Collective excitations on a surface of topological insulator
© Efimkin et al; licensee Springer. 2012
Received: 2 November 2011
Accepted: 29 February 2012
Published: 29 February 2012
We study collective excitations in a helical electron liquid on a surface of three-dimensional topological insulator. Electron in helical liquid obeys Dirac-like equation for massless particles and direction of its spin is strictly determined by its momentum. Due to this spin-momentum locking, collective excitations in the system manifest themselves as coupled charge- and spin-density waves. We develop quantum field-theoretical description of spin-plasmons in helical liquid and study their properties and internal structure. Value of spin polarization arising in the system with excited spin-plasmons is calculated. We also consider the scattering of spin-plasmons 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.
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). Three-dimensional topological insulators are insulating in the bulk, but have gapless surface states with numerous unusual properties. These states are topologically protected against time-reversal invariant disorder. When gap is opened in surface states by time-reversal or gauge symmetry breaking, a spectacular magnetoelectric effect arises [3, 4].
Recently a "new generation" of 3D topological insulators (the binary compounds Bi2Se3, Bi2Te3 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 spin-momentum locking leads to interesting transport phenomena including coupled diffusion of spin and charge , inverse galvano-magnetic effect (generation of spin polarization by electric current)  and giant spin rotation on an interface between normal metal and topological insulator . The spin-momentum locking offers numerous opportunities for various spintronic applications.
Collective excitations (plasmons) in helical liquid on the surface of topological insulator was considered in . It was shown that due to spin-momentum 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 spin-density waves and can be called "spin-plasmons". In  application of spin-plasmons in spin accumulator device was proposed. Also surface plasmon-polaritons under conditions of topological magnetoelectric effect were considered in .
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  and references therein). In particular, the properties of plasmons [18, 19] and hybrid plasmon-photon  and plasmon-phonon [21, 22] modes were investigated theoretically and experimentally. It was realized recently that graphene is a fertile ground for quantum plasmonics  due to very small plasmon damping.
In this article, we develop quantum field-theoretical formalism to describe plasmons in graphene and spin-plasmons 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 spin-plasmons 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 field-theoretical 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 spin-momentum 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 and are unit vectors directed along the momentum p and the z-axis, 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 = γvFp-μ is its kinetic energy measured from the chemical potential μ and V q = 2πe2/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 are the weights of intraband (γ = γ') and interband (γ ≠ γ') single-particle 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 electron-doped Dirac liquid at T = 0 (see also the remark at the end of this section), we have np+= Θ(pF - |p|), np-= 1, where pF = μ/vF is the Fermi momentum. In the case of hole doping, all characteristics of plasmons are the same due to electron-hole symmetry.
The degeneracy factor g is 1 for helical liquid on the surface of topological insulator and 4 for graphene. The angular factors 〈fp+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 rs, defined as a ratio of characteristic Coulomb interaction energy to kinetic energy. For the gas of massless particles, rs does not depend on electron density and equals to rs = e2/εvF (effectively ε = (ε1 + ε2)/2 when 2D electron layer is surrounded by two dielectric half-spaces with permittivities ε1 and ε2). For Bi2Se3, rs = 0.09 with ϵ = 40 for dielectric half-space, and applicability of the RPA is well established (the value of rs for another material Bi2Te3 is close to that for Bi2Se3). In the case of graphene, rs 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  and references therein, and also the work ).
Maximal achievable amounts of doping of helical liquid on a surface of Bi2Se3 are μ ~ 0.3 eV , 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 rs (the case of topological insulator) the plasmon dispersion law approaches very closely to the upper border of the continuum of intraband single-particle transitions (ω < vFq). On the contrary, at moderate and large rs, plasmon has well-defined square-root dispersion in the long-wavelength range. At q ≈ pF, the plasmon enters the continuum of interband single-particle transitions (ω > 2μ - vFq). Inside the single-particle continuum, energy and momentum conservation laws allow energy transfer between plasmon and single-particle excitations, so the plasmon acquires a finite lifetime.
At large values of parameter rs (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 〈fp+q,±|fp∓〉 suppresses interband forward scattering.
5 Charge- and spin-density waves
When a spin-plasmon is excited in the helical liquid, anisotropic distribution of electron-hole pairs of the type, depicted in Figure 3, arises. This distribution is shifted towards the plasmon momentum q. Due to the spin-momentum locking, the system should acquire a total nonzero spin polarization, perpendicular to q. A similar situation occurs in the current-carrying state of the helical liquid, which turns out to be spin-polarized .
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 pseudospin-unpolarized state this shift is zero on the average.
(only the in-plane 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 Spin-plasmon 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 spin-plasmon, 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 in-plane 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 Φe(q, θ) and magnetic Φm(q, θ) form-factors of spin-plasmon.
Here is the derivative of spin-plasmon dispersion law. It is convenient to project the vector Φm on directions, parallel and perpendicular to the initial plasmon momentum q to get and , 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 form-factor Φe(q, 0) = 0. Similarly, forward scattering on magnetic field probes the total spin of spin-plasmon, which is directed perpendicularly to q. Therefore, and . As for backscattering of plasmon, it is strictly prohibited, as for individual massless Dirac electrons .
As seen in Figure 6, the form-factors demonstrate two side lobes, rather sharp at small rs, which can be considered as a consequence of sharp peaking of the plasmon wave function (see Figure 3). At large rs, these lobes are much broader.
In the process of spin-plasmon scattering on some configurations of electric or magnetic fields, we can expect interplay between spatial structure of these fields and that of the spin-plasmon. 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 spin-plasmon form-factors, one can manipulate spin-plasmon 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.
The properties of collective excitations (plasmons or spin-plasmons) 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 field-theoretical formalism for comprehensive description of spin-plasmons as composite Bose particles in the RPA was developed. Internal structure and wave function of spin-plasmons were studied.
Signatures of spin-momentum locking in helical liquid were considered. In particular, it was shown that excitation of a spin-plasmon induces the total nonzero spin polarization of the system. Moreover, coupling between charge- and spin-density waves, accompanying a spin-plasmon, was demonstrated. It was shown that amplitudes of both of these waves are close by the order of magnitude for spin-plasmons of intermediate momenta. The results of this work can be confirmed by experiments involving spin-plasmon 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 ). The similar effect of coupling between charge and spin appears in electron gas with spin-orbit coupling , but amplitude of spin wave is considerable less in this case than that of charge wave for experimentally relevant parameters.
Elastic scattering of spin-plasmons in helical liquid on electric and magnetic external fields is considered. Angular distribution of scattered spin-plasmons depends on both the shape of the potential and the form-factor of the spin-plasmon, revealing its internal structure. It was shown that, due to the form-factor, the scattering occurs into two side lobes, while forward and backward scattering is suppressed. One can use this fact to manipulate spin-plasmon scattering via interplay between plasmon form-factor and shape of the external potential. It can be also concluded that scattering of spin-plasmons on long-range 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 spin-plasmon waves and thus create regions with high spin polarization. Spin-polarized electrons, accumulating in these regions, can diffuse to adjacent electrodes and be used to drive spin currents (similarly to ).
The quantum field-theoretical 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 form-factors (39)-(40) can be used to construct matrix elements of plasmon interaction with external electric and magnetic fields. The problems of 2D spin-plasmon optics, based on manipulations by inhomogeneities of the system and 3D environment, can be solved. Also the properties of hybrid modes (plasmon polaritons , plasmon-phonon modes [21, 22], plasmon-hole modes--plasmarons ) 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 field-theoretical approach, presented in this article, should be especially useful for the problems of quantum plasmonics, which seems to be rather feasible in graphene-based structures .
random phase approximation.
The work was supported by Russian Foundation for Basic Research (grants 11-02-12209-ofi-m and 10-02-92607-KO_a) and by Grant of the President of Russian Federation MK-5288.2011.2. Two authors (DKE and AAS) acknowledge support from the Dynasty Foundation.
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