Spin-Plasmons in Topological Insulator

Journal of Magnetism and Magnetic Materials 324 (2012) 3610–3612

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Journal of Magnetism and Magnetic Materials

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Spin-plasmons in topological insulator

D.K. Eﬁmkin a, Yu.E. Lozovik a,b,n, A.A. Sokolik a a Institute for Spectroscopy, Russian Academy of Sciences, Fizicheskaya 5, 142190, Troitsk, Moscow Region, Russia b Moscow Institute of Physics and Technology, Institutskii Per. 9, 141700, Dolgoprudny, Moscow Region, Russia article info abstract

Available online 28 February 2012 Collective plasmon excitations in a helical electron liquid on the surface of strong three-dimensional Keywords: topological insulator are considered. The properties and internal structure of these excitations are Topological insulator studied. Due to spin-momentum locking in helical liquid on a surface of topological insulator, the Plasmons collective excitations should manifest themselves as coupled charge- and spin-density waves. Helical liquid & 2012 Elsevier B.V. All rights reserved. Spintronics

1. Introduction calculate amplitudes of charge- and spin-density waves in the plasmon state. In recent years, topological insulators with a non-trivial topological order, intrinsic to their band structure, were predicted theore- tically and observed experimentally (see [1] and references therein). 2. Wave function of spin-plasmon Three-dimensional (3D) realizations of ‘‘strong’’ topological insulators (such as Bi2Se3,Bi2Te3 and Sb2Te3) are insulating in the bulk, but Low-energy effective Hamiltonian of the surface states of Bi2Se3 have gapless topologically protected surface states with a number of in the representation of spin states {9mS,9kS}isH0¼vF(pxsypysx) unusual properties [2]. These states obey two-dimensional Dirac E 5 for a surface in the xy plane, where the Fermi velocity pvFffiffiffi 6.2 10 equation for massless particles, similar to that for electrons in ipUr9 S m/s [2]. Its eigenfunctions can be written as e pfffiffiffipg = S,whereS is T graphene [3], but related to real spin of electrons, instead of 9 S ifp=2 ifp=2 thesystemareaand f pg ¼ðe ,ige Þ = 2 is the spinor part sublattice pseudospin in graphene. of the eigenfunction, corresponding to electron with momentum p

The consequence of that is a spin-momentum locking for (its azimuthal angle in the xy plane is fp)fromconduction(g¼þ1) electrons on the surface of strong 3D topological insulators, i.e., or valence (g¼1) band. Many-body HamiltonianP of electrons spin of each electron is always directed in the surface plane and þ populatingP the s urface of topological insulator is H ¼ pgxpgapgapg perpendicularly to its momentum [1,4]. The surface of topological þ þð1=2SÞ qV qrq rq,whereapg is the destruction operator for insulator can be chemically doped, forming a charged ‘‘helical electron with momentum p from the band g, xpg¼gvF9p9m is its liquid’’. 2 energyP measured from the chemical potential m, Vq¼2pe /eq; Collective excitation of electrons in such helical liquid were þ / 9 S þ rq ¼ pgg0 f p þ q,g0 f pg ap þ q,g0 apg isthechargedensityoperator considered in [5], where relationships between charge and spin for helical liquid. responses to electromagnetic ﬁeld were derived. It was shown The creation operator for spin-plasmon with wave vector q can that charge-density wave in this system is accompanied by be presented in the form: spin-density wave. Application of spin-plasmons to create ‘‘spin X þ g0g þ accumulator’’ was proposed in [6]. Also the surface plasmon- Q q ¼ Cpq ap þ q,g0 apg ð1Þ pgg0 polaritons under conditions of magnetoelectric effect in 3D þ topological insulator were considered [7]. This operator should obey the equation of motion ½H,Q q ¼ þ In the present article we consider the properties and internal OqQ q , where Oq is the plasmon frequency. We can get solution of structure of spin-plasmons in a helical liquid. Within the random- this equation in the random phase approximation at T¼0 (simi- phase approximation, we derive plasmon wave function and larly to [8]): 9 9/ 9 S g0g npgnp þ q,g0 f p þ q,g0 f pg Nq Cpq ¼ , ð2Þ n Corresponding author at: Institute for Spectroscopy, Russian Academy of Oq þxpgxp þ q,g0 þid Sciences, Fizicheskaya 5, 142190 Troitsk, Moscow Region, Russian Federation. 9 9 Tel.: þ7 496 751 0881; fax: þ7 496 751 0886. where npþ ¼Y(pF p ) and np are occupation numbers for E-mail address: [email protected] (Yu.E. Lozovik). electron-doped helical liquid (pF¼m/vF is the Fermi momentum).

The plasmon frequency is determined in this approach from 4 incorporated into rs) are also presented for comparison. It is seen the equation 1VqP(q,Oq)¼0, where that the undamped spin-plasmon consists mainly of intraband X transitions. When the dispersion curve enters the interband con- 0 9/ 9 S92 npg np þ q,g Pðq,oÞ¼ f p þ q,g0 f pg ð3Þ tinuum, the spin plasmon becomes damped and inter- and intra- 0 oþxpgxp þ q,g0 þid pgg band transitions contribute almost equally to its wave function. is the polarization operator of the helical liquid, different from that for graphene [3] only by degeneracy factor. The factor Nq in (2) can be determined from the normalization condition 3. Charge- and spin-waves X þ þ /09½Q ,Q 0 90S ¼ d 0 D 0 ¼ d 0 , 9 S 9 S q q qq g g qq The helical liquid in the state 1q ¼ Q q 0 with one spin- 0 gg plasmon of wave vector q has a distribution of electron–hole X excitations (2), shifted towards q. Due to the spin-momentum 9 g0g92 Dg0g ¼ Cpq ðnpgnp þ q,g0 Þð4Þ locking, the system acquires a total nonzero spin polarization, p perpendicular to q. A similar situation occurs in the current- 2 carrying state of the helical liquid, which turns out to be spin- (90S is the ground state), so that 9Nq9 ¼S½@Pðq,oÞ= 9 polarized [4]. @o o ¼ O . The quantities Dg0g in (4) can be considered as total q Introducing one-particle spin operator as s¼r/2, we can calcu- weights of intraband (Dþþ) and interband (DþþDþ¼1Dþþ) / S / 9 9 S electron transitions, contributing to the plasmon wave function late its average value in the one-plasmon state s ¼ 1q s 1q as X (1). Note that all these formulas are also applicable to the case of / S / 9 9 S tg s ¼ ½ f p þ q,g0 s f p þ q,t Cpq graphene. pgg0t Spin-plasmon dispersion Oq and contribution of intraband g0t/ 9 9 S g0g n Cpq f pt s f pg ðCpq Þ ðnpgnp þ q,g0 Þð5Þ transitions into its wave function are plotted in Fig. 1 at various 2 rs¼e /evF, where e is the dielectric susceptibility of surrounding If q is parallel to ex, only the y-component of /sS is nonzero. 3D medium. For Bi2Se3, rsE0.09 with eE40 for dielectric half- Its dependence on q at various rs is plotted in Fig. 2(a). space [5] (for such small rs, the corresponding dispersion curve Charge- and spin-density waves, accompanying spin-plasmon approaches very closely to the upper bound o¼vFq of the intraband with the wave vector q, can be characterized by corresponding þ continuum). The results for suspended graphene with rather large spatialP harmonics of charge- and spin-density operators: rq and E 6 þ / 9 9 S þ rs¼8.8 (for vF 10 m/s, e¼1 and with the degeneracy factor sq ¼ pgg0 f p þ q,g0 s f pg ap þ q,g0 apg. Using, similarly to [9], the

2 1

1.5 0.8 μ / ++ q 1 D Ω

rs 0.6 0.5 0.09 2.2 8.8 0 0.4 00.51 200.51 2

q/pF q/pF

Fig. 1. Dispersions of spin-plasmon (a) and contributions Dþþ of intraband transitions into its wave function (b) at various rs. Continuums of intraband (oovFq) and interband (oþvFq42m) single-particle excitations are shaded in (a).

0.4 0.4

rs 0.09 0.3 A 0.2 2.2 ρ 8.8 〈〉〈s〉 0.2 0

0.1 As 0.2

0 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 q p q/pF / F

Fig. 2. Total spin polarization /sS of the helical liquid in the one-plasmon state (a) at various rs and normalized amplitudes Ar and As of charge- and spin-density waves respectively in the many-plasmon state (b). 3612 D.K. Efimkin et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 3610–3612 unitary transformation, inverse with respect to (1), we can write: quantum-mechanical formalism can be applied for a number of þ n þ þ problems in spin-plasmon optics. r ¼ SN Pðq,OqÞQ þr~ , ð6Þ q q q q We calculated the average spin polarization, acquired by the n þ helical liquid in a spin-plasmon state, as well as mean-square s þ ¼ SN P ðq,O ÞQ þ þs~ , ð7Þ q q s q q q amplitudes of charge- and spin-density waves, arising in this state. þ þ where the operators r~ q and s~ q are the contributions of single- Coupling between these amplitudes, caused by spin-momentum particle excitations and are dynamically independent on plas- locking, was demonstrated. The interconnection between charge- mons. Here the crossed spin-density susceptibility of the helical and spin-density waves can be applied for constructing various spin- liquid [5] has been introduced: plasmonic and spintronic devices. X 0 / 9 S/ 9 9 S npg np þ q,g Psðq,oÞ¼ f p þ q,g0 f pg f pg s f p þ q,g0 þ 0 þ pgg0 o xpg xp þ q,g id ð8Þ þ þ Acknowledgment The averagep valuesffiffiffiffiffiffiffi of rq and sq in the nq-plasmon state þ nq 9nqS ¼½ðQ Þ = nq!90S vanish, therefore we consider their q The work was supported by Russian Foundation for Basic mean squares in 9n S after subtracting their background values q Research, by Grant of the President of Russian Federation in 90S, i.e., MK-5288.2011.2 and by the Federal target-oriented program / þ S / 9 þ 9 S / 9 þ 9 S rqrq nq rqrq nq 0 rqrq 0 ‘‘Research and scientific-pedagogical staff of innovational Russia’’ 29 n 92 for 2009–2013. One of the authors (A.A.S.) acknowledges support ¼ nqS NqPðq,OqÞ , ð9Þ from the Dynasty Foundation. / ? ? þ S / 9 ? ? þ 9 S / 9 ? ? þ 9 S sq ðsq Þ nq sq ðsq Þ nq 0 sq ðsq Þ 0 29 n ? 92 ¼ nqS NqPs ðq,OqÞ ð10Þ (only the in-plane transverse component s? of the spin s is References nonzero in these averages). The normalized amplitudes ArðqÞ¼ þ 1=2 ? ? þ 1=2 [1] M.Z. Hasan, C.L. Kane, Reviews of Modern Physics 82 (2010) 3045–3067. ½/r r S=nqSr and AsðqÞ¼½/s ðs Þ S=nqSr of charge- q q q q [2] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, S.-C. Zhang, Nature Physics 5 (2009) ¼ 2 and spin-density waves are plotted in Fig. 2(b) (r pF =4p is 438–442. the average electron density). The ‘‘continuity equation’’ for [3] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Reviews of density and transverse spin, following from the spin-momentum Modern Physics 81 (2009) 109–162. [4] D. Culcer, E.H. Hwang, T.D. Stanescu, S. Das Sarma, Physical Review B 82 locking [5], requires that OqAr(q)¼2vFqAs(q), in agreement with (155457) (2010) 1–20. our results. [5] S. Raghu, S.B. Chung, X.L. Qim, S.-C. Zhang, Biophysical Reviews and Letters 104 (116401) (2010) 1–4. [6] I. Appelbaum, H.D. Drew, M.S. Fuhrer, Applied Physics Letters 98 (023103) 4. Conclusions (2011) 1–3. [7] A. Karch, Physical Review B 83 (245432) (2011) 1–5. [8] K. Sawada, K.A. Brueckner, N. Fukuda, R. Brout, Physical Review 108 (1957) We have considered microscopically spin-plasmons in helical 507–514. liquid in the random phase approximation. The developed [9] R. Brout, Physical Review 108 (1957) 515–517.