IC/83/9INTERNAL REPORT

(Limited distribution)

1. IHTRODUCTIOH

* • . .

International Atomic Energy Agency

• • , and-

United-.a,iations Educational Scientific and Cultural Organization

• "-''•'' INTERNATIONAL CEMTRE FOR THEORETICAL PHYSICS

OH THE EQUIVALEHCE OF TWO APPROACHES IB THE EXCITOH-POLAEITOH THEORY *

*»Ha Vinh Tan

International Centre for Theoretical Physics, Trieste, Italy,

and

Hguyen Toan Tliang

I n s t i t u t e of Phys i c s , Nat ional Centre for S c i e n t i f i c Research of Vietnam,Mghia Do, Tu Liem, Ha n o t , Vietnam.

ABSTRACT

The polariton effect in the optical processes involving photons with

energies near that of an exciton is investigated "by the Bogolubov

diagonalization and the Green function approaches in a simple model of the

direct band gap semiconductor with the electrical dipole allowed transition.

To take into account the non-resonant terms of the interaction Hamiltonian

of the photon-exciton system the Green function approach derived "by

Nguyen Van Hieu is presented with the use of Green's function matrix technique

analogous to that suggested by Kambu in the theory of superconductivity. It

is shown that with the suitable choice of the phase factors the renormalization

constants are equal to the diagonalization coefficients. The dispersion of

polaritons and the matrix elements of processes with the participation of

polaritons are identically calculated by both methods. However the Green

function approach has an advantage in including the damping effect of polaritons.

MIRAMAEE - TRIESTE

February 1983

* Submitted for publication.

** Permanent address: Institute of Physics, National Centre for ScientificResearch of Vietnam, Hghia Do, Tu Liem, Ha noi, Vietnam.

In recent years many optical processes in semiconductors have "been

intensively studied In the resonant region: resonant Raman scattering ,

resonant scattering of light by light , resonant electronic Raman

scattering and high order harmonic generation at resonance. 8)-iO)

frequencies

Due to the photon-exciton interaction in semiconductors there arises

new elementary excitation, which is a mixture of the photon and the exciton:

excitonic polaritons, or briefly the polaritons. Polariton effect plays

a very important role at the resonance because in this region the dispersion

of the polaritons is rather different from ^hose of the photon and the

exciton.

The polariton theory was developed by many authors: method of

Bogolubov diagonalization , Green's function approach * and other

microscopic approaches (for example, see Hef.19)- Recently a new Green

function approach to the scattering of the polaritons was derived by20)Kguyen Van Hieu . By the use of the renormalization recipe, this method

allows us to construct the matrix element of all optical processes involving

photon at the resonant energies. Moreover in this Green function approach to

the theory of polaritons the effect of the damping of polaritons can "be in-

cluded In a rather elegant manner.

The purpose of this work is to compare two approaches on the polariton

problem: the Bogolubov diagonalization approach and the Green function20)approach . We show that these approaches lead to the same results, but

the second has an advantage in including the damping effect of polaritons.

To this purpose we take a simple model of the direct band gap semiconductor

with the electrical dipole allowed transition and with on-degenerate valence

and conduction bands whose extreraa are located at the centre of the Brillouin

zone. We suppose that the polarization state of al l photons are given. Here

only the n = 1 exciton state is taken into account.

In Sec.II we give a review of the Bogolubov diagonaliaation method.

In Sec.Ill the Green function approach derived by Hguyen Van Hieu is

presented with the use of the Green function matrix technique analogous to£1)

that suggested "by Haabu in the theory of superconductivity . Sec,IV is

devoted to the study of the amplitudes of some polariton interaction processes.

We shall use the unit system with -ft = c = 1.

-2-

I I . BOGOLUBOV DL/BQNJtLIZAIION, APPROACH'

In Ref.20 i t was shown t h a t t h e exei ton-photon ti-aiis. i t icins.can.be

described by the following effective interaction Hamiltonian:

(t) = •iA(r,t)

with the effective coupling constant

(2)

where e and m are the (free) electron charge and mass, E = E(p" = 0) is

the energy of the exciton with the total momentum p = 0, <y(0) * <p(r = 0)

is the external wave function of the exciton in point ? = 0. f[ denotes the

matrix element of the dipole transition 'between the valence and conduction

bands

± (3)

£! is the elementary cell volume, % is the unit vector characterizing the

polarization of the photon, A(r,t) and BfF.t) are the effective scalar quantum

fields for describing the photons and excitons in the given polarization state.

We write

A(r,t) = A Al'\r,t)

where

fit) = B(V,t) ,

B (r,t) = B lr,t)

r ?,^ U?r- E(pjt]

-3-

(5)

• a * . * '

"a(k.) and tr(p) are the annihilation operators of the photor,= and the excitons,

respeeti.vely, €„ is the background dielectric constant, (j(k) is tl'e energy

of the photon with raoraentuni k.

To apply the Bogolubov diagonalization method we write the Hamiltonian

of the photon-exciton system as;

H - Z

h =

HY~

E(p)b+(p}

According to (l) HY~ is written in the form

H

(6)

(T)

(8)

(9)

By the Bogolubov diagonalization method we can rewrite the Hamiltonian (6)

H = 21y, if.

do)

In this formula C (k) and C (k) are the annihilation and creation operators,

respectively, of the polariton with momentum k in the branch V , £ (k) is

the polariton energy and can be determined by the algebraic equation

It is easy to obtain the precise expression for e (k)

(12)

The annihilation and creation operators of polaritons are connected with

those of photons and excitons as follows:

(13)

with the transformation coefficients•

u*

where

(15)

The annihilation operators of photons and exr-itons are expressed in terms of

the annihilation and creation operators of polaritona by the inverse trans-

formation:

(16)

Using (16) one "an ror.struet matrix elements of each interaction

proopss with the parti^ipat, ion of polariton from the corresponding processes

involving photons or cxoitonn. :i";rU'. method wa.i applied to the atudy of

resonant Raman scatteriug ' , absorption of light, , scattering of light

hy light ' and non-linear optical effect ' ' .

I I I . GREEK'S FUBCTIOtl APPROACH

It is essential to notice that due to the presence of the terms

in the Hamiltonian (l) corresponding to the non-resonant interaction tetveen

exciton and photon we must apply the Delow matrices of Green's functions

analogous to those used by Nambu in the theory of superconductivity.

First we define the matrices of free field operators of photon and

exciton

We denote by T the following matrix:

Then we can rewrite Hamiltonian (1) in the form

, BC1r,t))(17)

(18)

+ J-afsV.(19)

Introduce the matrix of Green's functions of free fields

T

t r U D, f^.f^t,-

(20)

where

V'V. V V -c <°/ T [

Analogously we define

' (21)

Here we denote by |0 > the vacuum state of non-interacting photon-exciton

system. We also denote by |v(fc)^ or |E (p)^> the one-photon or one-

exciton state with definite energies cO(k) or E(p) and momenta k or J

respectively, in the interaction representation. In the Heisenberg repre-

sentation we have the interacting field operators determined as follows:

r,t) = S(o,±)

r,t) = 5 (o,t)

Slt,o)

where

5 ft,te) =

*. "to

and the quantum interacting (dressed) field states

u! J J l

(22)

( 2 3 )

In a similar manner we define the matrices of the interacting field operators

of photon and exciton.-

and the matrices of the Green functions of interacting fields

Here | $ n ^ denotes the vacuum state of pnoton-exoiton system in the Heisenberg

representation.

From (20) and (21) we have the expression for the Fourier transforms

of each element of the matrix of Green's function of free fields

(27)

3EC*)I EC-*)-1

(23)

- 2

(29)

From (19), (22), (£3) n.nd the perturbativt expansions of the Green functions

of the interacting fields i t is easy to verify that the Fourier transforms

of these matrices of Green's functions must satisfy the linear matrix equation:

= 5 {$, T 6UT 6U,co) T OH

From (30) we obtain

6.

and similarly

G-ViM =

(30)

(31)

(32)

Each polariton corresponds to a pole of the Green function

D.,(k,{j). From (31) an

satisfies the equation

A -

Its solutions are

-H ,-or

DH tfe^j). From (31) and (32) we can see that the pole of the Green functions

(33)

By a comparison of expressions (12) and (3M we conclude that the

dispersions of polaritons derived by Green function approach coincide with

the result obtained in the Bogolubov diagonalization approach.

It is easy to see that

From (31), (32) and (31*) we have

(35)

where 6^(k) is determined as in (15) and

+ -j- (37)

From (37) and (15) we have the identity

(38)

-10-

In;ordef to calculate, the matrix; elements of: the interaction- proceses with the

.participation of polar it ons", we define the matrix wave functions of bare (non-

interacting) photons and excitons as the matrix elements:

(39)

The matrix wave function of the dressed (interacting) photons and exeitons are

determined in a similar manner

> S 'IfIf.*)- -,

The matrix wave functions "u (q»£) and vH(q,£) satisfy the equations:

Expressions (kl) show again that the energies and momenta of the dressed photon

and exciton must obey the dispersion equations of the polaritons, i.e. they

are the polaritons.

From expansions (5) it follows that for the one-photon or one-exeiton

state with the definite energy w(k) or E(p) and momentum k or p we have

0,

-11-

Similarly for the polariton branch'

It f f

With the renormalized constants Z^(k), W^(-k),

From the spectral representation of the Green function we have1

By a c o m p a r i s o n o f - E g g . (lilt i a n d ( 3 K ! w e o b t a i n

(1*5)

From (lU), (15), (37) and (1*5) it follows that with a suitable choice of the

phase factors of the renormali2ation constants

-12-

they exactly coincide with the Bogolubov transformation coefficients

If we use the phase factor as In Etjs.(llt) we have the following identities

i+6b)

They will be used In Sec.IV.

IV. APPLICATION AM) DISCUSSION

Let as apply the above results to the study of interaction processes

with the part icipat i-on or" the polaritons, '7o explain the method we shall

consider two simple examples. As a first ex:jmpie we consider the process

py(i) + a + c

where a,b,c denote some elementary excitations different from polaritons,

P (k) is the polariton in the branch v witU the momentum k this process

has been considered in Ref.2O, and ve will rot ;p into details here. We

just vant to recall that besides the processes

-L3-

a

d) + a

4 + c + •••

+ c +

(la)

(It)

which give the main contribution to the matrix element of the process (I) there

are two other processes which can also give some contribution

a -* (Ic)

a(Id)

This contribution is negligible in comparison with that of (la) and (Ib) in

the resonant domain and is often negelected in the resonant approximation.

Let the matrix elements of these processes In the absence of the photon-

exciton interaction be T , T , TT , T , respectively.13 lu j-C Id

FTOIQ expression (*+6) we have

E*(1*7)

wBy o. manner analogous with that in P.ef.PO we obtain the following exact matrix

element of pro^nss ( I ) :

W'(-*)[Trc£,UJ(*J

According to C*T) 1*, (k) can be rewritten in an equivalent form

Ib

*tf> T I C TT

Before considering the second example ve formulate the recipe for the

construction of the matrix element of each interaction process with the

participation of polaritons in the initial and final states. From the above

line of reasoning we see that for this purpose it is enough to write down all

the lowest order matrix elements of this interaction process with the direct

absorption and (or) emission of photon or exciton without exciton-photon

transition. The sum of these lowest order matrix elements which are multipliedy

by the corresponding renormalization constants 2,,(k) for an external photonEx

and Z , (it) for an external exciton will be the main contribution to the total

matrix element of the polaritou interaction process. To obtain the non-

resonant terms of the matrix element of this process we must write down all

those lowest order matrix elements but instead of the absorption (or emission)

of one photon (or exciton) we have the emission (or absorption) of this photon

(or this exciton) with the momentum in the inverse direction. Each matrix

element is then multiplied by the corresponding renormalization constants:

Wy(-k) for a substituted external photon and W^X(-k) for a substituted external

exciton. The sum of these matrix elements will Toe the non-resonant contribution

to the total matrix element of this polariton interaction process.Now we consider the following process:

Pv (<) a (ID

Corresponding to the above recipe we have the following lowest-order

processes which give the main contribution to the total matrix element:

+ a

a

a

4 + c + (Ila)

(lib)

(He)

a + c + (lid)

and the other lowest-order processes which give the non-resonant contribution

a

a

ft

C

c

a

+ Ex(*1)+'«- - -f + c t

+*(*,,) + a -» -6- + C +

E x (TA) + iff-?,) + 4 + c t

+ Ex (-X) + a -» 4 t- c +

Then the total matrix element nan be written as

1

(He)

dim)

dm)

(no)

(UP)

- 1 5 - - 1 6 -

f t or •»> «

ACKNOWLEDGMENTS

The authors would like to thank Professor Hguyeu Van Hieu for Suggesting

the problem and fruitful discussions. One of them (Ha Vinh Tan) would like

to thank Professor Abdus Salam, the International Atomic Energy Agency and

UilESCO for hospitality at the International Centre for Theoretical Physics,

Trieste.

W W Tffff0

The matrix elements of processes (l) and (II) can also be calculated by the

Uogolubov diagonalization method. From the expression (16) we can transform

the interaction Hami Itonian of photons and excitons with other elementary"

excitations into the interaction Hamiltonian of polaritons with them. Then

by use of this interaction Haitiiltonian we easily obtain the same expressions

U S ! and (50) for the T^ol(k) and T^°i \^ ,~ . but instead of the ^normalization

constants Zo(k), Z y (K) , W^(-k), W^ (-k) ve have th-;> dlagonalisation constants

V'.ik), "J. (k), V y(k), ~'7j (k). Due la the suitable choice of the phase factors

;.t J.s iVio'.ni aoovs th:-t these constants exactly coincide'• The dispersions

of polar Icon derived by the two approaches also coincide. We conclude

tr,at these two approaches are equivalent for the simple model of direct band

gap semiconductor. Our result can be generalized to more complicated cases,

lr. conclusion we empha-sise once more that the Green function approach

which was derived in Eef.,?O can include the effect of the damping of polaritons.

The results of the treatment of the damping effect of polaritons will "be

published . elsewhere.

-17- -18-

1)

2)

3)

1»)

5)

6)

7)

8)

9)

10)

11)

12)

13)

Ik)

15)

16}

17)

18)

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IC/12/222 BRYAN W. LYNH - Order aG^ corrections to the parity-violatingelectron-quark potential in the Weinberg-Salam theory; parityviolation in one-electron atoms,

IC/82/223 G.A. CHRISTOS - Note on the m .,,„„ dependence of < qq, > from chiralIMT.RFP * yUAKK.

perturbation theory.IC/82/221* A.M. SKULIHOWSKI - On he optimal exploitation of subterraneanlii'T.EEP.* atmospheres.

IC/82/225 A. KOWALEWSKI - The necessary and sufficient conditions of theINT.REP.* optimality for hyperbolic systems with non-differentiable performance

functional.

IC/82/226 S. GUNAY - Transformav. \":n methods for constrained optimization.IHT.REP.*

IC/82/227 S, TANGMANEE - Inherent errors in the methods of approximations:INT.REP.* a case of two point singular perturbation problem.

IC/82/228 J. STRATHDEE - Symmetry in Kalu?,a-Klein theory.INT.REP.*

IC/82/229 S.B. KILADKIKAH and S.K. GUPTA - Megnf.r. Lc momentc of lii.-it baryo...;in harmonic model.

IC/82/230 M.A. HAHAZIE, ABDUS 3ALAM and J. STEAIHDEE - Finiteneac of brokenU = h super Yang-Mills theory.

IC/82/231 N. MAJLIS, S. SEL2ER, DIEP-THE-HUHG and H. PUEZKAKSKI - ..'urfaceparameter characterization of surface vibrations in lino-ir chains.

IC/82/232 A.G.A.G. BAEIKER - The distribution of sample egg-count and itsIHT.REP.* effect on the sensitivity of schistosomiasis tests.

IC/82/233 A. SMAILAGIC - Superconformal current multiple^.INT.REP.*

IC/82/231* E.E. RADESCU - On Van ier Waals-like forces in spontaneously brokensupersymmetries.

IC/82/235 G- KAMIEIIIAKZ - Random transvarse Is.lng model in two liir.enoions.

R.D. BAETA - Steady ; r:r,.rtz.

G.C. GHIRARDI, A. JUUi:;: i/iit i1. wn;.-.n - Valve prpccrv-ing quantummeasurements: impossibility uhoornj. :nl Lo.'fri1 to,nl3 for the distortion.

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