Abstract
We study collective excitations in a helical electron liquid on a surface of threedimensional topological insulator. Electron in helical liquid obeys Diraclike equation for massless particles and direction of its spin is strictly determined by its momentum. Due to this spinmomentum locking, collective excitations in the system manifest themselves as coupled charge and spindensity waves. We develop quantum fieldtheoretical description of spinplasmons in helical liquid and study their properties and internal structure. Value of spin polarization arising in the system with excited spinplasmons is calculated. We also consider the scattering of spinplasmons on magnetic and nonmagnetic impurities and external potentials, and show that the scattering occurs mainly into two side lobes. Analogies with Dirac electron gas in graphene are discussed.
PACS: 73.20.Mf; 73.22.Lp; 75.25.Dk.
1 Introduction
Topological insulator is a new class of solids with nontrivial topology, intrinsic to its band structure. Theoretical and experimental studies of topological insulators grow very rapidly in recent years (see [1,2] and references therein). Threedimensional topological insulators are insulating in the bulk, but have gapless surface states with numerous unusual properties. These states are topologically protected against timereversal invariant disorder. When gap is opened in surface states by timereversal or gauge symmetry breaking, a spectacular magnetoelectric effect arises [3,4].
Recently a "new generation" of 3D topological insulators (the binary compounds Bi_{2}Se_{3}, Bi_{2}Te_{3 }and other materials), retaining topologically protected behavior at room temperatures, were predicted and studied experimentally [57]. Band structure of the surface states of these materials contains a single Dirac cone, where electrons obey 2D Dirac equation for massless particles. Direction of electron spin in these states is strictly determined by their momentum, so these states can be called as "helical" ones. Surface of topological insulator can be chemically doped, forming charged helical liquid. The spinmomentum locking leads to interesting transport phenomena including coupled diffusion of spin and charge [8], inverse galvanomagnetic effect (generation of spin polarization by electric current) [9] and giant spin rotation on an interface between normal metal and topological insulator [10]. The spinmomentum locking offers numerous opportunities for various spintronic applications.
Collective excitations (plasmons) in helical liquid on the surface of topological insulator was considered in [11]. It was shown that due to spinmomentum locking responses of charge and spin densities to external electromagnetic field are coupled to each other. Therefore the plasmons in the system should manifest themselves as coupled charge and spindensity waves and can be called "spinplasmons". In [12] application of spinplasmons in spin accumulator device was proposed. Also surface plasmonpolaritons under conditions of topological magnetoelectric effect were considered in [13].
Properties of the states on a surface of 3D topological insulator are similar to those of electrons in graphene. Graphene is unique 2D carbon material with extraordinary electronic properties [1416]. Its band structure contains two Dirac cones with electrons behaving as massless Dirac particles in their vicinities. Graphene is a perspective material for nanoelectronics due to large carrier mobilities at room temperature. Electronic interactions and collective excitations in graphene have been extensively studied (see [17] and references therein). In particular, the properties of plasmons [18,19] and hybrid plasmonphoton [20] and plasmonphonon [21,22] modes were investigated theoretically and experimentally. It was realized recently that graphene is a fertile ground for quantum plasmonics [23] due to very small plasmon damping.
In this article, we develop quantum fieldtheoretical formalism to describe plasmons in graphene and spinplasmons on a surface of 3D topological insulator based on random phase approximation (RPA). Problems of excitation, manipulation, scattering and detection of single plasmons can be conveniently considered using this approach. Thus, this approach can be especially useful for problems of plasmon quantum optics and quantum plasmonics. We use our approach here to study internal structure and properties of spinplasmons in a helical liquid.
The rest of this article is organized as follows. In Section 2, we present a brief description of electronic states on a surface of topological insulator and in graphene. Next we develop the original quantum fieldtheoretical description of plasmons in Dirac electron gas in Section 3 and apply it further to study their properties. We consider internal structure of plasmons in Section 4 and important consequences of spinmomentum locking in Section 5. Scattering of plasmons on impurities and external potentials is considered in Section 6, and Section 7 is devoted to conclusions.
2 Dirac electrons
The lowenergy effective Hamiltonians for electrons in helical liquid [24] and graphene [16] are
where the Fermi velocities of electrons v_{F }are 6.2 × 10^{5 }m/s for the topological insulator Bi_{2}Se_{3 }and 10^{6}m/s for graphene; the Pauli matrices σ_{x }and σ_{y }act in the spaces of electron spin projections (helical liquid) or sublattices (graphene). The eigenfunctions of (1)(2) can be written as , where S is the system area and f_{pγ}) is the spinor part of the eigenfunction, corresponding to electron with momentum p from conduction (γ = +1) or valence (γ = 1) band:
where φ_{p }is a polar angle of the vector p (here and below we assume ħ = 1). Another difference between helical liquid on a surface of topological insulator and electron liquid in graphene is additional fourfold degeneracy g = 4 of electronic states in graphene by two spin projections and two nonequivalent valleys.
The value of electron spin in helical liquid and of pseudospin in graphene in the state f_{pγ}〉 is
where and are unit vectors directed along the momentum p and the zaxis, respectively. We see that, in helical liquid, the spin of electron lies in the system plane and makes an angle 90° (in counterclockwise direction in the conduction band and inversely in the valence band) with its momentum. In graphene, the sublattice pseudospin of electron is directed along its momentum in conduction band and opposite to it in the valence band. Physically, a definite direction of the pseudospin in the system plane corresponds to definite phase shift between electron wave functions on different sublattices.
A starting point for quantum fieldtheoretical consideration of plasmons on the surface of topological insulator and in graphene is the manybody Hamiltonian of electrons with Coulomb interaction between them:
where a_{pγ }is the destruction operator for electron with momentum p from the band γ, ξ_{pγ }= γv_{F}pμ is its kinetic energy measured from the chemical potential μ and V_{q }= 2πe^{2}/εq is the 2D fourier transform of Coulomb interaction potential screened by surrounding 3D medium with a dielectric permittivity ε.
3 Description of plasmons
To investigate the properties of plasmons in Dirac electron gas, we develop the equationofmotion approach similarly to the original works on plasmons in usual electron gas [25]. We treat a plasmon as a composite Bose quasiparticle, consisting of electronhole pairs with the common total momentum q. Thus, the creation operator for plasmon with momentum q can be written as:
Here the coefficients are the weights of intraband (γ = γ') and interband (γ ≠ γ') singleparticle transitions, contributing to the wave function of plasmon.
The plasmon creation operator should obey Heisenberg equation of motion
where Ω_{q }is plasmon frequency. We start from equation of motion for single electronhole pair, which can be derived using (7):
where we have introduced the Fourier transform of the density operator:
The righthand side of the Equation (10) contains products of four fermionic operators. To reduce it to products of two fermionic operators we use the RPA; its applicability will be discussed below. According to RPA [25], the operator products in the last two lines of (10) can be replaced by their average values in ground state 0〉 of the system:
where n_{pγ }is the occupation number for electronic states with momentum p from the band γ. For electrondoped Dirac liquid at T = 0 (see also the remark at the end of this section), we have n_{p+ }= Θ(p_{F } p), n_{p }= 1, where p_{F }= μ/v_{F }is the Fermi momentum. In the case of hole doping, all characteristics of plasmons are the same due to electronhole symmetry.
Thus, the equation of motion (10) for electronhole pair with taking into account the RPA assumption (12) takes the form:
Combining the definition of plasmon creation operator (8) with the equation of motion for plasmon (9) and single electronhole pair (13), we obtain the system of equations for the coefficients :
Introducing infinitesimal damping into the plasmon frequency and denoting
we can find the plasmon wave function from (14) as:
The normalization factor N_{q }can be determined from the commutation relation for plasmon operators, which should be satisfied on the average in the ground state:
Substituting (8) in (17), we get
where the total weights of intraband (D_{++}) and interband (D_{+ }+ D_{+ }= 1  D_{++}) electron transitions, contributing to the plasmon wave function (16), are:
To find the plasmon frequency Ω_{q}, we can substitute (16) in (15), arriving to the standard RPA equation (see it for the helical liquid [11] and for graphene [18,19]):
where the polarization operator of Dirac electron gas is introduced:
The degeneracy factor g is 1 for helical liquid on the surface of topological insulator and 4 for graphene. The angular factors 〈f_{p+q,γ'}f_{pγ}〉 are specific to helical Dirac electrons and arise in (21) as a result of summation over spinor components of electron wave function.
The RPA becomes exact in the limit of small values of dimensionless parameter r_{s}, defined as a ratio of characteristic Coulomb interaction energy to kinetic energy. For the gas of massless particles, r_{s }does not depend on electron density and equals to r_{s }= e^{2}/εv_{F }(effectively ε = (ε_{1 }+ ε_{2})/2 when 2D electron layer is surrounded by two dielectric halfspaces with permittivities ε_{1 }and ε_{2}). For Bi_{2}Se_{3}, r_{s }= 0.09 with ϵ = 40 for dielectric halfspace, and applicability of the RPA is well established (the value of r_{s }for another material Bi_{2}Te_{3 }is close to that for Bi_{2}Se_{3}). In the case of graphene, r_{s }is not small, but applicability of the RPA can be established due to smallness of the parameter 1/g, leading to selection of bubble diagrams (see [16] and references therein, and also the work [26]).
Maximal achievable amounts of doping of helical liquid on a surface of Bi_{2}Se_{3 }are μ ~ 0.3 eV [24], therefore at room temperature it can be degenerate electron liquid. Hence we assume that T = 0 in all calculations below.
4 Wave function of plasmon
Plasmon dispersions Ω_{q}, calculated numerically from (20)(21) at various r_{s}, are plotted in Figure 1. We also present results for suspended graphene with r_{s }= 8.8 (ε = 1) and for graphene embedded into SiO_{2 }environment with r_{s }= 2.2 (ε = 4). For convenience, the degeneracy factor g = 4 is included here into r_{s}, since the plasmon dispersion in RPA (20) depends only on the combination gr_{s}.
Figure 1. Plasmon dispersions. Plasmon dispersions Ω_{q }at different values of r_{s}. Continuums of intraband (ω <v_{F}q) and interband (ω > 2μ  v_{F}q) singleparticle excitations are shaded; p_{F }= μ/v_{F }is the Fermi momentum, μ is the chemical potential.
It is seen that for small values of r_{s }(the case of topological insulator) the plasmon dispersion law approaches very closely to the upper border of the continuum of intraband singleparticle transitions (ω < v_{F}q). On the contrary, at moderate and large r_{s}, plasmon has welldefined squareroot dispersion in the longwavelength range. At q ≈ p_{F}, the plasmon enters the continuum of interband singleparticle transitions (ω > 2μ  v_{F}q). Inside the singleparticle continuum, energy and momentum conservation laws allow energy transfer between plasmon and singleparticle excitations, so the plasmon acquires a finite lifetime.
The internal structure of plasmons in Dirac electron gas can be characterized by total weights (19) of intra and interband transitions in its wave function. The weight D_{++ }of intraband singleparticle transitions is plotted in Figure 2. It is seen that the undamped plasmon for all values of parameter r_{s }consists mainly of intraband transitions, thus an influence of valence band on its properties is rather weak. When the plasmon enters the singleparticle continuum, inter and intraband transitions start to contribute almost equally to its wave function, but the plasmon undergoes strong Landau damping.
Figure 2. Weight of intraband transitions in plasmon wave function. Weight D_{++ }(19) of intraband electronhole transitions in plasmon wave function at different values of r_{s}.
The detailed picture of the plasmon wave function in momentum space , showing the distribution of contributions of intraband or interband electronhole pairs into wave function of plasmon with given momentum q is presented in Figure 3. The results are calculated for q = 0.4p_{F }at r_{s }= 0.09 and 8.8. For other values of plasmon momentum q, the results are qualitatively similar. It is seen that for small values of r_{s }(the case of topological insulator) the distribution of intraband contributions is very sharply peaked in the forward direction, whereas the contribution of interband transitions is negligible. It can be understood from the reason that the plasmon dispersion is very close to the singleparticle continuum (see Figure 1) and thus the plasmon itself behave almost as single intraband electronhole transition.
Figure 3. Wave function of plasmon in momentum space. Squared modulus of plasmon wave function on the plane of the hole momentum p at q = 0.4p_{F}. Top row: intraband channel at r_{s }= 0.09 (a) and r_{s }= 8.8 (b). Bottom row: interband channel at r_{s }= 0.09 (c) and r_{s }= 8.8 (d).
At large values of parameter r_{s }(the case of suspended graphene), the broad range of electron intraband transitions in momentum space contributes to plasmon. The weight of interband transitions is small but not negligible. Contributions of interband transitions form two side lobes, since the angular factor 〈f_{p+q,±}f_{p∓}〉 suppresses interband forward scattering.
5 Charge and spindensity waves
When a spinplasmon is excited in the helical liquid, anisotropic distribution of electronhole pairs of the type, depicted in Figure 3, arises. This distribution is shifted towards the plasmon momentum q. Due to the spinmomentum locking, the system should acquire a total nonzero spin polarization, perpendicular to q. A similar situation occurs in the currentcarrying state of the helical liquid, which turns out to be spinpolarized [9].
Average of spin polarization in the state with a single plasmon excited is 〈s〉 = 〈1_{q}σ/21_{q}〉 and can be calculated using (8) as:
If q is parallel to e_{x}, only the ycomponent of 〈s〉 is nonzero. Its dependence on q for undamped plasmons at various r_{s }is plotted in Figure 4. At large enough r_{s}, the spin polarization of the system is comparable with the whole spin of a single electron.
Figure 4. Spin polarization of the system with an excited spinplasmon. Inplane transverse spin polarization of the system 〈s〉 (22) in the state with single spinplasmon excited as function of its momentum q at different values of r_{s}.
Note that in the case of graphene the isospin polarization of the system appears instead of spin polarization. Isospin polarization corresponds to nonzero average phase shift between wave function of electrons on different sublattices. On the contrary, in pseudospinunpolarized state this shift is zero on the average.
In ordinary electronic system without spinmomentum coupling, plasmon manifests itself as chargedensity wave. In helical liquid on surface of topological insulator, due to the spinmomentum locking, spinplasmon manifests itself as coupled charge and spindensity waves. These waves, accompanying spinplasmon with the momentum q, can be characterized by corresponding spatial harmonics of charge (11) and spindensity
operators.
To relate these quantities to the plasmon operators (8), we employ the unitary transformation, connecting operators of electronhole excitations in noninteracting system with operators of plasmons . For this purpose, we also need explicit expressions for operators of distorted singleparticle excitations in the system with Coulomb interaction (similarly to the work [27]). Denoting creation operator for such singleparticle excitation with total momentum q and energy as , we can use the equationofmotion method in the RPA and, similarly to the case of plasmons, get the explicit expression for it:
where
The expressions (8), (16), (24), and (25) establish the unitary transformation at given q from operators of electronhole excitations in noninteracting system to operators of excitations in Coulombinteracting system: plasmons and singleparticle excitations . We can easily derive the inverse transformation of the form:
According to (26), we can represent the operators (11) and (23) of charge and spindensity waves in the form:
where the parts and are the contributions of singleparticle excitations and are dynamically independent on plasmons. Here, along with the usual chargedensity susceptibility (21), the crossed spindensity susceptibility of the helical liquid [11] has been introduced:
The formulas (27)(28) show us that the average values of and in any state with a definite number of plasmons vanish (similarly to the mean value of coordinate or momentum in the simplest quantum harmonic oscillator). However, we can calculate their mean squares in the n_{q}plasmon state :
(only the inplane transverse component s^{⊥ }of the spin s is nonzero upon averaging). Since we are interested in plasmons only, we have subtracted the vacuum fluctuations of these quantities in the ground state 0〉.
Using (27)(28), the mean squares of charge and spindensity wave amplitudes (30)(31) can be easily calculated:
We can normalize the amplitudes to obtain dimensionless quantities and ( is the average electron density), plotted in Figure 5. As seen, the amplitudes of charge and spindensity waves are close quantitatively at moderate momenta and any r_{s}.
Figure 5. Amplitudes of charge and spindensity waves. Amplitudes of charge and spindensity waves. Normalized amplitudes of chargedensity A_{ρ }(a) and spindensity A_{s }(b) waves in manyplasmon state as function of plasmons momentum q at different values of r_{s}.
In [11], it was shown that, due to spinmomentum locking, electron density and transverse component of spin obey an analogue of "continuity equation". It requires that
Our results are in agreement with this equation.
6 Spinplasmon scattering
Nontrivial internal structure of plasmon in Dirac electron gas can reveal itself in a process of its scattering on external potentials or impurities. For the case of spinplasmon, it is also interesting to consider its scattering on magnetic field, acting on electron spins. Since we are interested in spin effects here, we will consider only inplane magnetic field, affecting only spins of electrons and not their orbital motion.
Hamiltonian of interaction of electrons in helical liquid with external electric U(r) and magnetic H(r) fields are, respectively,
where U_{q }and H_{q }are Fourier components of external electric and magnetic fields, μ_{B }= e/2mc is the Bohr magneton.
To calculate the probability of elastic scattering of the spinplasmon with initial momentum q to the state with the final momentum q', we can use the Fermi golden rule, corresponding to the Born approximation. Thus, the differential (with respect to the scattering angle) probabilities of this scattering on electric and magnetic fields can be presented in the following form:
where θ is scattering angle (i.e., the angle between q and q'); q is absolute value of both q and q'. Here we have introduced electric Φ_{e}(q, θ) and magnetic Φ_{m}(q, θ) formfactors of spinplasmon.
Using (35)(36), we can calculate the formfactors explicitly:
Here is the derivative of spinplasmon dispersion law. It is convenient to project the vector Φ_{m }on directions, parallel and perpendicular to the initial plasmon momentum q to get and , respectively.
We consider only the formfactors, revealing the specifics of spinplasmons, instead of the whole differential probabilities (37)(38), also dependent on the form of external field. In Figure 6, angle dependencies of squared modulus Φ_{e}(q, θ)^{2}, and of the formfactors at q = 0.6p_{F }are plotted; these angular distributions are normalized to unity. In their calculation, we have used the singleband approximation, since undamped plasmons consist mainly of interband transitions, according to Figure 2. Results for other values of plasmon momentum are qualitatively the same.
Figure 6. Scattering formfactors of spinplasmon. Scattering formfactors of spinplasmon. Polar graphs of the squared modulus of electric Φ_{e}(q, θ)^{2 }and components , of magnetic formfactor of spinplasmon with the momentum q = 0.6p_{F }at r_{s }= 0.09 (a) and r_{s }= 8.8 (b).
In the case of forward scattering with zero momentum transfer (at θ = 0), the external electric field probes the total charge of plasmon, which is actually zero. Thus, the corresponding electric formfactor Φ_{e}(q, 0) = 0. Similarly, forward scattering on magnetic field probes the total spin of spinplasmon, which is directed perpendicularly to q. Therefore, and . As for backscattering of plasmon, it is strictly prohibited, as for individual massless Dirac electrons [28].
As seen in Figure 6, the formfactors demonstrate two side lobes, rather sharp at small r_{s}, which can be considered as a consequence of sharp peaking of the plasmon wave function (see Figure 3). At large r_{s}, these lobes are much broader.
In the process of spinplasmon scattering on some configurations of electric or magnetic fields, we can expect interplay between spatial structure of these fields and that of the spinplasmon. Angular distribution of the scattered plasmons will incorporate both of these factors, according to (37)(38). Controlling the overlap of maxima of external potential and spinplasmon formfactors, one can manipulate spinplasmon scattering.
Magnetic or nonmagnetic impurities can also create the external field. If the characteristic radius of the impurity potential is R, the transferred momentum will be limited by R^{1 }by the order of magnitude. If qR ≫ 1, the plasmon scattering on impurities will be suppressed due to proximity to the regime of forward scattering. In the opposite limit qR ≪ 1, plasmons will be effectively scattered on considerable angles.
7 Conclusions
The properties of collective excitations (plasmons or spinplasmons) in 2D gas of massless Dirac particles were studied. Two physical realizations of such systems were considered: electron gas in graphene and helical liquid on the surface of topological insulator. Quantum fieldtheoretical formalism for comprehensive description of spinplasmons as composite Bose particles in the RPA was developed. Internal structure and wave function of spinplasmons were studied.
Signatures of spinmomentum locking in helical liquid were considered. In particular, it was shown that excitation of a spinplasmon induces the total nonzero spin polarization of the system. Moreover, coupling between charge and spindensity waves, accompanying a spinplasmon, was demonstrated. It was shown that amplitudes of both of these waves are close by the order of magnitude for spinplasmons of intermediate momenta. The results of this work can be confirmed by experiments involving spinplasmon excitation on the surface of topological insulator and independent measurements of charge and spin wave amplitudes (one of experiments of this type was proposed in [11]). The similar effect of coupling between charge and spin appears in electron gas with spinorbit coupling [29], but amplitude of spin wave is considerable less in this case than that of charge wave for experimentally relevant parameters.
Elastic scattering of spinplasmons in helical liquid on electric and magnetic external fields is considered. Angular distribution of scattered spinplasmons depends on both the shape of the potential and the formfactor of the spinplasmon, revealing its internal structure. It was shown that, due to the formfactor, the scattering occurs into two side lobes, while forward and backward scattering is suppressed. One can use this fact to manipulate spinplasmon scattering via interplay between plasmon formfactor and shape of the external potential. It can be also concluded that scattering of spinplasmons on longrange impurities should be very weak.
Coupling between charge and spin waves, demonstrated in this article, can be used for realization of various spintronic devices. One can perform controllable focusing of spinplasmon waves and thus create regions with high spin polarization. Spinpolarized electrons, accumulating in these regions, can diffuse to adjacent electrodes and be used to drive spin currents (similarly to [12]).
The quantum fieldtheoretical approach, presented in this article, can be used for theoretical description of an influence of various external factors (impurities, external fields) on plasmons in Dirac electron gas. The obtained explicit expression for plasmon wave function allows to derive and solve Hamiltonians of plasmons interacting with such external fields. In particular, the plasmon formfactors (39)(40) can be used to construct matrix elements of plasmon interaction with external electric and magnetic fields. The problems of 2D spinplasmon optics, based on manipulations by inhomogeneities of the system and 3D environment, can be solved. Also the properties of hybrid modes (plasmon polaritons [30], plasmonphonon modes [21,22], plasmonhole modesplasmarons [31]) can be studied using this approach.
The classical electrodynamic approach based on Maxwell equations and response functions cannot describe quantum effects, arising when individual plasmons are emitted and adsorbed. Therefore, the quantum fieldtheoretical approach, presented in this article, should be especially useful for the problems of quantum plasmonics, which seems to be rather feasible in graphenebased structures [23].
Abbreviation
RPA: random phase approximation.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
YEL formulated the problem, provided the consultations on key points of the work and helped to finalize the manuscript. AAS developed the mathematical approach, carried out analytical calculations and contributed to the manuscript preparation. DKE performed a part of analytical calculations, obtained all numerical results and wrote the manuscript draft. All authors read and approved the final manuscript.
Acknowledgements
The work was supported by Russian Foundation for Basic Research (grants 110212209ofim and 100292607KO_a) and by Grant of the President of Russian Federation MK5288.2011.2. Two authors (DKE and AAS) acknowledge support from the Dynasty Foundation.
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