Theory of Two-Dimensional Spatially Indirect Equilibrium Exciton Condensates
We present a theory of bilayer two-dimensional electron systems that host a spatially indirect exciton condensate when in thermal equilibrium. Equilibrium bilayer exciton condensates (BXCs) are expected to form when two nearby semiconductor layers are electrically isolated, and when the conduction band of one layer is brought close to degeneracy with the valence band of a nearby layer by varying bias or gate voltages. BXCs are characterized by spontaneous inter-layer phase coherence and counterflow superfluidity. The bilayer system we consider is composed of two transition metal dichalcogenide monolayers separated and surrounded by hexagonal boron nitride. We use mean-field-theory and a bosonic weakly interacting exciton model to explore the BXC phase diagram, and time-dependent mean-field theory to address condensate collective mode spectra and quantum fluctuations. We find that a phase transition occurs between states containing one and two condensate components as the layer separation and the exciton density are varied, and derive simple approximate expressions for the exciton-exciton interaction strength which we show can be measured capacitively.
Recent advances in the study of two-dimensional van der Waals materialsGeim and Grigorieva (2013) have opened up new horizons in condensed matter physics by allowing familiar properties, including those of metals, superconductors, gapless semiconductors, semiconductors, and insulators, to be combined in new ways simply by designing stacks of atomically thick layers. In this article we consider condensation of spatially indirect excitons in the case of a two-dimensional semiconductor bilayer formed by two group-VI transition metal dichalcogenides (TMD) that are separated and surrounded by an insulator, for example hexagonal boron nitride (hBN). The TMDs are in their 2H structure monolayer form. Two-dimensional material stacks of this type are promising hosts for exciton condensation, both because they host strongly bound excitons, Berkelbach et al. (2013); Qiu et al. (2013); Zhang et al. (2014); Chernikov et al. (2014); Ye et al. (2014); He et al. (2014); Wu et al. (2015) and because of recent progress in realizing flexible high quality TMD heterostructures.Fang et al. (2014); Hong et al. (2014); Cheng et al. (2014); Gong et al. (2014); Yu et al. (2014); Rivera et al. (2015)
In van der Waals heterostructures it is possibleLee et al. (2014); Kim et al. (2015) to tune the positions of the Fermi levels in individual layers over wide ranges while maintaining overall charge neutrality, either by applying a gate voltage between surrounding electrodes or a bias voltage between the semiconductor layers. When the indirect band gap between the conduction band of one layer and the valence band of the other layer is reduced to less than the indirect exciton binding energy, charge will be transferred between layers in equilibrium. At low densities, the transferred charges form spatially indirect excitons, and these are expectedLozovik and Yudson (1976); Zhu et al. (1995); Lozovik and Berman (1996); Fernández-Rossier et al. (1996); Vina (1999); Combescot et al. (2008a); Combescot and Snoke (2008); High et al. (2012) to form Bose condensates. The bilayer exciton condensate (BXC) state has spontaneous interlayer phase coherence and supports dissipationless counterflow supercurrentsEisenstein and MacDonald (2004); Su and MacDonald (2008) that could enable the design of low-dissipation electronic devices.Banerjee et al. (2009)
Exciton condensates in TMD heterostructures are similar to atomic spinor Bose-Einstein condensates because of the presence of both spin and valley degrees of freedom. The spin-valley coupling of conduction band electrons and valence band holes that are specific to TMD heterostructuresXiao et al. (2012) enriches the excitonic physics. In this paper we study the interplay between the exciton condensation and spin and valley internal degrees of freedom to construct an exciton condensate zero temperature phase diagram as a function of effective layer separation and exciton chemical potential , or equivalently exciton density. We demonstrate that there are two distinct condensate phases with different number of condensate flavors, as shown in Fig. 1.
Our paper is organized as follows. In Sec. II, we explain how we model the heterostructure, and present the mean-field phase diagram implied by Hartree-Fock theory. In Sec. III, we derive an effective boson model that incorporates exciton-exciton interaction effects and can be used to describe excitons in the low density limit. The difference between the strengths of the repulsive interactions between excitons with the same internal label and between excitons with different internal labels changes sign as the layer separation increases. This change drives the transition from phase-II, a phase with two condensate flavors present, to phase-I, a phase with only one condensate flavor. Both phases spontaneously break the symmetry of the model Hamiltonian, and the symmetry breaking pattern of each phase is analyzed. In this section we also explain how capacitance measurements can be used to study the exciton phase diagram experimentally and to extract the value of the exciton-exciton interaction strength within each phase. In Sec. IV, we use a time-dependent Hartree-Fock theory to study the stability of phase-I against small fluctuations, and to calculate the collective mode spectra of these exciton condensates. Finally in Sec. V, we present a brief summary, discuss issues related to experiments, and comment on the relationship between our work and previous studies.
Ii Mean-Field Phase Diagram
We consider two monolayer TMD semiconductors separated and surrounded by hBN (Fig. 1). Many of the points we make apply with minor modification, however, to any bilayer two-dimensional semiconductor system. Monolayer TMDs are direct-gap semiconductors with band extrema located at valleys and . Because these TMD layers lack inversion symmetry, spin degeneracy in the TMD bands is lifted by spin-orbit interactions. Because of differences between the orbital character of conduction and valence band states,Xiao et al. (2012) it turns out that spin splitting is large at the valence band maxima and small at the conduction band minima. As illustrated in Fig. 1, we therefore retain in our theory the two valley-degenerate valence bands labeled by , and four conduction bands with labels corresponding to spin and valley. We assume that exciton binding energies and densities are small enough to justify a parabolic band approximation for all band extrema. Our mean-field ansatz allows up to two types of excitons to be present; for examples pairs formed from holes in band and electrons, selected by spin-splitting, in band , can condense, along with pairs formed from holes in band and electrons in band . Although the unpaired conduction bands are only slightly higher in energy, this pairing ansatz is fully self-consistent at low exciton density, because of the substantial exciton binding energy. Our pairing ansatz is also justified by an interacting boson model, described in Sec. III, which allows for the most general possible pairing scenario.
The ansatz leads to the mean-field Hamiltonian:
where the prime in the first summation restricts the pair index to (11) and (22) contributions. and are fermionic creation and annihilation operators. The kinetic term accounts for the difference between conduction and valence band effective masses, and , and are Pauli matrices. The dressed energy difference between conduction and valence bands, , and the coherence induced effective interlayer tunneling amplitude, , are defined as:
where is the expectation value in the mean-field ground state,
In Eq. (2), is the reduced mass, and is the area of the system. The paramter is:
where is the chemical potential for excitons, and is the total charge density transferred between layers. Equations (2), (3) and (4) form a set of mean-field equations that can be solved self-consistently. Note that the (11) and (22) pairing channels are coupled through the dependence of on the total transferred density .
The exciton chemical potential can be tuned electrically by applying a bias potential between the electrically isolated layers: where is the spatially indirect band gap between the conduction band of the electron layer and the valence band of the hole layer. The band gap can be adjusted to a conveniently small value by choosing two-dimensional materials with favorable band alignmentsGong et al. (2013); Chiu et al. (2015).
and are the Coulomb interaction potentials within and between layers. The forms of Coulomb potentials are determined by solving the Poisson equation for our schematic experimental setup(Fig. 1). , where and are hBN dielectric constants perpendicular and parallel to the z-axis, is the effective dielectric constant due to insulator layer(hBN) between electron and hole layers. , where is the geometric layer separation between electron and hole layers, is the effective layer separation and slightly larger than .Cai et al. (2007)
Below we express lengths and energies in terms of the characteristic scales , and . Typical values for different material combinations are listed in Table 1.
|[Kormányos et al., 2015]||[Kormányos et al., 2015]||(Å)||(meV)||(eV)[Gong et al., 2013]|
The indirect exciton binding energy determines the value for at which excitons first appear. When no excitons are present. In this state each layer is electrically neutral and there is no interlayer coherence. Eq. (2) has nontrivial () solutions only for . We find two distinct types of BXC phase. In phase-I, only one type of exciton condenses (e.g. and ). In phase-II, excitons associated with both valence bands condense and have equal population (e.g. ). Both phases are allowed by Eq. (2). We obtain the phase diagram in Fig. 1 by comparing the total energy of phase-I and II as a function of . Below a critical layer separation , phase-II always has a lower energy, as illustrated in Fig. 2(a). Above , a transition from phase-I to phase-II occurs as the chemical potential increases(Fig. 2(b)). Typical quasiparticle energy bands in phase-II and I are depicted in Fig. 2(c) and (d), and show that the system is an excitonic insulator with a charge gap.
In our mean-field theory, condensation of one type or the other always occurs at when excitons are present. It is well known however that at high electron and hole densities a first-order Mott transition occurs Liu et al. (1998); De Palo et al. (2002); Nikolaev and Portnoi (2008); Asano and Yoshioka (2014); Fogler et al. (2014) from the gapped exciton condensate phase to an ungapped electron-hole plasma state. The electron-hole plasma state is preferred energetically because it can achieve better correlations between like-charge particles, reducing the probability that they are close together, while maintaining good correlations between oppositely-charged particles. The density at which the Mott transition occurs is most reliably estimated via a non-perturbative approaches.De Palo et al. (2002) No estimate is currently available for the TMD case, for which the valley degeneracy and the small spin-splitting in the conduction band will tend to favor plasma states over exciton condensate states. Based on existing estimatesDe Palo et al. (2002) we can conclude that the Mott transition density is below as and below for . Corrections to mean-field theory which go in the direction of favoring plasma states can be partially captured by accounting for screening of the electron-hole interaction which becomes stronger as exciton sizes increase and excitons correspondingly become more polarizable. The results reported here are intended to be reliable only in the low exciton density limit.
Iii Interacting Boson Model
To understand the phase diagram more deeply, we employ a boson Hamiltonian designed to describe weakly-interacting excitons in the low density limit. Our strategy to obtain the boson Hamiltonian is to construct a Lagrangian based on a al wavefunction which parametrizes a family of states with electron-hole coherence. The Berry phase part of the Lagrangian has the same form as that in the field-theory functional integral representation of a standard interacting boson model.Negele and Orland (1988) Appealing to this property, we promote variational parameters in the wavefunction to bosonic operators. The details of the derivation are presented in Appendices A and B.
The boson Hamiltonian is:
where is a bosonic operator for an exciton with a hole in valence band , an electron in conduction band , and total momentum . is the momentum transfer . For the TMD system, there are 8 possibilities for the composite index . The quadratic term in Eq. (5) accounts for exciton kinetic energy () and chemical potential. The quartic terms describe exction-exciton interactions. The prime on the quartic term summation enforces momentum conservation .
The two types of exciton interaction arise from the fermionic Hartree and exchange interactions respectively. In the exchange interaction, two excitons swap constituent electrons or holes. Analytic expressions for the coupling strength and are given in Appendix A.
We focus here on their zero-momentum limits and , which are more easily interpreted and capture much of the exciton-exciton interaction physics. For the case in which the exciton condensate is populated by a single flavor we find that for low exciton densities
where is the total exciton-exciton interaction, as expected from the mean-field theory for weakly interacting bosons. This behavior is illustrated in Fig. 3(a). We have verified that the interaction parameter obtained by examining the dependence of on in the fermion mean-field theory agrees with the analytic expression in App. A, as illustrated in Fig. 3(b) which plots as a function of layer separation . We find that and that , where and are both positive and originate from inter and intra layer fermionic exchange interactions respectively. The binding energy of isolated excitons is due microscopically to attractive inter layer exchange interactions. When excitons overlap and interact with each other, coherence between layers is reduced weakening inter layer exchange, but strengthening intra layer exchange. This explains the signs of the two contributions to . The overall sign of is positive at because the loss of interlayer exchange energy when excitons overlap is greater than the gain in intralayer exchange energy. In Fig. 3(c), we show that becomes negative beyond a critical layer separation . It turns out that although both and increase with layer separation , the rate of increase of is smaller than for . The difference in behavior can be traced to the exponential decrease in the momentum space inter-layer Coulomb interaction with layer separation as shown in Eq. (49) and (50).
To find the ground state in the realistic multi-flavor case, we assume that all excitons condense into states and introduce the following matrix:
Neglecting the small spin-orbit splitting of conduction band states, the total energy per area can be written in a compact form,
where . In Eq. (8) is the total density of excitons,summed over all flavors,and measures the flavor polarization of the exciton condensate. This energy functional is invariant under the following transformation:
Here captures the U(1) symmetry which originates from separate charge conservation in the individual layers. and are respectively and special unitary matrices, which capture the SU(2) symmetry of the valence bands and the SU(4) symmetry present in the conduction bands when their spin-splitting is neglected. The overall symmetry group of the system is U(1)SU(2)SU(4). When the conduction band spin-orbit splitting is included, the higher energy conduction band states in each valley are not occupied and the symmetry group is reduced to U(1)SU(2)SU(2), corresponding to separate charge conservation and rotations in both conduction and valence band valley spaces.
acquires a nonzero value in the ground state only if , . By minimizing the energy functional, we verify that the sign of determines the position of a phase boundary between two different classes of exciton condensate which we refer to as phase-I and II. When , phase-I is energetically favorable and a representative realization of the ground state is,
where is the exciton density. is invariant under the transformation:
where is a unitary matrix. Therefore, phase-I spontaneously breaks the U(1)SU(2)SU(4) symmetry down to U(1)U(3) symmetry.
When phase-II is realized. Energy minimization shows that a representative realization of the ground state in phase-II is,
where is the total exciton density in phase-II. is invariant under the transformation:
where is a unitary matrix. Phase-II spontaneously breaks the U(1)SU(2)SU(4) symmetry down to SU(2)U(2) symmetry. A similar analysis can be applied to identify the symmetry breaking pattern when the spin splitting of the conduction bands is considered. In phase-I, an application of an infinitesimal external Zeeman field lifts both conduction and valence band valley degeneracies, and selects a unique condensate ground state with a finite spin-polarization, as illustrated in Fig. 4. Phase-I therefore satisfies the definition of a ferromagnet, defined as a system with a finite spin-polarization in an infinitesimal Zeeman field, but has a distinct set of broken symmetries compared to the usual spin rotational symmetry breaking.
Based on this mean-field calculation, we conclude that although the system has 8 types of excitons in total, only one or two flavors condense in the ground state. The number of condensed flavors is in general limited by the number of distinct valence or conduction bands, which ever is smaller in number. Although the boson model correctly captures the phase transition position as a function of , it is important to emphasize that it is valid only in the low exciton density limit. For this reason, it fails to accurately predict the dependence of the phase boundary. In addition it fails to capture the tendency toward weaker electron-hole pairing at high exciton densities, which eventually leads to an electron-hole quantum liquid state with no interlayer coherence.
The relationship between the exciton density , the exciton chemical potential and the coupling strength makes it possible to extract the value of from capacitance measurement. The differential capacitance per area for the heterostructure is:
The Hartree coupling strength can be identified as the inverse of the geometric capacitance:
Therefore, capacitance measurement provides a simple way to determine the value of in the low-exciton density limit:
The sign of helps to distinguish phase-I and II.
Iv Fluctuations and Stability
The bilayer exciton condensate is a state with spontaneously broken continuous symmetries, and therefore hosts low-energy collective fluctuations. Theoretical studies of fluctuation properties are of interest in part because they can reveal mean-field stateFedorov et al. (2014) instabilities. The collective modes can be studied using the interacting boson model, which is described in detail in Appendix C. The interacting boson model admits analytic solutions for collective modes associated with exciton density, phase and flavor fluctuations in both phase-I and II. However, it is valid only in the low exciton density limit. Here, we study another approach that can be applied to any exciton density. This approach is based on the following variational wave function which captures exciton density and phase fluctuations in phase-I:
where is the phase-I ground state. and are respectively quasiparticle creation operators for occupied and empty quasiparticle states in associated with the ground state condensate and are defined as follows:
where and are parameters determined by self-consistent Hatree-Fock equations (2),
is a normalization factor,
and are complex parameters.
To study fluctuation dynamics, we construct the Lagrangian:
where is the harmonic Berry phase, and is the harmonic energy variation Giuliani and Vignale (2005):
Note that and are complex numbers, and that there is a redundancy,
In terms of the and fields, the Berry phase and energy variation are,
The kernel matrices are real and symmetric. The and fields in can be identified with exciton density and phase respectively, and the Berry phase contribution to the action captures the conjugate relationship between these fluctuation variables.
Stability of the mean-field ground states against small fluctuations requires that the matrices are nonnegative. We have verified that this condition is satisfied out to large by explicit numerical calculations like those summarized in Fig. 5(a) and (b). At , the matrix always has a zero-energy eigenvalue since,
which follows from the fact that ground state energy is independent of global interlayer phase.
For low exciton density (Fig. 5(a)), the lowest eigenvalues of and have similar behavior and are separated from the continuum. This is expected since and are identical in the limit . Fig. 5(b) demonstrates that the lowest eigenvalues of are close to the particle-hole continuum when the exciton density becomes large; the interacting boson model discussed above fails qualitatively in this limit.
The Euler-Lagrange equation for the Lagrangian in Eq. (21) gives rise to the equation of motion,
which leads to
It follows that the energy of the collective mode is given by the square root of the lowest eigenvalues of the matrix product , which is plotted in Fig. 5(c) and (d). The lowest energy collective mode is the gapless Goldstone mode of the exciton condensate. For low exciton density (Fig. 5(c)), the Goldstone mode has linear dispersion at small , becoming quadratic at large . This agrees with the Goldstone mode behavior predicted by the weakly interacting boson model(Eq. (5)). For large (Fig. 5(d)), the Goldstone mode deviates from quadratic behavior at large . The failure of the weakly interacting boson model in the high density limit originates from the internal structure of the excitons. When the typical distance between excitons is comparable to exciton size, excitations must be described in terms of the underlying conduction and valence band fermion states.Keldysh and Kopaev (1965); Comte and Nozieres (1982); Zhu et al. (1995)
V Summary and Discussion
By combining Hartree-Fock theory and an interacting boson model, we have shown that spatially indirect exciton condensates in group-VI TMD bilayers have two distinct phases. We have also studied the dynamics of exciton condensate density and phase fluctuations and calculated the associated collective mode spectra.
The topic of exciton condensation in semiconductors has a long history and our work is related to some earlier studies. For example, Berman et al.Berman et al. (2012) studied exciton condensation in bilayers formed from gapped graphene, although the possibility of two distinct condensate phases was not considered. The phase transition between the two condensate phases as a function of layer separation was studied previously Fernández-Rossier and Tejedor (1997); de Leon and Laikhtman (2001) for the case of quantum well bilayer excitons, and further explored in a very recent publication.Combescot et al. (2015) The TMD layers considered in this paper are distinguished from semiconductor quantum well systems by exciton binding energies that are an order of magnitude larger, and by spin-valley coupling which leads to two-fold degenerate valence bands and approximately four-fold degenerate conduction bands. Compared to Refs.de Leon and Laikhtman, 2001 and Combescot et al., 2015, we used a completely different approach to derive an interacting boson model. Our approach is physically transparent, and is based on a variatonal wavefunctions defined by parameters whose quantum fluctuations are characterized by using a Lagrangian formalism. The bosonic nature of excitons is automatically taken into account in the Lagrangian, and there is no need to calculate combinatorial factors arising from the indistinguishability of bosonic particles. Our approach provides a simple yet systematic way to model the exction-exciton interaction. We have also discussed a fermionic Hartree-Fock approach from which the exciton-exciton interaction strengths can be extracted with similar results, and proposed that the interaction strengths can be experimentally determined by performing capacitance measurement.
Because the hBN dielectric barrier in the systems of interest, must be thick enough to make interlayer tunneling weak, Fig. 1 implies that phase-I with a single condensate flavor is more likely to be realized in experiment than phase-II. Phase-I breaks the invariance of the system Hamiltonian under separate valley rotations in conduction and valence bands, and is ferromagnetic in the sense that infinitesimal Zeeman coupling leads to a spin-polarization that is proportional to the exciton density.
In spit of their large gaps, band edge states in TMDs have relatively large Berry curvaturesXiao et al. (2012). In monolayer TMDs momentum space Berry curvatures leadWu et al. (2015) to unusual exictonic spectra in which hydrogenic degeneracies are lifted. Although band Berry curvatures should be less important in spatially indirect exciton systems because weaker binding implies that the exciton states are formed within a smaller region of momentum space. In terms of its influence on quasiparticle bands, exciton condensation has the effect of preventing gaps between conduction and valence band states from closing. Since the host semiconductor materials are topologically trivial, and since transitions between trivial and non-trivial states can occur continuously only when the quasiparticle charged excitation energy vanishes, we argue that exciton condensation will not result in interaction-driven topologically nontrivial states in our system.
The critical temperature of spatially indirect exciton condensate is the Berezinskii-Kosterlitz-Thouless transition temperature, given at low exciton densities by the weakly-interacting boson expression
where is the electron-hole pair total mass and is the reduced mass. In the low exciton density limit scales linearly with exciton density.Filinov et al. (2010); Fogler et al. (2014) For the MoS/hBN/MoTe heterostructure and exciton densities in the cm range, and is about 10K. The maximum possible transition temperature is closely related to the critical density at which the Mott transition to an electron-hole plasma occurs, and this increases as decreases. Using the variational Monte Carlo estimate of DePalo et al.De Palo et al. (2002) the critical value of as . From this we conclude that cannot exceed around . Adjustment of exciton density by external bias voltage can be employed to search for the highest transition temperature and to study the Mott transition to an ungapped electron-hole plasma that is expected at high exciton densities. The most interesting regime is likely to be the case of very small layer separations of which current leakage driven through the tunnel barrier by an interlayer bias potential might be appreciable, requiring the bilayer to be treated as a non-equilibrium system.
The exciton condensate should be experimentally realizable in TMD bilayers provided that samples with sufficiently weak disorder can be achieved. The photoluminescence line width of an individual monolayer TMD is a particularly useful characterization of sample quality for this purpose. is currently dominatedMoody et al. (2015) by the position-dependence of exciton energies. Therefore, the narrower the line width , the weaker the disorder, and the better the sample quality. We expect bilayer exciton condensation to occur only in samples in which , since the excitons will otherwise simply localize near positions where they have minimum energy. Note that the inhomogeneous broadening of spatially indirect excitons will not be experimentally accessible since the corresponding transitions are optically inactive when the interlayer tunneling is negligible, but that it should be similar to the broadening of the readily measurable direct exciton energies. It should therefore be possible to judge on the basis of optical characterization when samples have achieved sufficient quality to study spatially indirect exciton condensate physics.
This work was supported by the SRC and NIST under the Nanoelectronic Research Initiative (NRI) and SWAN, and by the Welch Foundation under Grant No. F1473.
Appendix A Interacting boson model for excitons in the low density limit
In the low density limit, excitons can be approximated as interacting bosons. We take a BCS like variational wave function to describe excitons,
where and respectively denote a conduction and valence band state, and include internal indices such as spin and valley and also momentum label. is the vacuum state defined by . operator creates particle-hole excitations on top of the vacuum. is a normalization factor so that . is a set of complex variational parameters, which are small when the exciton density is low.
The density matrix with respect to is , where and can be conduction or valence states. We expand the density matrix to fourth order in ,
where is understood to be a matrix and is its Hermitian conjugate.
We introduce another matrix so that has a quadratic form without fourth-order correction,
Expanding up to fourth order of , we have that
The number of excitons is . Therefore, we verify that acts as the small parameter in the limit of low .
An important property of the density matrix is that
which indicates that can be approximated as a Slater determinant up to fourth order in .Giuliani and Vignale (2005)
parametrizes a family of states with electron-hole coherence, and also represents low-energy states in the low-exicton density limit. We choose to construct an effective interacting boson model using this variational wavefunction approach rather than a commonly used auxiliary field approach because of the necessity of consistently accounting for both exchange and Hartree mean-fields in spatially-indirect exciton systems. (See additional discussion below.) To study low-energy dynamics, we construct a Lagrangian based on
and again expand everything to . This Lagrangian provides an effective field theory for excitons. The Berry phase has the following form,
which does not have fourth order corrections.
where is quadratic in , and is quartic in with subscript and representing Hartree and exchange contributions. The explicit forms are below.
Here and are conduction and valence state energy including self-energy effects. The interaction kernel has the form
where the momentum dependence is now explicit, and denotes internal indices. is the area of the system. is the intralayer interaction if both and represent conduction or valence bands, and the interlayer interaction otherwise.
We now write in a more concrete form
Here and denote different valence and conduction bands. We approximate and by parabolic bands, and assume different valence (conduction) bands have the same hole (electron) mass (). In the case of TMDs, these are reasonable approximations, and and respectively take two and four different values.
can be reduced into a diagonal from by doing the following decomposition,
where is a complex field that depends on momentum but not on . , where the total mass . For notation convenience, we also introduce . is the wavefunction for a single exciton,
where the reduced mass , and is the binding energy for state. The normalization condition is that
Here we have chosen to be real. In Eq. (42), is the wavefunction for an exciton with center-of-mass momentum .