Helicoidal excitonic phase in an electron-hole double layer system
We propose helicoidal excitonic phase in a Coulomb-coupled two-dimensional electron-hole double layer (EHDL) system with relativistic spin-orbit interaction. Previously, it was demonstrated that layered InAs/AlSb/GaInSb heterostructure is an ideal experimental platform for searching excitonic condensate phases, while its electron layer has non-negligible Rashba interaction. We clarify that due to the Rashba term, the spin-triplet (spin-1) exciton field in the EHDL system forms a helicoidal structure and the helicoid plane can be controlled by an in-plane Zeeman field. We show that due to small but finite Dirac term in the heavy hole layer the helicoidal structure of the excitonic field under the in-plane field results in a helicoidal magnetic order in the electron layer. Based on linearization analyses, we further calculate momentum-energy dispersions of low-energy Goldstone modes in the helicoidal excitonic phase. We discuss possible experimental probes of the excitonic phase in the EHDL system.
One of the fundamental challenges in condensed matter physics is an experimental realization of excitonic condensation and excitonic insulator at the equilibrium mott61 (); knox63 (); keldysh65 (); jerome67 (); halperin68 (); wakisaka09 (); kogar17 (); werdehausen18 (). Early experiments report significant electron-hole Coulomb drag phenomena in bilayer quantum well structure lozovik75 (); comte82 (); datta85 (); xia92 (); zhu95 (); littlewood96 (); naveh96 () made out of semiconductors such as GaAs/AlGaAs croxall08 (); seamons09 (); yang11 () or Si sivan92 (). Recently, a strained layer InAs/AlSb/GaInSb heterostructure naveh96 (); wu19 () provides an ideal platform of Coulomb coupled electron-hole double layer (EHDL) system, that has a lot of advantages over the others. Thereby, a dual gate device enables a continuous change of both chemical potential and charge state energy. Large relativistic spin-orbit interaction (SOI) realizes a nearly isotropic Fermi contour of the heavy hole band as well as the electron band. In fact, a recent experiment reports a transport signature of the excitonic coupling in the charge neutrality regime of the strained layer heterostructure wu19 ().
The excitonic pairing in Coulomb coupled EHDL system is either of the spin triplet (spin-1) nature or of the spin-singlet (spin-0) nature zhu95 (); zhu96 (). An energy degeneracy between these two will be lifted by the large SOI in the electron layer of the EHDL system. Besides, due to the SOI, the spin triplet vector (spin-1 vector) could exhibit a non-trivial spatial texture, that breaks the translational symmetry in the two-dimensional plane. The spatial texture of the excitonic pairing field in EHDL system is generally free from charged and/or magnetic impurities in each layer, while it could be pinned by a spatial variation of the dielectric constant in the intermediate separation layer. To our best knowledge, it is entirely an open question at this moment how the spin-1 exciton forms spatial textures in the presence of the SOI, how the textures could be coupled with external magnetic (Zeeman) field, what kind of low-energy collective excitations would emerge from the translational symmetry breaking, and how the texture could be experimentally detected.
In this paper, we identify the spatial textures of the spin-1 exciton condensate in the presence of the SOI, clarify the natures of the low-energy collective modes in the excitonic phase, and propose possible experimental probes for detecting the spatial texture of the spin-1 exciton condensate. First, we derive a effective action for the spin-triplet (spin-1) excitonic pairing field and determine a form of the couplings among the exciton, the SOI and the magnetic Zeeman field. We then propose that helicoidal structures of the spin-1 exciton minimize the classical action. Based on the effective action and the classical configuration, we derive linearized EOMs for fluctuations of the spin triplet exciton field around the classical configuration. From the EOMs, we calculate the low-energy collective excitations in the helicoidal excitonic condensate. Using these theoretical knowledges, we propose possible experimental probes for detecting the helicoidal excitonic texture in the EHDL system.
Ii model and effective action
We begin with a non-interacting Hamiltonian for the two-dimensional EHDL system pikulin14 ();
with , , , and () are two by two unit and Pauli matrices. and denote the creation and annihilation operators of electron and hole bands with respective mass and . The chemical potential as well as the charge state energy (band inversion parameter) can be separately tuned by the dual gate device. The SOI in the electron band with doublet takes a form of the Rashba term with and the SOI in the heavy hole band with doublet takes a form of the Dirac term with . In the InAs/AlSb/GaInSb heterostructure, the Rashba term in the electron layer is much larger than that in the heavy hole band; , , Here and (thickness of the intermediate separation layer) define a length scale and energy scale of the EHDL system; . Typically, K and nm pikulin14 (); wu19 (). For simplicity, we take , unless dictated otherwise (e.g. Section. V). To study how the spatial texture of the spin-1 exciton condenate can be changed by an in-plane magnetic field, we include the magnetic Zeeman field along the in-plane direction ().
The electron and hole layers are coupled with each other through the long-range Coulomb interaction. For simplicity, we employ the short-ranged repulsive interaction;
with . The spin-rotational symmetry in Eq. (2) allows to decompose the interaction into the spin singlet and triplet parts. In terms of the respective excitonic pairing fields (), the interaction part takes a form of
Using the Stratonovich-Hubbard transformation followed by a standard procedure (appendix A), we derive a partition function and its action , that is a functional of the spin-triplet (spin-1) exciton pairing fields (‘-vector’); ,
with . and are the unit vectors along and axis. and are positive, and and are negative (appendix A). Since and are proportional to and respectively, we can always assume and without loss of generality. Note that the spin-1 exciton field has both real and imaginary parts, and ; , (). Note also that using a Talyor expansion, we took into account the lowest order in and . Within the lowest order, the SOI favors helicoid orders of both the real and imaginary parts of the -vector, where a rotational plane of the -vector is parallel to a propagating direction within the plane. The Zeeman field is linearly coupled with a vector chirality formed by the real and imaginary parts of the -vector.
The quadratic part of the action in Eq. (4) gives energy dispersions of the spin-1 exciton bands as a function of momentum . The bands are triply degenerate at the zero momentum point at the zero magnetic field (), while the degeneracy is lifted at non-zero momentum due to the SOI term yu14 (). In the zero field, the lowest spin triplet exciton band has a ‘wine-bottle’ minimum at a line (ring) of . The energy at the minimum is . When the energy minimum decreases on lowering temperature or on decreasing the charge state energy , the minimum eventually touches the zero energy. Thereby, the system picks up one from the line of , and undergoes Bose-Einstein condensation (BEC) pethick01 () of the spin-1 exciton band. The resulting phase is what we call in this paper helicoidal excitonic condensate phase. In the next section, we assume that and describe this helicoidal excitonic phase in details and explain especially how the helicoidal excitonic condensate can be controlled by the in-plane Zeeman field ().
Iii helicoidal excitonic condensate
The classical action at the zero magnetic field is maximally minimized by a helicoidal structure of the triplet pairing field;
The spatial pitch of the helicoid structure is determined by the SOI energy () and the phase stiffness energy () in the classical action. The propagation direction of the helicoid structure is arbitrary at the zero field, where the system is symmetric under the continuous rotation of spin and coordinate around the -axis; the U(1) phase is arbitrary. The momentum and the -axis subtend a rotational plane of the -vector, while the real and imaginary parts of the -vector are parallel to each other everywhere (Fig. 2(a)). The arbitrary U(1) phase represents a relative gauge degree of freedom; a difference between the two U(1) gauge degrees of freedom of the electron and hole bands.
Under the in-plane Zeeman field (appendix B), the direction of the momentum becomes perpendicular to the Zeeman field, so that the -vector can rotate around the in-plane Zeeman field (see Fig. 2(b,c)). The real and imaginary parts of the -vector form a finite vector chirality along the field direction. The vector chirality is spatially uniform. The amplitude of the vector chirality becomes larger for the larger in-plane Zeeman field. Meanwhile, an angle between the real and imaginary part, , saturates into at a critical field defined by
For an in-plane Zeeman field below the critical field (), the helicoid structure of the -vector is given by with
Without loss of generality, we always take the field along the -axis henceforth. The vector chirality formed by and is spatially uniform, and it increases on increasing the field;
A ratio between and is specified by , and the angle between and is specified by . and form an energy degeneracy line under a fixed vector chirality [Eq. (11)]. When the in-plane field reaches the critical field , the angle becomes and the ratio becomes the unit. Above the critical field (), the angle takes and the ratio takes one everywhere (Fig. 2(c));
Iv low-energy collective modes
The helicoidal excitonic condensations described in the previous section break the spatial translational symmetry, spin rotational symmetry and the relative U(1) gauge symmetry; the condensate phases are accompanied by gapless Goldstone modes. Experimental observation of the collective excitations would serve as future ‘smoking-gun’ experiment for the confirmation of the excitonic condensation at the equilibrium and therefore, it is important to characterize theoretically energy-momentum dispersion of the low-energy collective modes. To this end, we take a functional derivative of the effective action (Eq. (4)) with respect to and , and derive a coupled non-linear equation of motions (EOMs) for the spin-triplet pairing field. The helicoidal structures described in the previous section are static solutions of these coupled EOMs. Thus, we consider a small fluctuation of the excitonic pairing field around these static solutions, and and linearize the EOMs with respect to the fluctuation field, and .
As suggested by the Berry phase term in the effective action, , and are nothing but (Holstein-Primakov) boson annihilation and creation operator respectively. Accordingly, the linearized EOMs thus obtained must reduce to a generalized eigenvalue problem with bosonic Bogoliubov de-Gennes (BdG) Hamiltonian colpa78 ();
Here is the 2 by 2 diagonal Pauli matrix in the particle-hole space, that takes for the annihilation and for the creation operator. is a 6 by 6 matrix-formed differential operators, that are hermitian, . For , the BdG Hamiltonian thus obtained takes the following explicit form,
For , the BdG Hamiltonian is given by
, , and are already defined by Eqs. (7,13,8) respectively. and in Eqs. (20,23) must satisfy Eq. (11). Note that the two BdG Hamiltonians beome identical to each other at , where , , . Using bosonic Bogoliubov transformations colpa78 (), we diagonalize these Hamiltonians in the momentum space, to obtain energy-momentum dispersions of the low-energy collective excitations in the helicoidal excitonic phases (Appendix C). Fig. 2 shows the dispersions along the high symmetric line in the momentum space for , and respectively.
The helicoidal condensate phase at has two gapless Goldstone modes around ; translational mode and spin rotational mode (Fig. 2(a)). These two result from the spontaneous symmetry breakings of the continuous symmetries; translational symmetry and a combined symmetry of the relative gauge symmetry and the spin rotation symmetry respectively. The translational mode at the gapless point induces a simultaneous rotation of and by the same angle around the plane (Fig. 3(a)). The spin rotational mode induces a change of the amplitudes of and as well as the relative angle between these two vectors (Fig. 3(b)), but it does not change the total amplitude of the spin-1 exciton field, . When the field is above the critical field (), the relative angle between the real and imaginary part is locked to by the large in-plane field; the angle between these two are fully saturated (‘fully saturated phase’). Accordingly, the spin rotational mode acquires a finite mass and only the translational mode forms a gapless dispersion at (See Fig. 2(b)).
Finally, let us give several remarks on light scatterings for probing these low-energy collective modes in the EHDL system. In the layered heterostructure, the electron and hole layers are physically well separated by an intermediate separation layer with its thickness being typically 10 nm wu19 (). Meanwhile, single exciton is a pair of the electron creation and the hole creation. Thus, generally, the external electromagnetic field can couple with the exciton in the EHDL system only through 2-excitons process, 4-excitons process, and so on; one photon causes at least a pair of exciton creation and exciton annihilation in the EHDL system. A calculation in appendix D shows that the (static) helicoidal excitonic order parameter quadratically induces a uniform change of the electron density in the electron layer, . This suggests that photon can couple electrically with the low-energy collective modes in the helicoidal excitonic condensate in EHDL system.
V experimental probes
In the presence of a small Dirac term in the hole layer , the helicoidal texture of the spin-1 exciton condensate induces a helicoidal texture of local magnetic moment in the electron layer, whose spatial pitch is instead of . At the zero magnetic field, however, the helicoid structure given by Eq. (5) is symmetric under the time-reversal symmetry combined with a gauge transformation for the electron or hole band. Thus, the local magnetic moment in each layer is quenched at .
The helicoidal magnetic texture appears when the in-plane Zeeman field is applied to the helicoidal excitonic phase. Thereby, the local magnetic moment as well as the spin-1 exciton field rotate around the in-plane Zeeman field;
Here and are real-valued and odd functions in the magnetic field ; , . ‘’ in the right-hand side denotes the higher order harmonic contributions (, , components). In the leading order in small , and are proportional to (Appendix D).
Having a finite out-of-plane (-component) magnetization, the spatial magnetic texture given in Eq. (24) could be experimentally seen by magnetic optical measurements hubert98 (). For example, when the electron layer is sufficiently reflective, a spatial map of the magnetic Kerr rotational angle in the two-dimensional layer must show a stripe structure. According to our theory prediction, the magnetic stripe appears in parallel to the in-plane Zeeman field, and it disappears when the field is set zero. The Kerr rotation angle changes its sign when the in-plane field is reversed.
Vi conclusion and discussion
A recent transport experiment on a strained layered InAs/AlSb/GaInSb heterostructure reports resistive signatures of the excitonic coupling at low temperature around the charge neutrality line of the BEC regime wu19 (). In this paper, we show that due to the large Rashba interaction in the electron layer, energy degeneracy among three spin-1 (spin-triplet) exciton bands are lifted at finite momentum. On lowering the temperature or on changing the charge state energy, the lowest spin-1 exciton band can undergo the BEC at finite momentum, resulting in a helicoidal structure of the spin-1 exciton field (helicoidal excitonic condensate). The helicoidal plane of the spin-1 exciton can be controlled by the in-plane Zeeman field.
Based on the linearized coupled EOMs of the spin-1 exciton, we calculate momentum-energy dispersions of the low-energy collective modes in the helicoidal excitonic phase. For future possible light scattering experiments, we show that these low-energy modes can couple electrically with one photon through the two-excitons processes. We also demonstrate that due to small Dirac term in the heavy hole layer, the helicoidal structure of the spin-1 exciton condensate results in a helicoidal magnetic structure in the electron layer. Having a finite out-of-plane magnetization, the helicoidal magnetic structure could be visualized by a spatial map of the magnetic Kerr rotation angle in the two-dimensional layer. Our theory predicts that the magnetic stripes appear in parallel to the in-plane Zeeman field and it disappears at the zero Zeeman field.
In a Coulomb coupled EHDL system without the SOI, the spin triplet (spin-1) excitons and spin singlet (spin-0) exciton are energetically degenerate due to the independent spin rotations of electron spin and hole spin zhu96 (). Meanwhile, the analyses in this paper has ignored a coupling between the spin-0 exciton and spin-1 excitons. The coupling does exist at the first order in the Rashba term in the electron layer as well as at the first order in the in-plane Zeeman field ;
Here and denote the spin-1 and spin-0 exciton fields respectively. The first and third terms are nothing but the last three terms in Eq. (4). and are proportional to the in-plane field , . At the quadratic level of the effective action at the zero field (), the four-fold degenerate exciton bands at the zero momentum (one spin-0 and three spin-1) are split into two doubly degenerate exciton bands at finite momentum ;
with . The upper two exciton bands have an energy of , and the lower two exciton bands have an energy of . One of the lower two bands is a mixture of purely three spin-1 excitons, , whose BEC induces the helicoidal excitonic phase [discussed in this paper]. On the one hand, the other of the lower two bands is a mixture of the spin-0 and spin-1 exciton bands, , whose BEC induces an in-plane collinear texture of the spin-1 exciton field. The collinear texture and the helicoidal texture are energetically degenerate at the zero Zeeman field in the absence of the Dirac term in hole layer (). The energy degeneracy is due to the spin-rotation around the -axis only in the hole band; in Eqs. (1,2). The degeneracy is lifted by the Dirac term in the hole band as well as the in-plane Zeeman field. In the presence of these perturbations, a mixture of these two textures will be selected as a true classical ground state. By construction, the mixture has lower symmetries than the helicoidal excitonic phase discussed in this paper and thereby it has essentially the same physical response as the helicoidal phase [Sec. VI and Sec. V]. Nonetheless, detailed physical properties of the mixed phase need more theoretical studied and will be discussed elsewhere.
RS thank Rui-Rui Du for helpful information and discussion. This work was supported by NBRP of China (Grant No. 2014CB920901, Grant No. 2015CB921104 and Grant No. 2017A040215).
Appendix A derivation of the effective action
In this appendix, we derive an effective action for the spin-triplet (spin-1) exciton field in the presence of the SOI and the Zeeman field. We begin with the partition function for Eq. (1,2,3) with ,
where the effective action is given by
Here non-interacting temperature Green function, Rashba and Zeeman field parts take forms of,
with , , and . Note that we used the following Fourier transformation for , and ,
is an inverse temperature, , for and , for , and .
A decomposition of the interaction by the Stratonovich Hubbard (SH) variables fradkin91 () gives out a quadratic form of the and fields,
A gaussian integration over the and fields leads to a functional of the SH variables. A Taylor expansion of the functional with respect to the SH variables gives,
In the expansion, we took into account only the first order in the small and . We also expand Eq. (32) in small , to keep up to in , up to in , and up to in the other terms. This gives Eq. (4) as the effective action for the spin-1 exciton field. The coefficients in Eq. (4) are calculated in the followings,
From these expressions, we can see that both and are positive. Since and are proportional to and respectively, we can assume that and are positive without loss of generality.
Appendix B minimization of the classical action
In this appendix, we minimize the classical energy in Eq. (4) with respect to the real and imaginary parts of the spin-1 exciton field, . Take and with unit vectors and ; . In the absence of the SOI, the exciton field takes a spatially uniform solution, because . Thereby, the real and imaginary parts are parallel to each other in the spin space;
The solution breaks the two global symmetries. The U(1) symmetry associated with is nothing but a difference between the U(1) gauge degree of freedom of the electron and that of the hole. The SO(3) symmetry associate with represents the global spin rotational symmetry. In the following, we study how the uniform solution would be deformed in the presence of finite SOI ().
In the presence of the SOI, the spin-1 exciton field takes a spatially dependent solution. Spatial gradients of the amplitudes, and , do not lower the SOI energy. Accordingly, without loss of generality, we can assume that the amplitudes are spatially uniform and the unit vectors depend on the space coordinate ;
This gives the following functional for the classical action, where and are the volume of the system and the inverse temperature respectively;
Here are are functionals of the unit vectors,
For the later convenience, let us rotate and by around the -axis;
In the rotated frame, and are given by
In the following, we first minimize the classical action for fixed and (Eq. (37)). To this end, we have only to maximize , and with respect to and , because . These three functionals can be simultaneously maximized.
To see this, let us first maximize under the normalization condition of for any . In the momentum-space representation, the Fourier series of comprises of two real-valued vectors, and ,
with and . In terms of these vectors, takes a form of