Collective excitations on a surface of topological insulator
 Dmitry K Efimkin^{1},
 Yurii E Lozovik^{1, 2}Email author and
 Alexey A Sokolik^{1}
DOI: 10.1186/1556276X7163
© Efimkin et al; licensee Springer. 2012
Received: 2 November 2011
Accepted: 29 February 2012
Published: 29 February 2012
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 [5–7]. 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 [14–16]. 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
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.
where $\widehat{\mathbf{p}}$ and $\widehat{\mathbf{z}}$ 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.
where a_{ p γ }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
Here the coefficients ${C}_{\mathbf{pq}}^{{\gamma}^{\prime}\gamma}$ are the weights of intraband (γ = γ') and interband (γ ≠ γ') singleparticle transitions, contributing to the wave function of plasmon.
where n_{ p γ }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.
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_{ p γ }〉 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
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.
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].
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.
operators.
(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〉.
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.
where U_{ q }and H_{ q }are Fourier components of external electric and magnetic fields, μ_{ B }= e/2mc is the Bohr magneton.
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.
Here ${\Omega}_{q}^{\prime}=d{\Omega}_{q}/\mathrm{dq}$ 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 ${\Phi}_{\text{m}}^{\parallel}$ and ${\Phi}_{\text{m}}^{\perp}$, respectively.
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, ${\Phi}_{\text{m}}^{\parallel}\left(q,0\right)=0$ and ${\Phi}_{\text{m}}^{\perp}\left(q,0\right)\ne 0$. 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.
Declarations
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.
Authors’ Affiliations
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