Bose condensation and superfluidity of excitons in a high magnetic field

PHYSICAL REVIEW B VOLUME 50, NUMBER 19 15 NOVEMBER 1994-I

Bose condensation and superfluidity of excitons in a high magnetic field

A. V. Korolev and M. A. LibermanCondensed Matter Theory Group, Department ofPhysics, Uppsala University, Box 530, S-751 21, Uppsala, Sweden

(Received 2 August 1993; revised manuscript received 23 May 1994)

In a high magnetic field, such that the distance between the Landau levels exceeds the binding energyof an exciton, an exciton gas in a semiconductor is capable of forming the Bose-Einstein condensate aswell as a superfluid state even at a relatively high temperature. We consider the problem of excitonic in-

teraction in a semiconductor in its multielectron formulation, starting from the second-quantization rep-resentation of the Hamiltonian of interacting electrons and holes in a high magnetic field. The expres-sions for the ground-state energy, the chemical potential, and the spectrum of elementary excitations ofthe system are obtained in a linear approximation in the concentration of excitons. It is shown that asystem of excitons in a high magnetic field is similar to a weakly nonideal Bose gas. The existence andthe stability of the Bose condensate due to an essential decrease of the interaction between excitons andan increase of their binding energy in a high magnetic field are established at a high density of excitons.The results obtained show that the excitation spectrum vanishes linearly, with a slope equal to a macro-scopic speed of sound which depends on the direction of the magnetic field. Thus this spectrum satisfiesthe Landau criterion for superfluidity. The possible observable effects of the Bose condensate and thesuperfluidity of excitons in a high magnetic field are also discussed. It is shown that Bose condensationleads to unusual optical properties of a semiconductor, e.g., anomalous absorption of light, and to anoth-er mechanism of light amplification.

I. INTRODUC;T1ON

A demonstration of the Bose condensate of excitonswould give another example of a quantum liquid underthe laboratory conditions. As is well known, almost allgases have a phase transition into a liquid phase when ei-ther the pressure is increased or the temperature islowered. A transition into a solid phase takes place withfurther lowering of the temperature. The only known ex-ceptions are the isotopes of helium He and He, whichremain liquid even at zero temperature. Such exclusivebehavior of helium is due to the extremely weak interac-tion between helium atoms and the great magnitude ofzero-point oscillations because of the small mass of a heli-um atom. Only in the case of helium is the energy ofzero-point oscillations large in comparison with the depthof the potential well of interaction between the heliumatoms. Thus quantum effects dominate in liquid heliumat low temperature close to the temperature T„wherethe de Broglie wavelength is of the order of the averagedistance between the atoms. The concept of a quantumliquid is of great importance in the quantum many-particle theory and solid-state physics. That is why theattention of many physicists is attracted to He and Heand also to obtaining a new quantum state of matter suchas the Bose condensate of excitons.

The possibility of creating the Bose condensate andeven the superfluid state of an exciton gas without a mag-netic field has been discussed by a number of authors. '

The well-known expression for the critical temperature ofan ideal Bose gas is"

g2N 2/3k, T, =3.31

where N is the concentration of the system and M is themass of the particle. If we assume that excitons obeyBose statistics, we see that the small effective mass of anexciton can make the condensation temperature quitehigh. For instance, large-radius excitons, so-calledWannier-Mott excitons, have the total effective massM =m, +m& —10 -10 g and the condensation tem-perature would become quite noticeable, T, —100 K, atthe concentration N-10' cm . In principle, there isno diSculty in creating the electron and the hole concen-tration of this order by optical excitation. ' ' Neverthe-less, clear experimental evidence for the Bose condensateof excitons has not appeared yet. To get insight, we shallrecall briefiy the main facts related to physics of excitons.

Evaluation of the binding energy of the Wannier-Mottexciton by the Bohr formula yields

4

2c. A

where c—10 is the dielectric constant andm =m, mhl(m, +mt, )-10 —10 g is the reducedeffective mass of the electron and the hole. Typical ener-gies of interaction between excitons are much higher atthe concentrations mentioned above. On the other hand,the exciton radius is quite large

ao = —10 cmme

and becomes of the same order as the average distancebetween excitons N ' at N-10' cm

Taking into account the smallness of the binding ener-gy of the exciton and the fact that at these concentrationsthe excitons greatly deform each other both by the direct

0163-1829/94/50(19)/14077(13)/$06. 00 50 14 077 1994 The American Physical Society

14 078 A. V. KOROLEV AND M. A. LIBERMAN

dynamic interaction and by virtue of the Pauli principle(for the excitons are made up of the Fermi particleswhich obey the Pauli principle), we cannot obviously re-gard the excitons as structureless Bose particles. Thus,carrying over the results of the theory of a nonideal Bosegas to the case of excitons needs special consideration.Keldysh and Koslov showed that the exciton operatorsobey the Bose commutation relations with accuracy toterms of the order of Rap. That implies the restrictionfrom above on the concentration of the exciton gasRap &&1. At such a concentration, the system of exci-tons does behave like a gas of Bose particles. The "non-Bose nature" of excitons shows itself only in the expres-sion for the scattering amplitude (the correction to thechemical potential of the system) defined as the total am-plitude for the forward scattering of two excitons. Thisamplitude can be obtained by solving the Schrodingerequation for two electrons and two holes with boundaryconditions corresponding to two excitons at infinity.However, if the exciton density is high enough, i.e.,10' —10' cm, the "interatomic" interaction of the ex-citons becomes so important that we can no longer de-scribe the situation in terms of excitons and the excitongas turns into an electron-hole liquid. Thus we cannotexpect a more or less high transition temperature of exci-tons into the Bose condensate. Besides, the kinetic ener-

gy of the exciton is approximately equal to the excitonbinding energy soon after the binding of an electron witha hole and the excitons are therefore rather "hot." So thetime needed to "cool them down" with transfer of theirenergy to the lattice, together with the finite exciton life-time, gives another serious limitation from above on theconcentration of excitons and therefore on the possibletransition temperature. Other restrictions come from thepossibility of formation of an excitonic bound state of thehydrogen-molecule type (biexciton). Such a state arisesmostly in semiconductors of the type A"'8, where themass of one of the particle (hole) is much larger than theother. Higher concentrations easily destroy biexcitonicmolecules because the kinetic energy of relative motion ofthe excitons is of the same order as the energy of interac-tion between the excitons. In the case when these massesare of the same order, the formation of excitonic mole-cules is not likely to happen either, due to similarreasons. Surnrnarizing all this, we find the following re-striction on the concentration of the exciton gas:

R p (&X (&ap (4)

i.e.,

where R p is the characteristic radius of the interactionbetween the excitons. This inequality gives us the con-centrations of 10' —10' crn and, accordingly, extreme-ly low critical temperatures. Thus observation of theeffects of quantum statistics still remains undecided. '

The situation is dramatically changed in a high rnag-netic field 8, such that the distance between the Landaulevels eke /mc exceeds the exciton Coulomb unit of ener-

gy (2R,„,).' ' In the fields

8 &&8 =m e c/E fl

4o(r )=- — exp

1 p 1

&2~k 4X' Qg aexp —,(7)

QpCL

where we chose the z axis in the direction of themagnetic field, k=&cfi/eB =aol+B, a = 1/lnB ', and

p =x +y . So the size of each exciton becomes smallerby the factor ln8' in comparison with the Bohr radius inthe direction of the magnetic field and smaller by the fac-tor +B' in a plane perpendicular to this direction. Themotion of the particles in this plane is almost suppressedby the field, so that the exciton looks like a thin needlestretched out along the magnetic field. Thus we deal withquite a curious situation of a three-dimensional systemmade of almost one-dimensional particles. The bindingenergy of the exciton is related to the parameter u andcan be written with logarithmic accuracy as

4exc me

~2 2~2@2

In the presence of a high magnetic field, the binding ener-

gy of the exciton is increased to a great extent as a resultof deforming a Coulomb potential by the magnetic fieldand becomes 10—15 meV (which is large in comparisonwith 0.6 meV for InSb without the field).

The pair interaction between the Wannier-Mott exci-tons is qualitatively similar to the case of hydrogenlikeatoms. In a high magnetic field, the spins of all particlesare fixed along the direction of the magnetic field (anti-parallel for electrons and parallel for holes). It is there-fore evident that the lowest state of a couple of excitonsin a high magnetic field corresponds to the total electronspin equal to unity, i.e., to the triplet state. Under thecircumstances, the pair interaction reduced by the ex-change interaction of the spins is a few orders of magni-tude smaller (at the distances between the excitons whenthere is no strong overlap of their wave functions) thanthe interaction energy of two excitons in the singletground state without any magnetic field. ' This ensuresthat the scattering amplitude of two excitons as a func-

8*=8/8, »1,the main interaction parameters of the exciton system arecompletely changed and this change may give a realchance of observing phenomena related to Bose conden-sation and superAuidity of excitons in different types ofsemiconductors. This scale of a magnetic field may actu-ally be a few orders of magnitude smaller than for hydro-gen atoms. For instance, the characteristic "atomic"field 8, is about 0.9 T for Ge and 0.2 T for InSb, so thatthe magnetic field in the range 10-20 T is already strongenough for these semiconductors. In such fields, as a first

approximation, we can treat the motion of the electronand the hole inside a single excitonic atom as one-dimensional motion in a Coulomb potential, while con-sidering the motion in the magnetic field in a perpendicu-lar plane. All the particles are considered to be at thezeroth Landau level. In accordance with the results ofRef. 17, the expression for the ground-state wave func-tion at a zero excitonic momentum can approximately bewritten as

50 BOSE CONDENSATION AND SUPERFLUIDITY OF EXCITONS. . . 14 079

tion of their energy has no poles on a real axis, i.e., thereis no "molecule" of two excitons (biexciton}. The experi-mental results obtained by Kulakovskii, Kukushkin, andTimofeev' and the numerical simulation of the problemof the exciton-exciton interaction in a high magneticfields' also definitely demonstrate the substantial de-crease of the exciton-exciton pair interaction and the ab-sence of biexcitonic states in a high magnetic field. Atthe concentration of excitons X-10' cm, the averagedistance between excitons is approximately ten times aslarge as their effective size. Taking into account thesmallness of the pair interaction between the excitons, wecome to the conclusion that the exciton gas under suchconditions may presumably be looked upon as a low-density gas of weakly interacting particles even at thedensities when the collective effects, such as the forma-tion of an electron-hole liquid, dominate in a system ofexcitons without a magnetic field at low temperature.Therefore, formation of the superfluid state becomes per-fectly realistic in a high magnetic field even at relativelyhigh temperature.

The problem of the Bose-Einstein condensation and thetransition of the exciton gas into the superfluid state in astrong magnetic field is considered in the present paper.In Sec. II we consider the problem of exciton interactionin a semiconductor in its multielectron formulation, start-ing from the second-quantization representation of theHamiltonian of interacting electrons and holes in a highmagnetic field. Expressions for the ground-state energy,the chemical potential, and the spectrum of elementaryexcitons are obtained in a linear approximation in theconcentration of the excitons. The existence and the sta-bility of the Bose condensate are established. It is shownthat a system of excitons becomes similar to a weaklynonideal Bose gas and that the excitation spectrum van-ishes linearly, with a slope equal to a macroscopic speedof sound which depends on the direction of the magneticfield. Finally, in Sec. III we discuss possible experimentalmanifestations of the Bose condensate and the superfluityof excitons in a high magnetic field. Section IV concludesthe paper.

II. THE CONDENSED STATE OF EXCITONSIN A HIGH MAGNETIC FIELD

We proceed now to the quantitative investigation ofthe problem at zero temperature. However, the resultsobtained below are also valid at finite temperature not tooclose to the critical temperature of a Bose gas. We re-strict the consideration that follows by neglecting the ex-cited states of excitons. It can be shown that this approx-imation is legitimate if the interaction energy between theexcitons is small compared with the energy of transitionto the excited states. For the case of a high magneticfield this condition is valid due to the strong decrease ofthe interaction between the excitons. We consider belowonly large-radius excitons, that is, "Wannier-Mott exci-tons. " Strictly speaking, these states exist in the two-band model if the exciton radius is large in comparisonwith the lattice constant. This is the situation that takesplace in the most typical semiconductors with the large

+g P &p[& p& p'+p'+&~ p —p +b pb p bp'+Qbp —k1

p~p r~

(9}p' p'+~ p —~~ '

where V&=4me A jek is the Coulomb interaction, apand b pt are the Fermi operators describing the creation ofelectrons and holes, while a and b describe electronand hole annihilation, respectively. The chemical poten-tial of the electrons JM,, and the chemical potential of theholes p& are determined by the conditions

g(ata )=g(btb )=n, (10}p p

where n is the dimensionless exciton number density. Wecan write n as n =NVO, where N is the number density ofthe system and Vo=ao. The dependence of the electron(or hole) energy on the momentum p in a high magneticfield is

~(p}= p+ A1 e

2' e C

el (p}= p — A27tl g

fe/aa2NleC

Jeff2mgc

(12}

where, assuming that one can only consider the fixed pro-jections of spins on the direction of the field under suchconditions, we set the electron-spin projection equal to—1 and the hole-spin projection equal to +1, respective-ly. This spin configuration corresponds to the lowest-energy state of an electron and a hole in a high magneticfield.

This choice of the Hamiltonian is determined by thefollowing reasons. First, in this consideration we focusour attention on so-called direct-gap semiconductors, i.e.,semiconductors in which the minimum of the conductionband and the maximum of the valence band are at thesame value of the wave vector. Second, this form of theHamiltonian implies that the electrons and holes are con-sidered as two independent types of particles. Thus weneglected the possible transitions of the electrons fromone band to the other and excluded their matrix elementsfrom the Hamiltonian. An important assumption madehere is the orthogonality of the wave functions of thedifferent bands. We can do that because the correspond-ing matrix elements of the Coulomb interaction are assmall as the ratio of the exciton binding energy to thewidth of the forbidden zone. Therefore, the energies ofelectrons and holes are reckoned from the edge of thecorresponding band and the exciton energy is reckonedfrom the width of the forbidden zone. In the samemanner, the momenta of electrons and holes are reckonedfrom their values at the bottom of the energy band.

As we expect, the ground state of the system has to be

dielectric constant and the small reduced effective massof the electron and the hole.

The Hamiltonian of the system of electrons and holesin a high magnetic field has the form

&= g t[e, (p}—V, ]~'p&, +[&~(p}—Vi ~b,bpI

A. V. KOROLEV AND M. A. LIBERMAN

8,„,„„=U+A', +8,"+8,'+Sf+Sf+A'f, (14)

where U is a numerical functional of uP and UP, which

appears after reducing the transformed Hamiltonian tothe normal form

U= X [&.(P) P +&h(P) Ph)vP

—g V .(upv u vp +vpv, ) . (15)P~P

The operators @,8z, and 83 include matrix elementsof the processes in which we have the combinations oftwo Fermi operators. The process related to the propa-gation of an electron through the medium is representedby the operator 8,

@1=y [~,(p) —p, ]u', —le'h(p) —ph]v',

made up of excitons in a high magnetic field, at least inthe case of effectively low density. This is exactly thecase discussed earlier in connection with the reconstruc-tion of the excitonic system in a high magnetic field. Theanalysis of the properties of the ground state and its sta-bility in the presence of the condensate of excitons ismost simply carried out by defining a new set of creationand annihilation operators which is known as the Bogo-liubov canonical transformation. Assuming the existenceof an excitonic condensate without a magnetic field, Kel-dysh and Kozlov carried out a similar analysis for thecase B =0. %e shall go through similar manipulations,but for the case of an electron-hole system at 8%0. Solet us introduce a new set of operators

a =u a +vpb, b =u b —v a, (13)

where a and b are operators of quasielectrons andquasiholes, respectively, and Q

P+U

P1 These new

quasiparticles being elementary excitations of the systemunder consideration go over into ordinary electrons andholes as the density of the system n goes to zero. Thefunctions u and UP must be determined from the condi-tions of minimum energy and of the stability of theground vacuum state of the system. All operator expres-sjons below are written through the new operators aP and

b„. Therefore, we shall omit the tilde in what follows. Itis not difficult to check that such a transformationpreserves all the commutation properties of the Fermioperators. Then, using these Fermi commutation proper-ties of the new operators and reducing the operator ex-pressions to the normal form, we can rewrite the Hamil-tonian in the form

FIG. 1. The graphic representation of the matrix element ofa, related to the propagation of the electron.

P

[~h(P) I h—]u,' —[~,(p) —V, ]v,'

+ g Vp p [(vp up)vp +2upvpup vp ] bpbp.P

It is important to note that we use a right-pointing arrowfor an electron and a left-pointing arrow for a hole inFeynman diagrams throughout the paper.

Besides the scattering processes of the electron and thehole, there are also the virtual creation and annihilationof electron-hole pairs from vacuum, shown in Fig. 3. Thematrix elements responsible for such processes are in-cluded in the operator 83,

e, (p) —p, +~h(p) ph—2X—Vp-p vp upvp

P

—(u2 —v2) g Vuv„, (a .b +b pap) .

P

The operators A'f&, Bf2, and Pf3 contain all possiblecombinations of four Fermi operators. The matrix ele-ments in P~& (Fig. 4) are similar to the matrix elements ofelectron and hole scattering processes described by theinitial Hamiltonian

&f—iii 1 2 X Vk Yp, p —k Yp', p'+k

X[apa ap+kap k bpbpbp+kbp

—2apbpbp+kap k] . (19)

For example, the diagram of Fig. 4(a) describes thescattering of two electrons, the diagram of Fig. 4(b) givesthe same process for two holes, and the scattering of theelectron by the hole is shown in Fig. 4(c). Here we haveintroduced the notations yp P QP QP +Up Up

QP P QPUP VPQPThe matrix elements in the operator Qf2 describe the

processes of creating or annihilating an electron-hole pairfrom vacuum and scattering a Fermi particle:

+ g Vp p. [(vp —up)vp. +2upv u .v, ] a aP

(16)

Its graphic form is given in Fig. 1.A similar expressions also holds for the propagation of

a hole (see Fig. 2)

FIG. 2. The graphic representation of the matrix element ofm2 related to the propagation of the hole.


Af tt t tt t t tLz 2 + V&1 p p pPp p +Q[upup b p pap „b—

pbp a p „bp „—apap pb p Qp +p +bpbp pb p up +& ]P~P ~&

(20}

The graphic representation of difFerent terms of this operator is given in Figs. 5(a}—5(d). The vertex corresponding tothe scattering a Fermi particles has the factor yp p Q and the vertex corresponding to the creation or annihilation ofelectrons and holes has the factor yp p +Q.

All possible combinations of the matrix elements corresponding to the creation and the annihilation of two pairs aregathered in the operator Bf3..

Pf i m y — —u

tent

t3 z ~ p1 p p —p1 p p +p[upQpb p pb p+p+Qpup b p pb p+p +20 pb p+pb p+&0 p ]

P~P ~&

(21)

The Feynman diagram of Fig. 6(a) describes the creation of two electron-holes pairs from vacuum. Figure 6(b) showsthe opposite case, i.e., the annihilation of two electron-hole pairs. Finally, the process in which one electron-hale pair iscreated and the other is annihilated is shown in Fig. 6(c). Accordingly, each vertex in the diagrams representing theseprocesses has the factor yp p.

Using the Bogoliubov transformation (13), we can rewrite normalization condition (10) in the form

—,' y ((u ut+v b )(u a +v bt )+(u bt —v u )(u b —v u ))

P

=—,' g (2vp~+(up vp)—(~pap+bpbp)+2upvp(atbt p+b a })=n,

P

remembering that we omitted the tildes over the newoperators of the new Fermi quasiparticles, and finally geta new normalization condition

g [vp+ —,'(u —v~)(ata +bpb )P

+upv (atpbt +b ap)]=n . (22)

The functions (a ap) and (blab ) in (22) describe thepropagation of an electron and a hole, respectively,through the medium. The chemical potentials of elec-trons and holes are obviously negative (p, +ps =p), forthe chemical potential of excitons p must be near the en-ergy level of a free exciton in the case of the low-densityexciton system in a high magnetic field. Therefore, thereis no Fermi sphere (the radius pz of which is usuallydefined by the equation pp /2m, =p, ) and all the levels ofsingle-particle Fermi excitations have to be empty. Thismeans that the values (atap ) and (blab ) should vanishin the normalization condition (22).

As is known, in the choice of the parameters of thecanonical transformation, one has to keep in mind that itis necessary to guarantee the mutual compensation of allprocesses which lead to the virtual creation or annihila-tion of electron-hole pairs with opposite momenta andspins of an electron and a hole from vacuum becausethese processes immediately lead to divergences. Suchprocesses are described by the mean values (a pb p ) and(b pap) and arise in all orders of the perturbation-theory series. The principle of compensation of

(atbt )=(b a )=0. (23)

The normalization condition for vP therefore becomes

V =PlP (24)

All the matrix elements responsible for the creation orannihilation of electron-hole pairs are gathered in the

p p p'+ k

I

dangerous diagrams, first postulated by Bogoliubov, canin fact be derived from a number of difFerent and quitegeneral criteria, such as the assumption that the totalnumber of quasiparticles in the true ground state isminimum, the expectation value of an arbitrary operatoris simplified by diagonalizing its quadratic part, and thestarting point for dressing quasiparticles is chosen in themost convenient way. ' Making use of the principle ofcompensation of dangerous diagrams, and choosing vPand up such that the matrix elements in the transformedHamiltonian related to such processes are equal to zero,we should set the last term in (22) equal to zero. This, asis well known, should in fact define the function vP. Thusthe condition of the stability of the ground state of theexciton system takes the form

—p (c)

FIT&. 3. The processes of virtual creation and annihilation ofthe electron-hole pair included in 83.

FIG. 4. All the possible processes of scattering of the elec-tron and the hole included in 8,.


I—p —k

p'+ k y)'+ /.-

(b)

citon phase in the low-density limit to the electron-holeliquid at high density. At intermediate densities thesetwo limiting phases would be connected smoothly if thedensity fluctuations were suppressed. Due to the densityAuctuations, the excitonic phase and the electron-holeliquid should coexist for some range of intermediate den-sities. Since the excitonic system in a high magneticfield is believed to be an effectively low-density system atthe densities considered above, this region of intermedi-ate densities is pulled up to 10' —10 cm

This equation reduces in the first approximation in v

to the Schrodinger equation of the single exciton boundstate in a high magnetic field

[~,(p)+ ~h(p }—p]u, o—X V,—,v, 0 =o . (26)

FIG. 5. The Feynman diagrams for the matrix elements of8, describing creation or annihilation of an electron-hole pairand scattering a Fermi particle.

Hamiltonian 83 (18). It follows directly from (24) thatu -~n. Furthermore, as will be con5rmed by the sub-sequent analysis below, v -(1/'}/B'1nB')~n, which isin fact a small parameter of the problem. Having chosenthe exciton system in a high magnetic field as a final solu-tion, we shall see that the perturbation theory in vz isquite legitimate and should provide correct results evenin a linear approximation in n(u ), at densities up to10' —10' cm, contrary to the case of the electron-holeliquid. Thus the condition of the stability of the groundstate (23) (see also Comte and Nozieresz2), with accuracywithin terms of the order of v inclusively, immediatelyleads to the equation

e, (p)+@i,(p) —p —2g V v u u

upo=&n %o(p)

exp( —A, [p„+p~ ]/iii )

n 4 2m.A, aoaI+aoa p, /fi

(27)

where %o(p) is the momentum representation of theground-state wave function of an exciton in a high ma-netic field. We now see that vp-(I/+B*lnB') n,which is indeed a small parameter of the problem, as stat-ed above. The chemical potential in this approximationcoincides with the energy of the ground state

Po Eo . (28)

Proceeding to the next order in perturbation theory,we can obtain the correction to the chemical potential.Thus we write

The solution of this problem has already been discussedabove [see (7)]. Normalizing the wave function in accor-dance with normalization condition (24), and going over

to the momentum representation, we get v„ in the form

P p=v o+&v ~ p= Eo+&p (29)

—(up —up) g Vp p up vp =0. (25)P

Moreover, variational calculations ' show that theequation of type (25) remains valid over the whole densityrange and correctly describes the transition from the ex-

with 5up being orthogonal to upo. Taking (29) and (26)into account, we rewrite, accurate to terms linear in

5up and terms of the order of n (note here that

&un'~ and -5v nan-d they have an extremelysmall factor [see (27)]), the condition of the stability ofthe ground state (25) in the form

—p p'+ k [e,(p)+e„(p)+Eo]5up —g Vp p 5up = —Pp+u&5p,

where

(c)

FIG. 6. All possible combinations of creation and annihila-tion of two electron-hole pairs described by the matrix elementsof the operator m 3.

wf

P =—,' g V u o[4v o 4upovpo+upo] .

P

Projecting (30) onto v&, we obtain a formula f« the

correction to the chemical potential


5JM =—g V p +o(p')pp(p)2 p/

fip= A, a (5I,34—I22),32ll 4 2

X [4'po(p) —4+o(p')p, (p)+q (p')] . (32)

Using expression (27) for the wave function %o(p), we canwrite down this integral as

where the Coulomb system of units was introduced, i.e.,we set e /s=k=c=m =1, though the final result will be

written out in the usual units. Here the integrals I,3 and

I&2 are

exp( —jk [x' +y' ]—kA, [x +y ])([x—x']z+[y —y']z+[z —z'] )(1+a z' )~(1+a z )"

16nR,„,5p=, 4(B'), (33)

where 4(B) is a dimensionless function of B '4(B')=f dx exp( —x /5 )Ei( —x /6 )

0

x +20 x +9x +30(x +4) (x +3}

where Ei( —x) is the exponential-integral function andC =9~3/64 =0.243 57. The parameter 6 =B '/ln B '(=a /A, in the Coulomb units) and B'=B/B, . Theplot of 4(B') as a function of B is shown in Fig. 7. Asis seen from the figure, 4(B ) is quite a slowly varyingfunction of the magnetic field. Indeed, this functionchanges from 0.42 at B'=10 to 0.6 at B*=100. Withhigh accuracy, one can use the approximate expression4(B')=0.30943[hz]o s6 at the fields B' in the range10-1000.

Now we can proceed to the calculation of the excita-tion spectrum of the excitonic system. The excitationspectrum of excitons in the condensed state is determined

by the poles of a two-particle Green's function. A con-sistent approach to the problem in terms of the Green's-

where j and k are integers (j =1 and k =3 for I,3 and

j=k =2 for I22). After the straightforward but quite la-borious calculation of these integrals, we obtain the finalresult for the correction to the chemical potential

function technique- needs two Green's functions to be in-volved in the consideration: the normal Green's functionG2(P;p, p') describing the propagation of an electron-hole pair through the medium and the anomalousGreen's function G2(P;p, p') describing the appearanceof two electron-hole pairs from the condensate. ' HereP =(P,E) is the summary momentum and frequency ofthe exciton and p =(p, co} and p'=(p', co} are the relativemomentum and frequency of the exciton. The Green'sfunctions satisfy the system of equations which can begraphically represented in the form shown in Fig. 8. Thisexpression of the Green's function through the irreduc-ible self-energy parts X and X is known as the Dysonequation. It is worth pointing out again that, if the pairinteraction of excitons corresponds to the triplet groundstate in a high magnetic field, the formation of a molecu-lar state can happen under no circumstances. The vertexX(P;p,p') (which describes the scattering of an excitonby excitons of the condensate) and X(P;p,p') (which de-scribes the creation of two excitons with opposite mo-menta from the condensate and their departure to thecondensate) therefore have no pole character and the per-turbation theory can still be applied in the present case.The analysis shows that, accurate to terms linear in v&,the vertex X(P;p,p') is

1.4-@(~)

1.2—

0.8—

0.6—

100 10(X)B*= B/B

FIG. 7. The dependence of the function 4(B) on the magni-tude of the magnetic field.

FIG. 8. The diagram representation for the two-particleGreen's functions.


,p p V —'7 +P/2, '+P/2Y —P/2, ' —P/2

+4(P'p p'» (34)

gjp /I I ' ' p —p' Vp+P//2, —p'+P/2Vp' —P//2, —p+P j2

+4(P;p p'} .

where the first term corresponds to the processes shownin Fig. 4(c} and the second is the sum of all other (linearin UP } scatterings of two electrons and two holes by oneanother. In a similar fashion, we have

Here the first term is an analytic expression of the matrixelements of Fig. 6(a) and the second includes the sum ofall the diagrams describing the creation of two pairs fromvacuum in higher orders of the perturbation theory.

In an analytical form, the system of Fig. 8 can be writ-ten down as

ld piG2(P;p, p') =G, (p+P/2)Gl, ( —p+P/2) I ~ X(P;pl,p'}G2(P',pl»')

(2lr)

ld p)+G, (p)Gl, (—p) ~ X(P;p„p')G2(P;p„p')

(2n )

+G, (p+P/2)Gl, ( —p+P/2)X(P;p, p')G, (p'+P/2)Gh( p'+P/—2), (36)

ld p)G2(P;p, p') =G, (p P/2)Gl, —( —p P/2) J— X(P;p l,p')G2(P;p „p')

ld pi+G, (p)G&( —p) ~ X(P;pl, p')G2(P;pl, p')+G, (p)G&( p)X(P;p,—p')G, (p')Gz( —p') .(2m )

(37)

Here G, (p) and Gl, (p) are the Green's functions of a free electron and hole in a high magnetic field. In the presence ofan external field, space becomes inhomogeneous and, as a matter of fact, G, (p), for example, is a function of two mo-menta

D (ply p2i)e(p ) Ge(pl&p2~~) Ge(pz~p»&pz&p2l&~)

co —(p, /2m, —p,, ) +i 5

where

(38)

1 2 2 lD(pig p2J) = d'r»d'r»exp ——[r„p»—r».p»] exp( —[r» —r»]'/4A, '+2l [x, —~2][y, +y2]/A, ) .

2m-k'

Here rl = ~rl ~

=+x +y and pl =~pj ~ =Qp„+p . In obtaining this expression, we used the assumption that the elec-

tron is at the zeroth Landau level in a high magnetic field. As seen from expression (38), the motion of the electron is

almost suppressed by the magnetic field in a plane perpendicular to the magnetic field direction and can thus be con-sidered as almost one-dimensional motion along the magnetic field. A similar expression also holds for G„(p). We note

that the co dependence of the Green's functions of a free electron and a hole in a high magnetic Seld is the same as forthe case of a free particle without the field. Calculation of the integral yields

D(p„,p», p»)=~&~~exp —,[(p»+p2, )'+ —,', (pl, —p2y } +lp (ply

—p2y)14A

Because of the smallness of U in a high magnetic field [see (27)], the solution of the system (36) and (37) can be

obtained for a wide range of densities in the framework of the perturbation theory in U . To begin with, we note that

X= VP P +O(vP } and X(p,p')- V .u =0. As a result, we can use X= V ~ and X=0 in the zeroth order of the

perturbation theory. Assuming also that the total frequency E and the kinetic energy P /2M of the exciton are small in

comparison with corresponding parameters of the internal motion, accurate to terms linear in all three parameters u~,

E, and P /2M, we rewrite the system (36) and (37) in the following form (note that y = I —y» ):


ld PiG2(P;p,p')= —G, (p+P/2)Gh( —p+P/2) f 4 VP P G2(P;pi, p')

(2m. )

ld pi+G, (p)Gh( —p)f, [Vp —p Fp p +4(p pi)]G2(p pi p')

(2 )4 P P( P P(

ld P)+G, (p)Gh( —p)f, [V, , y'P, P +4(p,pi)]G2(P Pl P(2 )4 P Pi P Pi

+G, (p +P/2)Gh( —p+P/2)X(p, p')G, (p'+P/2)Gh( p'+—P/2),ld Pi

G2(P;p, p')=G, (p —P/2)Gh( —p P/2—) 4 Vp p G2(P;p„p')(2~)4

ld Pi+G, (p)G„(—p), [V, , 7', ,, +0(P Pi)]G2(P P1P(2 )4 P P) P~P)

ld Pi+G, (p)Gh( p) —[Vp Y +$(p,pi )]G2(P;pi,p')(2 )4 P P) P~P)

+G, (p)G„( —p)X(p, p')G, (p')Gh ( —p') .

(40)

(41)

If we neglect the dependence of the electron and hole Green's functions upon the summary excitonic momentum P(negligible in this approximation), it is clearly seen from the structure of Eqs. (40) and (41) that the two-particle Green sfunctions can be sought as

G2(P;p, p') =G, (p)Gh( —p) & (P;p, p')G, (p')Gh( —p'),

G2(P;p, p') =G, (p)Gh ( —p) & (P;p, p')G, (p')Gh( —p') .(42)

The Green s functions describe the propagation of two particles through the condensed state, their interaction, and theformation of a bound state of two particles. The pole terms of these functions are related to the wave function and tothe energy of the bound state of two particles (exciton) in the usual way

& (P;p, p') = [Eo+e,(p)+eh(p)]+0(p)g(p)PO(p')[Eo+ e,(p')+eh(p')],

A(P;P, P')=[Eo+ee(P)+eh(P)]+0(P)g(P)%'0(P')[Eo+ee(P )+eh(P )] .

Substituting (42) and (43) into (40), we find

(43)

(44)

The same substitution into (41) gives

ld Pig(P)[EO+ee(p)+~h(p)]11'0(p)= g(P) f 4 V, P G.(pi)Gh( —pi)[EO+ee(pi)+eh(pi))eo(pi)

)4 P P

ld P)+g(P) 4[Vp —p 1'p, p +4(p~pi)]G (pi)Gh( pi)[EO+e (pi)+eh(pl)]+0(pl)(2 )4 P P1 PiP1

ld Pi+k(p) 4 [V, , y'P, P +p(p, pi)]G. (pi)Gh( —pi)[EO+~, (pi)+eh(pi)]po(pi)(2~)4 P Pi P~P1

&(p,p')+0(p') [Eo+e (p') + eh (p') ]

g(P)[EQ+e, (p)+eh(p)]%0(p)

ld Pi= —g(P) f 4 VP P G, (pi )Gh( —pi)[EO+e, (pi)+eh(pi)]40(pi)(2m. )

ld Pi+g(P), [V, , yP'P +p(p, p, )]G,(p, )Gh( —p, )[EO+e' (p, )+eh(p, )]%0(p, )(2 )4 P P) P P)

ld4P,, [ VP —P 7P P +4(p Pi ) ]Ge(pi )Gh( —pi )[EQ+&.(pi)+eh(pi) 1'4(pi)

(2 )4 P P) PP)

%'0(p') [Eo+e,(p')+ eh(p') ](45)


E +5p, +e( P) 1++t'g(P) =

Ez —E~(P } E —E(P)Xp

E+E(P) ' (46)

E —E (P)(47)

where the distribution function with respect to P of thesupercondensate excitons is

1 5@+e(P}(48)

2 E(P)

Thus the two-particle Green's functions of an excitonicgas in a high magnetic field formally coincides with thesingle-particle Green's function of a dilute nonideal Bosegas. The correction to the chemical potential 5pstrongly depends on the magnitude of the magnetic fieldand is given by expression (33}. The excitonic spectrumin the presence of the condensate has the usual form for alow-density Bose gas

E(P)=+25@a(P)+e (P) . (49)

However, the dependence of the energy of a free excitonin a high magnetic field e(P ) on its momentum is drasti-cally changed in comparison with the case B=O. Al-though such a dependence was calculated in principle byGor'kov and Dzyaloshinski, their calculations contain anumber of mistakes and therefore do not give the correctresult at small values of P (P' in the notation of Ref. 27)and at reasonable values of B not equal to infinity. To ob-tain a correct solution, we note that the parameter po in-

troduced in Ref. 27 should be

P'XB .e8

(50)

Then, going through calculations similar to those of Ref.27, we obtain, for the exciton ground state,

W'=——E=— a0 0 22mQ o

where a is the solution of the equation

(51)

aocxa '=2ln

&2A,

where

ma Q"2A M

(52)

After integrating the system (44) and (45) over the fre-quency, we obtain a system of equations where the in-tegration takes place over the three-dimensional rnornen-tum p and in the zeroth order [X= V ~ and

X(p,p')- V .y -U =0] both equations are reducedto a Coulomb equation of type (26). Besides, we notehere that actually this system has become purely algebra-ic with respect to g(P) and g(P). Moreover, the first,second, and fourth terms on the right-hand side of Eq.(44) describe nothing but propagation of an electron-holepair through the "normal" system without the conden-sate. Therefore, it is likely to look for g(P) as the sum of[E —E(P)] ' and some function of P, whereE(P)=e(P)+5@. Then, going through the simple ma-nipulations, we obtain the final results

P = +2M 2MO

where M=m, +m& is the total effective mass of the exei-ton and the finite transverse mass appears as a result ofthe Coulomb interaction in the excitonic atom:Mo=Maoa/A. =MB'/lnB'. We see that the depen-dence of the energy of a free exciton in a high magneticfield on its momentum has strongly anisotropic behavior.

Thus the spectrum of an exciton gas in a high magneticfield becomes

5p, p 1cos 8+ sin 8 P

M

P4+ cos 8+ sin 84M' aB* (54)

where 8 is the angle between the direction of the rnagnet-ic field and the momentum P. The excitation spectrumvanishes linearly as P ~0, with a slope equal to a macro-scopic speed of sound, i.e., this spectrum satisfies theLandau criterion for superfluidity. The macroscopicspeed of sound in the superfluid state of the exciton gasdepends on the direction of the magnetic field

' 1/2&u

U, = cos 8+ sin 8M ~p' (55)

This function has a maximum at 8=0 and a minimum at8=m/2. Thus we have made sure that the superfluidityarises in the excitonic system in a high magnetic fieldand, as could be expected, it is likely to appear in direc-tions close to the direction of the magnetic field.

III. THE OBSERVABLE EFFECTSOF THE EXCITONIC CONDENSATION

In this section we discuss the necessary conditions forthe observation of Bose condensation and superfluidity ofexcitons in a high magnetic field and also some of thepossible experimental consequences. Photoexcitation bythe laser light with an intensity of the order of 10 —10W/cm may be the most convenient way to create ahigh-density exciton gas in a semiconductor. In this ease,exeiton concentrations of the order of 10' —10' emare quite realistic. ' As mentioned, soon after thecreation of excitons by photoexcitation, the excitonie sys-tem is not in thermal equilibrium. The point is that thetime needed to establish the thermal equilibrium of theexcitonic system and the lattice must be much moreshorter than the lifetime of an exciton, i.e., the time of ex-citon annihilation due to electron-hole recombination.At the concentrations of the electrons and holes of theorder of 10' cm, the characteristic time for the bind-ing of the electron with the hole into the exciton and thetime of cooling of the resultant excitons to the lattice

XA(x}= dy e ~in —=x (as x~0) .0

The calculation of the P dependence of a yields the finalresult


temperature are of the order of 1 nsec. The typical life-time of the exciton in a semiconductor such as InSb is,however, of the order of 10-100psec. '

A bound state of an electron and a hole within theframework of the two-band model used throughout thepaper can be considered only if the radius of such a stateis much larger than the lattice constant. Without a mag-netic field, this is exactly the case we have in most III-Vsemiconductors such as InSb, InAs, or GaAs where thedielectric constant is quite large (e,-10—20) and the re-duced effective mass of the electron and the hole is rathersinall (m 0. 1 of the free-electron inass). So the Bohr ra-dius of the exciton ao turns out to be of the order ofseveral tens of the interatomic distance. Although thesize of the exciton is greatly decreased in a high magneticfield, in particular, in a plane perpendicular to the direc-tion of the magnetic field (A, =&cfi/eB =ao/+B'), itstill remains several times greater than the typical intera-tomic distance, at least in the fields B' —10-100.

The formulation of the problem and particularly thechoice of the Hamiltonian imply a certain class of semi-conductors where the results obtained could be applieddirectly. As is known, semiconductors can be dividedinto two classes. Those in which the conduction-bandminimum and the valence-band maximum occur at thedifferent values of the wave vector are known as indirect-gap semiconductors. The most typical examples of thisclass are germanium and silicon. The second class in-cludes semiconductors in which the minimum of the con-duction band and the maximum of the valence band areat the same value of the wave vector. They are known asdirect-gap semiconductors. Here we have GaAs andInSb as the typical examples of this class. Thus the situa-tion considered in this paper corresponds to the direct-gap semiconductors. Perhaps InSb may be the bestchoice because the transition of the excitons into the Bosecondensate takes place there at relatively high tempera-ture. In the case of InSb the "atomic" scale of the mag-netic field is 0.19 T. Thus we already have 8*-100 inthe magnetic field 8 =20 T. In this case, the binding en-ergy is ED=12 meV. At the same time, for the excitonicconcentration of the order of 10' cm, the transitiontemperature into the Bose condensate is near room tem-perature.

Let us now consider the observable effects due to theexciton condensate and the superfluidity. The system ofexcitons allows the formation of the Bose condensate in ahigh magnetic field and behaves as an almost ideal Bosegas. This happens at the densities when the system of ex-citons without a magnetic field already turns into anelectron-hole droplet . In this system, a superfluid flowthrough the crystal can exist. However, it must be em-phasized here that neither a flow of matter nor an electriccurrent through the crystal exists, if excitonic gas be-comes superfluid, because an exciton is an electron exci-tation migrating through the crystal, relating to neithermass transfer nor charge transfer. The excitons cantransfer their excitation energy and also their angular,electric, and magnetic moment if there is any. Therefore,the superfluidity of excitons could imply, for instance, theexistence of a superflow of energy in the absence of a tem-

y(a) P)-— 1+1Vp

fico ii,,i,E—(P—

) +i 5

Np

Ac@ p,,h+—E (P )+i 5

where co is the frequency of an external electromagneticfield and p, I, =Eg —Eo is the energy of the exciton at rest(Eg is the energy gap). The imaginary part of the suscep-tibility, which determines the absorption, is

y"(ri), P)-(1+Np)5[fico —p, i,—E(P)]

Np5[Rco—p, p,+E—(P), ] . (57)

It is easy to see that we have three main features in thepresence of the condensate. First, there is an increase inthe absorption at the frequency A'm=p, i, +E(P). Thisabsorption comes from the process in which the energy ofthe absorbed quantum goes into the formation of an exci-ton together with exciting a condensate acoustic wave.Besides the increasing absorption, amplification of thelight at the frequency fico=@,i, —E(P) takes place, whereE(P) is given by expression (54). The amplification of thelight comes from stimulated annihilation of excitons inthe presence of the condensate. Finally, there is a regionof frequencies where the dielectric constant is equal tounity: fuo=p, & 5p e(—P) —In eff'e.ct, the crystal wouldseem to be totally transparent. The "laser" effect, super-transparency, and other exotic properties of semiconduc-tors would take place only in the presence of the excitoncondensate, which is likely to be created only in a highmagnetic field.

IV. CONCLUSION

The problem of Bose-Einstein condensation, being in-teresting in itself, has become of special interest in con-nection with experimental efforts to observe Bose conden-sation in systems with finite lifetime. Here one shouldmention the experimental work devoted to excitons' ' 'and biexcitons. Some other possibilities have also beendiscussed, such as condensation of polaritons and spin-

perature gradient and a chemical-potential gradient, thatis, the existence of superthermalconductivity. One canalso expect that after the transition into the superfluidstate, the anomalous diffusion of the excitons would takeplace. The expansion rate of the excitons from the pointwhere they were created can be 2—3 orders of magnitudegreater than that predicted by simple diffusion. The exci-tons may penetrate into the crystal (along the direction ofthe magnetic field) over a distance which is a few ordersof magnitude larger than the classical diffusion length.

Besides the interesting kinetic and thermal properties,the appearance of the exciton condensate must give riseto new optical properties of a semiconductor. The de-tailed discussion of these phenomena will be given in aseparate paper. Now we are going to make a few qualita-tive remarks. The optical susceptibility in the presence ofthe exciton condensate turns out to be proportional to theexcitonic Green's function

A. V. KOROLEV AND M. A. LIBERMAN

polarized hydrogen. In all the cases, the density has tobe low enough to guarantee the existence of a weakly in-

teracting Bose gas. That means that we cannot expect amore or less high critical temperature of Bose condensa-tion. Normally the Bose statistics of excitons is diScultto observe, at least in the indirect-gap semiconductors(Ge, Si, etc.), due to the instability of the condensate: thestrong interaction between excitons, together with theweak binding energy, results in condensation in theelectron-hole liquid instead of Bose condensation of exci-tons. The obvious advantage of the excitonic system in ahigh magnetic field is an opportunity to create a systemat much higher densities and accordingly with highercondensation temperature.

In our previous publications, ' ' we analyzed theproperties of hydrogenlike atoms in an ultrahigh magnet-ic field. It was shown that the properties of matter aredramatically changed in a high magnetic field. Under thecircumstances, due to a small characteristic size of theatoms, a substantial decrease of the pair interaction be-tween the atoms in the triplet ground state, and an in-crease of the binding energy of the atoms in an ultrahighmagnetic field, the behavior of a gas of hydrogenlikeatoms becomes similar to helium at low temperature.The Bose condensation and the transition into asuperfluid state for the hydrogenlike gas is therefore quiterealistic in an ultrahigh magnetic field at relatively lowtemperature.

In this paper we considered the possibility of Bose con-densation and superfluidity for hydrogenlike %annier-Mott excitons in a high magnetic field. Starting from thesecond-quantization representation of the Hamiltonian ofthe system of interacting electrons and holes in a highmagnetic field, we obtained the excitation spectrum of thesystem. The existence and the stability of the Bose con-densate due to an essential decrease of the interaction be-tween excitons and an increase of their binding energy ina high magnetic field are now established. Furthermore,the excitation spectrum satisfies the Landau criterion forsuperfluidity. Thus the Bose condensation and the

superfluidity of the hydrogenlike excitons become quiterealistic even at room temperatures because these phe-nomena are not suppressed in a high magnetic field bythe collective e8'ects leading to an electron-hole liquidmithout a magnetic field at a high density of the system.The Bose condensation and superfluidity of a high-density exciton liquid may cause a number of new proper-ties of semiconductors: both their kinetic and opticalproperties, such as, for example, the "laser" effect and su-pertransparency.

So far, helium has appeared to be the only practical ex-ample of a quantum superfluid liquid. Studying the prop-erties of matter in high magnetic fields, however, shedsnew light on the problem. Now we can, in principle,study and create a great number of phenomena related tothose hydrogenlike quantum liquids and having similarnature in both terrestrial laboratories and space. Itshould be mentioned that superfluidity and related phe-nomena are not all new properties of the excitonic systemin a high magnetic field. The pair interaction of hydro-genlike excitons in the singlet term is much stronger thanthe pair interaction of excitons without a magnetic field. '

The lifetime of this tight molecular state for a couple ofexcitons must apparently be comparable with the lifetimeof a free exciton. Therefore, if a gas of excitons mereprepared in the excited singlet state, metastable struc-tures such as long polymeric molecules or regular latticestructures would arise due to the extremely strong aniso-tropic interaction of excitons. "Crystallization" of exci-tons being in this state is an interesting problem, whichwill be considered in a separate paper.

ACKNO%'LED GMKNTS

We are pleased to acknowledge stimulating numerousdiscussions of experimental observation of the excitonicBose condensation with Professor Borje Johansson, Pro-fessor V. B. Timofeev, and Dr. Per Omling. This workwas supported by the Swedish National FoundationResearch Council, Grant No. FF-1731.

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Bose condensation and superfluidity of excitons in a high magnetic field