

~X/~m

e(q, 0) = 1 + z~rre ~ , ~ )

~

.F

1

,~=o[EF-- (n + ½)hoocl~'

(2.15)

where the radicant under the s u m m a t i o n is a s s u m e d to be positive.

3. Physical r e s u l t s 3.1. A c o u s t i c

plasma

resonance

Let us consider long-lived collective oscillations in the electron gas. We can then take the limit 1/~-~0 in the expression for the dielectric function e(q, oJ), and have I

Re

E ( q , o))

1

1

[ E F - - ( n + ~)hwc]2 ,~-% to 2 - ( 2 q Z / m )[ E F -- ( n + ½)htoc]'

A S"~"

(3.1)

where the principal value is to be taken in the summation. And Im e(q, 60) = A N/~~--q,=o

~-

,

(3.2)

with • 2[ e H it = 4 r re ~ 4 - - ~ c

"~ 2~v/2~m ) m

The plasma dispersion is given by E(q, to.) = 0.

(3.3)

We shall investigate the collective oscillations under two approximations. It is found that in both these cases the plasma dispersion is of acoustic type i.e. for low w a v e vector q, oJ ~ c q where c is some constant. This difference in the dispersion f r o m the usual quadratic type w o~ q2 for zero magnetic field has its origin in the following: the electron gas which is a single c o m p o n e n t plasma in the absence of the external magnetic field, behaves, under the field, as a multi-component plasma with electrons at different L a n d a u levels. It is known [see, for example, Pines and SchriefferT)] that a t w o - c o m p o n e n t plasma having charged particles of unequal effective masses (e.g. an electron-hole plasma) has an acoustic type dispersion for the collective mode. In our case the different L a n d a u levels act as different constraints of motion and hence simulate the effect of unequal effective masses. We shall now calculate the plasma dispersion, first and mainly for illustration, by considering only two Landau levels. Case 1. T w o L a n d a u

levels

e ( q , oJq)= 0 will then result in

580

A.K. DAS

A(Ev - ~h~o~)~

A(EF - ~hoJ~)~

lq- w2_(2q2/m)(EF_½htoD + w2_(2q2/m)(EF_3hw.. )

0,

(3.4)

w a n d q a r e a s s u m e d small such t h a t q u a n t i t i e s like o~ a n d ~o~,q2 etc., can be n e g l e c t e d . It then t a k e s s o m e s i m p l e a l g e b r a to s h o w t h a t ~o~ = cq, w h e r e explicitly

c

X/m

[(E~ - ..hw~.)~(Ev 3 [ ( G - ~ho~)~ + ( G -

~ ~ 3 I 1 ~ho~)q-"

_ ~hw.)]~

(3.5)

C a s e II. All Landau levels from 0 to nF L e t us first c o n s i d e r the s u m I

~ ( w ) = .~=,,w ~ - (2q~/m)[EF

•

(n + ~)hto~]"

(3.6)

N o t e t h a t w e a r e c o n s i d e r i n g the s y s t e m at T = 0 K. T o get a p h y s i c a l l y r e a s o n a b l e r e s u l t w e c a n a p p l y the E u l e r s u m m a t i o n f o r m u l a nF+ I

~. fin + ½)~

f i n ) dn.

(3.7)

n =o o

This w a y of s u m m i n g m i s s e s the de H a a s - v a n A l p h e n o s c i l l a t o r y t e r m s [Ref. 5a)]. A n i m p r o v e d s u m m a t i o n p r o c e d u r e is n e c e s s a r y to o b t a i n the l a t t e r t e r m s . After the summation

4,(,.o)=~ ±q~( X / E F mVmm

1

nFh,oc - ~/E-~)

COin

\'~ 2 q

q - X/ EF - nFho~

qq

(3.8) A g a i n f r o m the r e l a t i o n E(q, ~o.) = 0 a n d e x p a n d i n g the l o g a r i t h m i c f u n c t i o n for small q, a n d n e g l e c t i n g o)~ etc., 1 { [(X/EF - nFh,oc) ~ -- ( X / ~ ) ~

o% ~--m ~[(---~v =-~

~ ~ ) q "

(3.9)

T h i s is a t y p i c a l a c o u s t i c p l a s m a d i s p e r s i o n . T h e e x p e r i m e n t a l o b s e r v a t i o n of t h e a c o u s t i c t y p e p l a s m a o s c i l l a t i o n s has so f a r r e m a i n e d elusive. P l a t z m a n a n d W o l f f " ) h a v e r e v i e w e d the e x p e r i m e n t a l s i t u a t i o n . D e g e n e r a t e s e m i c o n d u c t o r s a r e s o m e of the p r o m i s i n g c a n d i d a t e s . W e w o u l d like to m e n t i o n that o u r p a p e r is not n e c e s s a r i l y the first to r e p o r t

AN ELECTRON GAS IN A QUANTIZING MAGNETIC FIELD

581

such an acoustic type plasma oscillation. Wolff") and others 9) have discussed it. But we believe that we have presented a physically more transparent formalism in our 'quasi-classical' model. 3.2. The wake behind a moving ion As an interesting example of the application of the dielectric function E(q, to) given by eq. (2.14), we have calculated the potential of an ion moving longitudinally i.e. parallel to the direction of the external magnetic field. The screened potential ~(z, t) is given by

(3.10)

We should like to draw attention to the one-dimensional Fourier transform used above. It will turn out that the specific form of the potential ~(z, t) has its origin in the one-dimensional Fourier transform that deserves some comments. As we have already emphasized, the problem has been effectively reduced to one dimension. This can be pictorially visualised as follows: Consider the jellium (electron gas + neutralising positive background) in the form of a cylinder having an infinite length. The magnetic field is assumed placed along the axis of the cylinder. The model we are considering describes well that part of physics which can be viewed along the cylindrical axis. The ion, say positively charged, is moving along the axis and we are interested in the spatial behaviour of the ionic potential along the same direction. We shall be content with the linear response of the electron gas to the moving ion. The screened potential ~6(q, w) is then simply 4~(q, w) - ~bo(q, to) E(q,w) ' Oo(q, to) has already been defined. We shall be concerned only with the real part of the potential. It can be shown that if the particle velocity v is either > p F ( n D / m or pF(nF)/m because in this range of velocity the linear response is a good approximation. From eqs. (3.1) and (3.10) we find

d~(z, t ) = - ( 4 1 r Z e 2) f q2e(q, ei"C....qv) ' dq,

(3.11)

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A.K. DAS

i.e.

& ( z ' t ) = - ( 4 r r Z e 2 ) f q2[1

eiq(= ~t) dq 1 1 "~ [EF-- (n + ~)ho&]2 -- A ,=0 y~ q2v2 -- (2q2/m)[Ev-- (n + ~)hoo~]

Therefore 4~(z, t) = - 81rZe 2 f

cos q ( z q 2

vt)

~:2 (nF, v) dq

(3.12)

0

with

"~ [Ev--(n + ~)h~oc]~" ' ~2(nF, v) = A ,~-'..,,v2 _ (2/m )[ EF - (n + ½)h~o~]"

(3.13)

Considering the real part

eb(z, t) = -41r2Ze 2 sin £(z - vt) Viewing the wake of the potential from the position on the ion 4,(R, t) =

41r~Ze 2 . ~

sin sCR.

(3.14)

This form of the potential is to be contrasted with the spatial behaviour of the wake behind a moving ion, recently studied by Neelavathi et al. '°) and by Day"). These authors do not consider the effect of the quantizing magnetic field on the spatial form of the wake. The potential in our case behaves sinusoidally and has no decay factor like I[R. Such a d e c a y factor is present in the potential derived by Neelavathi et al. The potential given by eq. (3.14) is in fact much like a Mathieu potential'2-'3). It is well known that such a potential gives rise to electron energy band structurel3). One should then expect a transient band structure when a stream of ions are moving through a metal in the direction of the applied magnetic field. We see that a wake of rather long range will be generated behind a moving ion. There could also be electrons trapped in this periodic wake. Neelavathi et al. have discussed some consequences of this kind of trapping. The potential minima can trap electrons and produce wake-riding states. Neelavathi et al. point out that it might be possible to detect electron capture into these states by observing the associated Auger electrons, or the soft X-rays generated in the capture process. We would like to add that in the case with the quantizing magnetic field, the above mentioned effect will be enhanced. When an ion having a wake like eq. (3.14) behind it emerges from a metal into vacuum the sinusoidal wake would vanish. Then the electrons which were following the ion in the wake will tend to appear outside the metal

AN ELECTRON GAS IN A QUANTIZING MAGNETIC FIELD

583

as a group of electrons having velocities centred around v, the ionic velocity. These electrons should then contribute a c o m p o n e n t to the electron distribution associated with the e m e r g e n c e of positive ions f r o m metals in a strong magnetic field. We believe that the potential as derived in this section has certain interesting c o n s e q u e n c e s and experimental observation of these effects should be worthwhile. 3.3. Energy loss by a moving ion Another quantity of interest is the energy loss by a moving ion in an electron gas in a magnetic field. Again the one-dimensionality of the problem in our model gives it certain specific feature. Let us briefly rederive the well-known expression for the polarization energy loss by a charged particle in an electron gas. (It may be recalled that in addition to the polarization loss, the particle can suffer a 'fluctuation loss' due to its fluctuating motion in the electron gas. The latter loss can be calculated in the f r a m e work of the F o k k e r - P l a n c k equation.) The energy loss by a particle of charge Z,e moving with velocity v in the electron gas medium is Energy loss = Force • Distance

- Z~e O~7 ~

• dR r=vt

T h e r e f o r e energy loss dE

dR -

Z,e

,

(3.15)

r=vt

where d~(r, t) is the potential of the moving particle as screened by the medium. In the linear response approximation we immediately arrive at d_EE dR

x" (ik) 1 k 2 e(k, k v)"

(3.16)

This is polarization energy loss. The a b o v e result has been derived for the three-dimensional case. In our case we are considering the charged particle moving collinearly with the direction of the magnetic field. As discussed earlier, we then regard the problem effectively reduced to one dimension. We therefore write

d~_)o,od~.... ,on=_4~.Z~e2i f dk k e(k,1kv)'

(3.17)

whence

dE

dR =

4zrZ21e2f do) e2(k, ~o) ~

e~+ E~'

(3.18)

584

A.K. DAS

where • = E, + i e 2 ,

to = kv.

We now see from the structure of the function E2(k, to) [eq. (3.2)] that the energy loss integral vanishes unless to = k p F ( n ) / m ;

v = pF(n)/m

for a particular Landau level n. In other words there is no polarization loss. This is an interesting result, and seems peculiar to the one-dimensionality of the model. In reality some reminiscence of the transverse direction will of course be present. Nevertheless we feel that a noticable dependence of the energy loss on the magnetic field can be observable. For this purpose the cyclotron energy htoc should be comparable to the plasmon energy htop for the medium in question. A mono-energetic beam (E = ½my 2) of charged particles can be bombarded into some semi-metals or degenerate semi-conductors and in the direction parallel to an applied magnetic field. One can then observe the energy loss as a function of the strength of the magnetic field. Alternatively one can keep the strength of the magnetic field fixed and can vary the energy of the beam.

3.4. Light-scattering in a m a g n e t i c field Light scattering from an electron gas in a quantizing magnetic field can provide some interesting information on the properties of the system. The formula for the differential cross-section for the scattering of light of f r e q u e n c y wo to frequency to,, by an electron in the plasma, resulting in a momentum transfer of the amount q is well known [see e.g. Platzman and Wolff'~)].

d2cr _( e2 "~2(to,") dto d ~

1

, , m c - / ,,7,,/ rr(l - e

~ ) Im E~, to)"

(3.19)

Arguing along similar lines as in the previous section we see that the differential cross section varies noticably with magnetic field. It is to be noted that this contribution arises from the single particle scattering and not from the scattering by the collective modes.

4. C o m m e n t s

We have missed the 'logarithmic singularity' at q = 2pv in the dielectric function E(q, w) because we started from the quasi-classical equation (2.6) instead of the ' R P A ' equation (2.9). In fact there will now be a series of the logarithmic singularities at q = 2pv(n), one for each Landau level n. As is well

AN ELECTRON GAS IN A QUANTIZING MAGNETIC FIELD

585

known, the logarithmic singularity in E(q, ~o) leads to the Kohn anomaly in phonon spectra. In the magnetic field case this gives rise to the longitudinal magneto-acoustic oscillations. A b o u t the wake behind a moving ion, we would like to remark that in the case of an ion moving in an actual metal, the system is of course three dimensional. In the quasi-classical model some trace of the transverse part of the system is included not in the form of the wave function (the Landau wave functions) but in the summation over the L a n d a u levels. We believe that the situation in a real metal ought to have some reminiscence of the onedimensional calculation of section 3. Similar comments can be made about the energy loss by a moving charge. In recent years a number of essentially one-dimensional ' c o n d u c t o r s ' have been discovered'%. Some examples are the T T F - T C N Q and the K r o g m a n n Salt K2Pt(CN)4Bro~. 3H20 (nick-named KCP). K C P consists of an array of a chain of Pt-atoms. The chains are assumed to be weakly coupled to each other. The electrons behave as nearly free electrons along each chain. Suppose a strong magnetic field is placed along the chain direction. Let us imagine a cylindrical tube representing each chain, the diameter L of each tube being smaller than the separation between the tubes. We have one more length in the problem, the diameter R of the lowest cyclotron orbit of the electrons. The tube diameter L can be comparable or even somewhat larger than R for very strong magnetic field. Admittedly this is a stringent condition on the applied magnetic field, but now-a-days fairly strong magnetic fields can be realized in a laboratory'6). Under these circumstances many of the results derived in this paper will be applicable to one-dimensional conductors. In particular we would like to mention the longitudinal sound wave propagation along the chain. (The magnetic field, so to say, will contract the wave function in the transverse direction and hence will contribute to make the inter-chain coupling even weaker. Therefore by varying the applied magnetic field the strength of the inter-chain coupling can be studied and hence the degree of one-dimensionality of the system can be enhanced.) The sound velocity is given by the B o h m - S t a v e r formula '~)

s2=ng

1

q2 e(q, sq)'

(4.1)

where /2p is the bare ion plasma frequency = 4~rNe2/m. It is clear that the acoustic sound velocity will exhibit an oscillatory behaviour as a function of the magnetic field. The question of a finite relaxation time enters at this stage. If the sound velocity is such that oJ~-~< 1 where ~o = sq, the dielectric function in eq. (4.1) should be derived for a finite relaxation time. A simple method to this end is described in the appendix. The formalism of this paper can be applied, in a straightforward manner, to a multi-component plasma e.g. the electron-hole plasma in a semi-conductor.

586

A.K. DAS

Appendix

Gauge-invariant formulation of the dielectric response The calculation in section 2 has b e e n d o n e a s s u m i n g essentially an infinite relaxation time ~- for the electrons. As long as we are interested in the range of values of ~o for w h i c h w~- >> l, the finiteness of the relaxation time could be ignored. H o w e v e r , for f r e q u e n c i e s ~o such that w~- ~< 1, one should regard r as finite. It is well k n o w n that the ~ - ~ f o r m u l a t i o n does not c o n s e r v e the local particle number. This is particularly i m p o r t a n t for longitudinal r e s p o n s e e.g. in the longitudinal s o u n d w a v e p r o p a g a t i o n in the f r e q u e n c y range ~o~-~< 1. W e have d i s c u s s e d elsewhere 'e) h o w a r e a s o n a b l e m i c r o s c o p i c f o r m u l a t i o n of the dielectric r e s p o n s e can be given, where this limitation is c i r c u m v e n t e d . In this appendix we shall closely follow that p r o c e d u r e within our quasiclassical model. T h e p r o c e d u r e is centred on the use of a p h e n o m e n o l o g i c a l relaxation time r. It should be m e n t i o n e d that the introduction of relaxation time in the p r e s e n c e of a m o d e r a t e l y strong magnetic field involves certain complications. T h e s e have been the subject of discussion by a n u m b e r of authors in recent years'~-2°). But there does not seem to be any u n a n i m o u s and s a t i s f a c t o r y c o n c l u s i o n r e a c h e d so far. W e have t h e r e f o r e decided to refrain f r o m m e r e l y c o n t r i b u t i n g to this c o n t r o v e r s y , and w o u l d remain c o n t e n t with our simple procedure. We a s s u m e , w i t h o u t d e m o n s t r a t i o n which m a y turn out to be fairly nontrivial, that the kinetic equation a p p r o a c h i.e, the equation of m o t i o n for the one particle density matrix, or the B o l t z m a n n transport equation is valid in the p r e s e n c e of the magnetic field even w h e n the relaxation time r can be a f u n c t i o n of the magnetic field strength. In that case one calculate such a r ( H ) , say in the Born a p p r o x i m a t i o n along the line of the w o r k of Kahn2'). For a dilute (non-interacting) distribution of & i m p u r i t y scattering potentials, the exact w a v e f u n c t i o n s can be obtained and used in calculating ~-(H). The idea behind our gauge-invariant formulation is to introduce a local equilibrium distribution f u n c t i o n )q, to which f, relaxes slowly. Then the c o r r e s p o n d i n g B o l t z m a n n - P o i s s o n equations are

c?f~+p~Of, q Orb O f ` , c~t

m cgz

f'-~'

(A.1)

cgz cgp~

~,o

dz 2 ~b(z, t) = -4rre-

f, dp:+ 4rrZe26(z

vt),

(A.2)

f,, = f,,(tx. 61z) w h e r e 6/x(z, t) is the local chemical potential and is proportional to fb(z, t). Fourier t r a n s f o r m i n g eqs. (A.1) and (A.2) we obtain

(

- iw +

iqp~) [, m

+ iq

Ofo ~h ~p:

f, r

6 , m Of,, r p:Op:"

(A.3)

AN ELECTRON GAS IN A QUANTIZING MAGNETIC FIELD

587

, 2[~4~_Y~c ) e l l \ ~ Jf f,(pz, q, n, 09) dp~ + 4zrZe:8(z - vt). - q26( q, 09) = -,~zre --2

(A.4) In eq. (A.3))~, has been expanded (see ref. 18) and the term proportional to 8/x has been retained. 3/x will be determined from the condition that the local particle number be conserved. This is given by the continuity equation div j

+ aN -~- = 0,

qj(q, 09) = 09N(q, 09),

eH N(q, 09)= (4---WT~c)~ f f, dpz, e

eH

j(q, w ' = m ( ~ )

(A.5)

(A.6)

~, f p~f' dp~.

(A.7)

The above equations are actually the analogues, in the present problem, of those discussed elsewhere recently by the author. From the above equations and after some algebra

iq[(eq/m)x3- wX,] 6tz(q, 09) = [(m09/r)X2 _ (eq/.r)X,],

(A.8)

where

x,=~f

a[,,lapz

[-i09 + lit + (iqpdm)] dpz

(A.9)

afo/ ap~

p~[- i09 + l/'r +

dp~,

(iqpdm)]

pz(afo/apz)

.

[-i09 T]7~-; C~qpz/m)]apz.

(A.IO)

(A.11)

Substituting for 3/x in eq. (A.3), solving for f, and d~ we finally obtain for the dielectric function ,(q, 0 9 ) = l + i 4 7 rqe 2 ( ~ eH +i47re:(

q

~

eH )

) x,(q,,o) (eqx3-m092(,) xdq, 09) (09X-----2~eqx,--~"

(A.12)

The quantities X~, X2 and X3 are not explicitly evaluated here. The evaluation is actually straightforward.

588

A.K. DAS

Acknowledgements The author gratefully acknowledges the hospitality extended to him by Professor C.P. Enz at Geneva, and a Fonds National Suisse Fellowship.

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