NGUYEN VAN TRONG and HA VINH TAN: Anomalous Electromagnetic Fluctuations

phys. stat. sol. (b) 84, 345 (1977)

Subject classification: 6 and 13

34 5

Laboratory of Theoretical Physics, Institute of Physics, State Committee for Science and Technology, Hanoi

On the Quantum Theory of Anomalous Electromagnetic Fluctuations in Electron-Phonon Systems

BY NGUYEN VAN TRONG and HA VINH TAN

The fluctuations in electron-phonon systems in the presence of a long-wavelength electro- magnetic radiation field are investigated. For obtaining the correlation functions of electromagnetic fluctuations a microscopic formalism is proposed in the framework of the random phase ap- proximation. It is shown that the level of collective fluctuations increases anomalously when approaching the borderline of the parametric instability region.

Die Fluktuationen in Elektron-Phonon-Systemen werden bei Anwesenheit eines kangwelligen elektromagnetischen Strahlungsfeldes untersucht. Zur Erzielung der Korrelationsfunktionen der elektromagnetischen Fluktuationen wird ein mikroskopischer Formalismus im Rahmen der "random-phase"-Naherung vorgeschlagen. Es wird gezeigt, da0 das Niveau der kollektiven Fluktuationen anomal zunimmt, wenn es die Grenzlinie des paramagnetischen Instabilitltsbe- reichs erreicht.

1. Introduction The electrodynamic properties of statistical charged particle systems can be

described by consideration of fluctuations. I n equilibrium systems, the correlation functions of fluctuations are related with

the response function by the fluctuation-dissipation theorem. I n the case of a classical collisionless plasma with non-equilibrium distributions a microscopic theory has been developed by Sitenko and co-workers [ 1, 21.

I n this paper, we consider an electron-phonon system in the presence of a long- wavelength electromagnetic radiation field and study the correlation functions of fluctuations in the framework of the random-phase approximation. Here is the case, when the Hamiltonian of the system depends on time explicitly and a suitable for- malism is proposed for the calculation of correlation functions. The expression for the electron density correlation function is obtained. The spectral distributions of fluctu- ations are characterized by the sharp maxima a t frequencies of two coupled modes (plasmon + phonon), which could be excited by the radiation field [3]. It is shown that the level of collective fluctuations increases anomalously when approaching the borderline of the parametric instability region.

2. Basic Equations Our system is described by the following Hamiltonian H:

H(t ) = ffo(t) + H i n t ;

346 NGUYEN VAX TBOXO a n d HA Vrsrr TAS

Here up,.Jui, s), and b k ( b i ) are the electron and phonon annihilation (creation) operators, respectively. Q,, = 4ne2/q2 and z'k repW3ent the Colllolllh and the electron- phonon interaction, respectively. The second tern1 in ( I b) descarihes the interaction of electrons with the ion background. 't'he condition of charge neutrality in the mean permits to express the ion density via the electron one. The symbol (...) means statistical averaging; .Ao([) == (cE0/o)) cos mot is the vector potent ial of the radiation field.

C'sing the Heisenberg representation we can define

(2) where

The averaged value of the dynaniical operator O(t ) niay he expressed via the density matrix 6 of the system a t the time to < t in the following way:

( l j ( t ) > t r { 6 ( t o ) ii(t)) . (V By choosing to -+ -a we can use the adiabatic hypothesis [a], according to which

the interaction of the particles as well as the radiation field vanish in the limit t + -+ -m. Thus ;(-a) may be replaced hy the density matrix of non-interacting par- ticles. Itestricting ourselves to the consideration of Gibbs distributions one can write

;(-a) = exp [ -~ T H 7 , where

and the symbol (...) will be understood as follows:

(...) : tr { exp [ ;?] ...}. (3')

On the Quantum Theory of Anomalous Electromagnetic Fluctuations 347

Averaging (4) one obtains the following equation:

The right-hand side of ( 5 ) describes the electron-eiectron and electron-phonon collisions. In the collisionless approximation the electron distribution function is time independent :

where Ep = p2/2rn and p, is the chemical potential of the electrons.

follows: The equations of motion for the operators b k ( t ) and b f _ k ( t ) may be represented as

Here the phonon relaxation time zp is formally introduced. However, in final ex-

As the averaged values of b k ( t ) and b f k ( t ) are equal to zero, the operators bk(t)

Using the random-phase approximation we neglect quadratic terms of fluctuations

pressions we will take zP -+ co.

and b + k ( t ) represent fluctuations of phonons.

in (4) and arrive at the following equation for S f :

where 1 = e q Eolmui.

equal to co in the final results.

follows :

Here the electron relaxation time z, is also introduced formally and will be taken

Let us introduce the operators of non-interacting particles, which are defined as

G&(t) X 0 - l ( t ) aP,,So(t) , b i ( t ) So-l(t) b k S 0 ( t ) , (9) where

So(t) == exp 1- i f dt' H,,(t')]. L -02 -.

348 NGUYEN VAN TRONC and Ha VINII TAN

The equations of motion for operators 6f:(p, p + q, t ) = aifs(t) a;+,,,(t) - fa@) 6,,0, b i ( t ) and bo_+,(t) may be represented in the forni

Solving (10) one obtains

Taking into account (3’) one can get the expressions of the correlation functions for operators of non-interacting particles Sfs, b i ( t ) , and bYk(t) :

Let us introduce the following notations

Then taking into account the initial conditions -

lini f ( t ) = lim &(t ) = lim ?;?k(t) = o t - t - -a , t+- -m t+-w

one obtains a set fo integral equations

349 On the Quantum Theory of Anomalous Electromagnetic Fluctuations

Substituting (17) into (16) one gets the following equation for h ( q , t ) :

t t‘ Bn(q, t ) - i J dt’ Q q ( t , t‘) [pq h ( q , t’) + 1vql2 J dt“ I‘q(t’ - t”) &(q, t”)] =

-w -w

1

= 8ao(q, t ) + i J dt’ Q,(t, t ’ ) vqQ:(t’) , (18) -W

where Tq(t’ - f r ) ~ i e-(t’-t”)/rp [eiwq(t’-t”) --im ( t ’ - t ” ) - e q 1 .

Using relations (12) we can determine the correlation function of the electron density

As a consequence of medium homogeneity, the correlation function is diagonal {&+(q’, t’) 8n(q, t ) ) with the help of equation (18).

with respect to q and q’, i.e.

Supposing the finiteness of the correlation function in the limit t , t’ --+ co we can use a Fourier transformation. For convenience, we transform the following function, but not the correlation function itself:

Let us multiply equation (18) termwise by the Hermitian conjugated equation, and perform statistical averaging. Using the expansion (lY), it is not difficult to obtain an infinite set of coupled equations for N,(w + swo, w‘ + s’wo) :

cq(w + swo) €X(W’ + s’wo) N,(w + swo, w’ + doo) -

350 NGUYEN VAN TRONG and Ha VINH TAN

where

3. Correlation Function for an Uncoupled Electron System

We first consider the uncoupled electron system in the presence of a radiation field. Thus we put vq = 0 in (20) and find the following expression for N,(w, w ‘ ) :

Consequently one obtains

One remarks that with the influence of radiation field the correlation function thus depends on the times t and t’ separately, but not on their difference ( t - t’).

4. Collective Fluctuations in the Absence of Radiation Field

Let us consider electron-phonon systems in the absence of a radiation field. There- fore, putting 2 = 0 in (20) we find the following expression for Nq(w, w ’ ) = = 2n 6(w - w ’ ) (6n2)am:

where Ao(u3 w ) = &q(W) - IvqI2 Pq(w) Dq(w) *

Ido(q, ~ ) l - ~ = lInl Ld0(q , w)I- l ns[Re d O ( q 7 w ) ] .

In the transparency domain of systems near eigenfrequencies, where the imaginary part of do(y, w ) is very small we can write

(23)

On the Quantum Theory of Anomalous Electromagnetic Fluctuations 35 1

Substituting (23) into (22) we obtain

(Sn2),, = 2 /(ewIT - l)-lI p-l[l CI - Re &,(w)] n 6[Re do(q, w ) ] . (24)

Let us examine the right-hand part of (24) in the long-wavelength region when Re F,(OI) takes the form

where wp(q) is the eigenfrequency of plasma oscillations. For a degenerate electron gas w:(q) = wz( 1 + f- qzw$/w2) = 02 + $ q2v& Here wL and 7iF are, respectively, Langmuir frequency and Fermi velocity.

Therefore Re do(q, w ) may be represented as follows:

where

are quadratic frequencies of plasmon-phonon coupled modes. Substituting (25) into (24) we obtain

<6n2),u = l(ew’T - 1)-lI A,(% w ) {6[w - W l ( 2 ) I + 6[w - wz(q)l + + 6 b + %(2)1 + 6 [ 0 + %42)1> , (26).

where

I n the T = 0 limit the spectral distribution of fluctuations vanishes in the region of positive values of w.

5. Collective Fluctuations at the Borderline of the Parametric Instability Region

The plasnion-phonon coupled modes should be excited by the radiation field, when the resonance condition wo = w1 + w2 is satisfied and the expression A = e q Eo/mw: exceeds some threshold value [3 ] . Phenomena of this kind are well known in nonlinear optics as parametric instabilities.

Let us consider the collective fluctuations when the condition wo = w1 + w2 is satisfied, but I takes stability values.

Assuming the two-mode approximation one takes into account only the frequencies w , 0) - coo, w’, and w’ - w,, in (20) for positive values of w and w‘. Therefore (20) may be reduced to a set of equations for N,(w, w’ ) , N,(w - wo, w’), N,(w, w’ - wo), and

One can also consider the region of negative values of w and 0)’ by taking into

The solution for collective fluctuations is found to be of the form

N,(w - wo, w’ - wo).

account only the frequencies w , w + wo, w’ , and w’ + wo in (20).

352 NGIJYEN VAN TRONG and HA VINH TAN: Anomalous Electromagnetic Fluctuations

where

I n the absence of a radiation field ( 1 2 = 0) one finds S(q , w , T , 0) = I(ew/T - 1)-11,

In the T = 0 limit (29) gives in which the expression (28) reduces to (26).

for w > 0 ,

for w < 0 .

I2

(30) F ( q , w , 0, 12) =

It is interesting to note that in the presence of a radiation field the spectral distribu- tion of fluctuations does not vanish for positive w-values.

Near the borderline of the parametric instability region, in which 1 - I 2 = 0, the function P ( q , w , 1') 12) increases anomalously as compared with the equilibrium function.

The analogous phenomena are well known in the collisionless plasma case, using the non-equilibrium distribution [5, 61. The indefinite growth of the fluctuation oscillation amplitude requires the necessity of taking into account nonlinear effects

Finally one must note here that the correlation function of the electron density is only represented. However, the correlation functions of other magnitudes may be also determined from their expressions related to the electron density (for example, the operator Qn(t) = bq(t) + b t g ( t ) is related with &(q, t ) by equation (17)).

c71.

References

f13 A. G. SITENKO and Yu. A. KIROCHKIN, Soviet Phys. - Uspekhi 9, 430 (1966). 121 A. G. SITENKO and 0. D. KOCRERGA, Preprint ITP, No. 76-10P, Kiev 1976. f3] N. TZOAR, Phys. Rev. 164,518 (1967). [4] M. GELL-MANN and F. E. Low, Phys. Rev. 84,350 (1951). 151 S. ICHIMARU, D. PINES, and N. ROSTOKER, Phys. Rev. Letters 8, 231 (1962). [S] S. ICHIMARU, Ann. Phys. 20, 78 (1962). 171 A. G. SITENKO, Physica scripta i , 193 (1973).

(Received April 11, 1977)