Light-induced Bose-Einstein condensation in two-dimensional systems of charge carriers with different masses
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Light-induced Bose-Einstein condensation in two-dimensional systems of chargecarriers with different masses
O. V. Kibis, ∗ M. V. Boev, and V. M. Kovalev
Department of Applied and Theoretical Physics, Novosibirsk State Technical University,Karl Marx Avenue 20, Novosibirsk 630073, Russia
It is demonstrated theoretically that the circularly polarized irradiation of two dimensional degen-erate electron systems can produce the composite bosons consisting of two electrons with differenteffective masses, which are stable due to the Fermi sea of normal conduction electrons. As aconsequence, the Bose-Einstein condensation of the charged bosons and the light-induced supercon-ductivity based on this effect can take place in modern nanostructures.
The modification of electronic properties of variousquantum systems by an off-resonant high-frequency elec-tromagnetic field (the Floquet engineering) became theestablished research area of modern physics, which re-sulted in many fundamental effects studied both theo-retically and experimentally [1–12]. Recently, it wasdemonstrated that a high-frequency circularly polarizedelectromagnetic field crucially modifies the Coulomb in-teraction in two-dimensional (2D) systems, inducing theattractive area in the core of the repulsive Coulomb po-tential [13]. As a consequence, the quasi-stationary elec-tron states bound at repulsive scatterers [14, 15] and, par-ticularly, the pairing of electrons with different effectivemasses [13] appear. However, the single electron pair isinstable with very short lifetime that prevents to observeeffects arisen from the pairing. In the present Letter,we developed the many-body theory of the light-inducedelectron coupling in the 2D degenerate electron gas andfound that the Fermi sea of conduction electrons stabilizethe pairs. Therefore, the stationary quantum phenomenaoriginated from the Bose nature of paired electrons, in-cluding the possible Bose-Einstein condensation of thepairs and the corresponding light-induced superconduc-tivity, can exist in state-of-the-art 2D nanostructures.Let us consider a two-dimensional (2D) nanostruc-ture containing charge carriers with different effectivemasses, where the energy spectrum of the two groundsubbands is ε ( k ) = − ∆ / ~ k / m and ε ( k ) =∆ / ~ k / m , ∆ is the energy distance between thesubbands, k = ( k x , k y ) is the momentum of charge car-rier in the 2D plane, and m , are the effective masses inthe subbands. For definiteness, we will develop the the-ory for conductivity electrons, which can be easily gen-eralized for 2D hole nanostructures. In the presence ofa circularly polarized electromagnetic wave incident nor-mally to the 2D structure, the Coulomb interaction of twoelectrons from the subbands ε ( k ) and ε ( k ) is describedby the Hamiltonian ˆ H ( t ) = ˆ H + ˆ H + U ( r − r ),where ˆ H , = (ˆ p , − e A ( t ) /c ) / m , are the Hamilto-nians of free conduction electrons irradiated by the wave,ˆ p , = − i ~ ∂/∂ r , are the plane momentum operators ofthe electrons, r , are the plane radius vectors of the elec-trons, A ( t ) = ( A x , A y ) = [ cE /ω ](sin ω t, cos ω t ) is the vector potential of the wave, E is the electric field am-plitude of the wave, ω is the wave frequency, U ( r ) = e /r is the potential energy of the repulsive Coulombinteraction between the electrons, and r = r − r is theradius vector of relative position of the electrons. If thefield frequency ω is high enough and lies far from char-acteristic resonant frequencies of the 2D structure (theoff-resonant dressing field), the time-dependent Hamil-tonian ˆ H ( t ) can be reduced to the effective stationaryHamiltonian [13, 14], ˆ H = ˆ p / m + ˆ p / m + U ( r ),where U ( r ) = 12 π Z π − π U (cid:0) r − r ( t ) (cid:1) d ( ω t )= (2 e /πr ) K ( r /r ) , r /r ≤ e /πr ) K ( r /r ) , r /r ≥ K ( ξ ) is thecomplete elliptical integral of the first kind, r ( t ) =( − r cos ω t, r sin ω t ) is the radius vector describing theclassical circular trajectory of a free particle with thecharge e and the mass m = m m / ( m − m ) in the cir-cularly polarized field, and r = | em | E /ω is the radiusof the trajectory.Introducing the radius vector of the center of mass ofthe electrons, R = ( m r + m r ) / ( m + m ), the ef-fective Hamiltonian ˆ H can be rewritten asˆ H = − ~ m + m ) ∂ ∂ R − ~ m ∂ ∂ r + U ( r ) , (2)where m = m m / ( m + m ) is the reduced effec-tive mass. Since the dressed potential (1) has thelocal minimum at r = 0 (see Fig. 1a), the solu-tion of the Schr¨odinger problem with the Hamiltonian(2) results in the wave function of coupled electrons ϕ K = (1 / √ S ) e i KR ϕ ( r ) and their energy spectrum ε ( K ) = ε + ~ K / m + m ), where S is the area of the2D structure, K is the momentum of the center of mass ofthe electron pair, ε is the ground energy of the pair and ϕ ( r ) is the wave function describing relative motion ofthe paired electrons with the characteristic radius of the FIG. 1: Electronic energy structure of the system underconsideration: (a) The light-induced effective potential ofelectron-electron interaction, U , with the local minimumwhich confines the electron pair with the wave function ϕ (the red curve) and the energy ε (the brown horizontal line).Normally, the electron pair is instable because of the tunnel-ing from the local minimum through the potential barrier intothe states of decoupled conduction electrons with the planeelectron wave e i qr (the green wave arrow); (b)–(c) The en-ergy spectrum of the subbands of conduction electrons ε , ( k )(the two lowest green curves) and the coupled electrons ε ( k )(the upper brown curve) for different Fermi energies, ε F . Thelarge brown circle marks the ground state of coupled elec-trons with the energy ε , whereas the two small green circlesmark the states of decoupled electrons with the energies ε , satifying the condition ε + ε = ε . pair r . It should be noted that the energy of coupledelectrons found from the spinless Hamiltonian (2) corre-sponds to the degenerate singlet and triplet spin states.Although the spin-spin interaction of coupled electronslifts this degeneracy, the corresponding spin splitting,∆ s ∼ ( e ~ /m e c ) /r , is relativistically small and can beomitted as a first approximation. Since the light-inducedelectron pairs have the integer spin, they can be consid-ered as composite bosons. It should be stressed also thatthe potential (1) turns into the usual repulsive Coulombpotential U ( r ) = e /r for m = m , that leads to dis-appearance of the electron coupling. Physically, this fol-lows from the fact that the oscillating dressing field A ( t )does not change the distance r between charge carri-ers with the same mass and, therefore, does not modifytheir Coulomb interaction. Thus, the different effectivemasses, m = m , are crucial for the electron pairingunder consideration.The state of coupled electrons with the energy ε is im-mersed into the continuum of normal electron states ofthe subbands ε ( k ) and ε ( k ) and separated from themby the potential barrier pictured in Fig. 1a, which con-fines the wave function ϕ ( r ) near the local minimumof the dressed potential U ( r ). Therefore, the coupled electron pair is quasi-stationary and can decay by tunnel-ing electrons through this barrier into the normal electronstates of the subbands. The energy of the two decou-pled electrons can be written as ε ( q , K ) = ~ k / m + ~ k / m = ~ q / m + ~ K / m + m ), where k = m K / ( m + m ) + q and k = m K / ( m + m ) − q are the momenta of the two decoupled electrons in thesubbands ε ( k ) and ε ( k ), respectively, q is the momen-tum of relative motion of the decoupled electrons, andthe corresponding electron wave function of the decou-pled electrons reads ψ q , K = (1 /S ) e i qr e i KR . Becauseof the translational invariance of the system, the dressedpotential (1) does not depend on the center-of-mass po-sition R and, therefore, it does not change the totalmomentum of the pair K . As a consequence, the tun-nel decay of the electron pair corresponds to the tran-sition ϕ K → ψ q , K , which depends only on the relativemomentum q and is pictured schematically in Fig. 1a.The probability of this transition under the conditionΓ ≪ ε − U (0) reads W q = ~ Γ Sm [( ε q − ε ) + (Γ / ] , (3)where Γ is the tunneling-induced broadening of the en-ergy level ε and ε q = ~ q / m is the kinetic energyof relative motion of the decoupled electrons [16]. Theenergy ε and the energy broadening Γ , which definethe probability (3), can be found numerically from theSchr¨odinger equation with the Hamiltonian (2) withinthe conventional approach [17] and are plotted in Fig. 2as functions of the dimensionless reduced effective mass m/m e = m m /m e ( m + m ) and the characteristic ra-dius of the electron pair r = | em | E /ω , where m e is theelectron mass in vacuum. Since the radius r depends onboth the field amplitude E and the field frequency ω ,the plots pictured in Fig. 2 describe also the effect of theirradiation on the pairs. FIG. 2: Dependence of the electron pair energy ε (a) andthe energy broadening Γ (b) on the dimensionless reducedelectron mass m/m e = m m /m e ( m + m ) and the charac-teristic radius of the electron pair r = | em | E /ω . Direct summation of the probabilities (3) results inthe unit total probability of the decay of the coupledelectrons, P q W q = 1, i.e. the single electron pair isinstable. To consider the effect of degenerate gas ofconduction electrons on the pair, it should be stressedfirst of all that the wave functions of coupled electrons, ϕ K , and normal conduction electrons, ψ q , K , are sepa-rated by the potential barrier pictured in Fig. 1a and,correspondingly, overlap of them is negligible small un-der the condition [16] Γ ≪ ε − U (0). Therefore, thePauli principle does not prevent the coexistence of cou-pled electrons and normal electrons in the same area ofspace. However, the Pauli principle crucially effects onstability of the pair since it forbids the tunnel decay (de-coupling) of the two coupled electrons into the states oc-cupied by normal electrons. As a consequence, the totalprobability of the decay of the electron pair with the to-tal momentum K in the presence of normal conductionelectrons reads W − K = P q W q (1 − f k )(1 − f k ), where f k , = 1 / [exp[( ε , ( k , ) − ε F ) /T + 1] are the Fermi-Dirac distribution functions for the states of normal elec-trons with the momenta k = m K / ( m + m ) + q and k = m K / ( m + m ) − q in the subbands ε , ( k ), ε F is the Fermi energy for conduction electrons in thesesubbands, and T is the temperature. Since the Pauliprinciple decreases the number of normal electron stateswhich are accessible for the decoupled electrons in thedecay process, the decay probability decreases as well,i.e. W − K <
1. Thus, the Fermi gas of normal electronsacts towards to stabilize the coupled electrons. More-over, due to the Fermi gas, the process of production ofcoupled electrons from normal electrons (which is inverserelative to the considered decay process) appears. Sincethe decay and production processes are reversible, theirprobabilities are described by the same Eq. (3). There-fore, the total probability of production of the coupledelectrons from normal electrons is W + K = P q W q f k f k .As a result, the production of electron pairs from theFermi gas of normal electrons takes place under the con-dition W +0 > W − , where W ± = X q W q { exp[ ± ε ( q ) ∓ ε F ] /T + 1 } − × { exp[ ± ε ( q ) ∓ ε F )] /T + 1 } − (4)are the probabilities of production ( W +0 ) and decay ( W − )of the electron pair in the ground state.For definiteness, we will restrict the following anal-ysis by the case of infinitesimal broadening Γ andtemperature T , when the probability (3) esquires thedelta-function singularity, W q = (2 π ~ /Sm ) δ ( ε q − ε ),and the thermal fluctuations can be neglected. Thenthe ground coupled state with the energy ε dissoci-ates/arises to/from the two states of normal electronsin the subbands ε , ( k ) with the momenta k = q and k = − q corresponding to the electron energies ε = ε ( q ) = − ∆ / ε m / ( m + m ) and ε = ε ( − q ) =∆ / ε m / ( m + m ), respectively, which are markedby the green circles in Fig. 1(b)–(c) and satisfy the con-dition ε + ε = ε . The direct calculation with usingEqs. (3)–(4) results in the inequalities W +0 > W − for ε F > ε / W +0 < W − for ε F < ε /
2. This means that the normal conduction electrons are instable withrespect to their coupling for ε F > ε / ε F < ε / W +0 = W − ) for ε F = ε / ε F > ε /
2, let us neglect the interactionbetween the electron pairs, considering them as an idealBose gas with the Bose-Einstein distribution function, p K = 1 / [exp( ε ( K ) − µ ) /T − p K is the averageddensity of the pairs, and µ is their chemical potential.Then the processes of decay and production of electronpairs can be considered formally as a chemical-like re-action in the mix of the Fermi gas of normal electronsand the Bose gas of coupled electrons, which turns onegas into other. Applying the known condition of chemi-cal equilibrium [18] to the considered system, we arriveat the equilibrium condition of the mix, ε F = µ / ε F is the equilibrium Fermi energy of the subsys-tem of normal electrons, µ is the equilibrium chemicalpotential of the subsystem of coupled electrons, and thefactor 1 / T = 0, the equality µ = ε takesplace. Therefore, the condition of thermodynamical equi-librium of the mix is ε F = ε / W +0 = W − , con-sidered above. Since the production of coupled electronsis accompanied by decreasing the Fermi energy of nor-mal electrons, the nonequilibrium state of the mix withthe Fermi energy of normal electrons ε F > ε / ε F = ε / n = n + 2 p . Here n is the electron den-sity corresponding to the initially instable system of nor-mal electrons with the Fermi energy ε F > ε /
2, whereas n and p are the densities of normal electrons and cou-pled electrons, respectively, which correspond to the equi-librium state of the system with the Fermi energy of nor-mal electrons ε F = ε / µ = ε . Particularly, in the case of ε > ∆ , the equilibrium density of the Bose-Einsteincondensate is p = ( m + m )∆ ε π ~ , (5)where the energy difference ∆ ε = ε F − ε / ε ≫ Γ in order to protect stability of the coupled electrons. Itshould be noted also that the overlap of the wave func-tions of neighbour electron pairs, ϕ ( r ), must be negli-gible small in order to apply the model of ideal Bose gasto them as a first approximation. Therefore, the den-sity (5) should be small enough to satisfy the condition p r ≪
1, where r is the size of the pair (see Fig. 1a).These two conditions can be satisfied simultaneously ifΓ ≪ ~ / ( m + m ) r . Since this criterion conforms tothe condition Γ ≪ ε − U (0) used above to derive theprobability (3), the present theory is self-consistent.It follows from the aforesaid that the circularly polar-ized irradiation of 2D structures can produce the hybridBose-Fermi system consisting of the Fermi gas of normalconduction electrons and the Bose gas of coupled elec-trons. Physically, the hybrid Bose-Fermi systems sub-stantially differs from pure Bose systems [19–22]. Partic-ularly, the Fermi component changes the interaction be-tween Bose particles, which is responsible for the disper-sion of collective modes in the Bose-Einstein condensate, ω K , where K is the wave vector of the mode. Gener-ally, the frequency of the mode, ω K , is defined by theconventional condition ǫ ( ω, K ) = 0, where ǫ ( ω, K ) =1 − g K Π( ω, K ) is the dielectric function, Π( ω, K ) = P q [ p q − p q + K ] / [ ~ ω + ε ( q ) − ε ( q + K )] is the polar-ization operator and g K is the Fourier transform of theinteraction between the Bose particles [24, 25]. Since asmall number of electron pairs ( p r ≪
1) is immersedinto the Fermi sea of conduction electrons, their Coulombinteraction is screened by many normal electrons and,therefore, can be described by the Fourier transform g K = 8 πe / ( K + 2 /r s ), where r s = ǫ ~ / ( m + m ) e isthe effective screening radius assumed to satisfy the con-dition r s /r ≫
1, and ǫ is the static dielectric constant.Substituting the single-pair energy spectrum ε ( K ) = ε + ~ K / m + m ) and the condensate density (5) intothe polarization operator, we arrive at the energy spec-trum of the collective mode, ~ ω K = p p g K E K + E K ,where E K = ~ K / m + m ) is the kinetic energy ofthe composite boson. As expected, this collective modehas the sound-like dispersion ω K = v s K for small wavevectors K ≪ /r s , where v s = p p g / ( m + m ) is thesound velocity. Therefore, the considered Bose-Einsteincondensate is superfluid if the velocity of its flow, v , satis-fies the Landau criterion [23], v < v s . As a consequence,the light-induced superconductivity appears.To finalize the Letter, the applicability conditions ofthe dressed potential (1) to describe the electron pairingshould be formulated. For the applicability of Eq. (1),the field frequency, ω , must satisfy the two conditions.Firstly, ω τ e ≫
1, where τ e is the mean free time ofcharge carriers. Secondly, the field frequency ω lies farfrom resonant frequencies corresponding to the optical transitions between different states of the paired chargecarriers. Under the first condition, scattering processescannot destroy the paired carriers, whereas the secondcondition allows to neglect the effect of the oscillatingterms omitted in the dressed potential (1) on the pair-ing (see the detailed discussion of the applicability lim-its of the dressed potential approach in Refs. 13–15).Among various 2D nanostructures containing charge car-riers with different effective masses, quantum wells basedon hole semiconductors look perspective to observe thediscussed effects. Particularly, it is remarkable that ef-fective masses in the subbands of heavy and light holescan be tuned in the broad range by a confining poten-tial of the quantum well and a mechanical stress [26–29].Since the mobility of charge carriers in modern semicon-ductor quantum wells is of 10 − cm / V · s, we have ω τ e ∼
10 near the high-frequency boarder of the mi-crowave range, ν = ω / π = 100 GHz. Normally, such amicrowave frequency lies far from the resonant frequen-cies of paired electrons as well. Therefore, the two above-mentioned applicability conditions can be met simulta-neously in quantum wells. Assuming that the reducedeffective masses are m, m ∼ . m e , we arrive at the pairradius r ∼
10 nm, the ground energy of coupled chargecarriers ε around 100 meV and the energy broadening Γ of meV scale for the relatively weak irradiation intensity I ∼ W/cm with the frequency ν = ω / π = 100 GHz(see plots in Fig. 2). Correspondingly, the electron pair-ing can take place for the Fermi energy ε F of tens meV,which is attainable in modern semiconductor quantumwells.In conclusion, we demonstrated theoretically that a cir-cularly polarized irradiation can turn a 2D system con-taining degenerate electron gas with different effectivemasses into the hybrid Bose-Fermi system consisting ofthe Fermi gas of normal conduction electrons and theBose gas of electron pairs (composite bosons) coupled bythe irradiation. The found conditions of the electron pair-ing can be realized in modern 2D nanostructures and itspossible manifestations, including the Bose-Einstein con-densation of the pairs, can be observed in state-of-the-artmeasurements. Acknowledgments.
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TUNNEL TRANSITIONS BETWEEN THESTATES OF COUPLED AND NORMALELECTRONS
The tunnel decay of the discussed electron paircorresponds to the transition ϕ K → ψ q , K , whichdepends only on the relative momentum q and ispictured schematically in Fig. 1a (see the main text ofthe Letter). To find the probability of this transition,let us construct the two-electron tunnel Hamiltonian. Inthe absence of the tunneling, the wave functions of thecoupled two electrons, ϕ K , and the decoupled twoelectrons, ψ q , K , satisfy the orthogonality condition sincethey are separated by the impermeable potential barrierand, correspondingly, overlap of them is zero.Therefore, we can use them as a basis of the tunnelHamiltonian which readsˆ H = | ϕ i ε h ϕ | + X q | q i ε q h q | + X q [ | q i| T q |h ϕ | + H . c . ] , (A1)where | ϕ i = ϕ ( r ) is the state of coupled electronpair with the energy ε , | q i = (1 / √ S ) e i qr is the stateof decoupled electron pair with the relative momentumof the decoupled electrons q and the energy of theirrelative motion ε q = ~ q / m , S is the area of theconsidered 2D system, and the last term couples thesestates with the tunnel matrix elements T q = h q | ˆ H| ϕ i .The wave function satisfying the Schr¨odinger equationwith the Hamiltonian (A1) can be written as | Ψ i = a ( t ) e − iε t/ ~ | ϕ i + P q a q ( t ) e − iε q t/ ~ | q i .Substituting this wave function into the Schr¨odingerequation with the Hamiltonian (A1), i ~ ∂ t | Ψ i = ˆ H| Ψ i ,we arrive at the equations for the expansion coefficients, i ~ ˙ a ( t ) = e i ( ε − ε q ) t/ ~ ) X q T ∗ q a q ( t ) , (A2) i ~ ˙ a q ( t ) = e i ( ε q − ε ) t/ ~ ) T q a s ( t ) . (A3)Let an electron pair be in the coupled state at the time t = 0, i.e. a (0) = 1 and a q (0) = 0. Then theintegration of Eq. (A3) results in a q ( t ) = − iT q ~ Z t e i ( ε q − ε ) t ′ / ~ a ( t ′ ) dt ′ . (A4) Since the considered system is axially symmetrical, thematrix element T q depends only on the electron energy, ε q and, therefore, can be denoted as T q = T ε q .Substituting Eq. (A4) into Eq. (A2), we arrive at theexpression˙ a ( t ) = − Sm π ~ Z ∞ dε q | T ε q | Z t e i ( ε − ε q )( t − t ′ ) / ~ a ( t ′ ) dt ′ . (A5)This is still an exact equation since we just replaced twodifferential equations (A2)–(A3) with one lineardifferential-integral equation (A5). Next, we make theapproximation. Namely, let us assume that thetunneling between the states | ϕ i and | q i is weak tosatisfy the condition Γ ≪ ε − U (0), where Γ is thetunnel-induced broadening of the coupled state energy ε . Then the quantity | T ε q | varies little around ε q = ε for which the time integral in Eq. (A5) is not negligible.Therefore, the energy of the decoupled electron pair, ε q ,is near the energy of the coupled pair, ε . As aconsequence, one can make the replacement | T ε q | → | T ε | and replace the lower limit in the ε q integration with −∞ . As a result, it follows fromEq. (A5) that a ( t ) = e − Γ t/ ~ , whereΓ = 2 π P q | T q | δ ( ε − ε q ). Substituting the foundamplitude a ( t ) into Eq. (A4), the amplitude of thetunnel transition ϕ K → ψ q , K during time t reads a q ( t ) = − T ε e i ( ε q − ε ) t/ ~ − Γ t/ ~ − ε q − ε + i Γ / . (A6)Correspondingly, the sought probability of tunnel decayof the two coupled electrons into the state of twodecoupled electrons with the relative momentum q is W q = | a q ( ∞ ) | = ~ Γ Sm [( ε q − ε ) + (Γ / ] ..