Pis'ma v ZhETF, vol. 97, iss. 2, pp. 88-92 © 2013 January 25
Photogalvanic effects in topological insulators
S. N. Artemenko1, V. O. Kaladzhyan Kotel'nikov Institute of Radio-engineering and Electronics of RAS, 125009 Moscow, Russia
Moscow Institute of Physics and Technology, 141700 Dolgoprudny, Russia Submitted 30 November 2012
We discuss optical absorption in topological insulators and study possible photoelectric effects theoretically. We found that absorption of circularly polarized electromagnetic waves in two-dimensional topological insulators results in electric current in the conducting 1D edge channels, the direction of the current being determined by the light polarization. We suggest two ways of inducing such a current: due to magnetic dipole electron transitions stimulated by irradiation of frequency below the bulk energy gap, and due to electric dipole transitions in the bulk at frequencies larger than the energy gap with subsequent capture of the photogenerated carriers on conducting edge states.
DOI: 10.7868/S0370274X13020045
The rich new physics and unusual properties of topological insulators (TI) attract much attention in recent years, for review see [1-3]. The spectrum of TI in the bulk has a finite energy gap EG, while there are topo-logically protected conductive surface states inside the gap. The states with somewhat similar properties were predicted also long ago at the surface of a conventional narrow-gap semiconductor [4] and at the interface of heterojunctions made of two insulators with inverted energy bands [5] without appealing to topological reasons. The chiral states studied in recent works on TI [1-3] and the chiral states considered in old papers [4, 5] were derived using different theoretical models and the states have different spin structure. Derivation of the surface state of TI was essentially based on quadratic dispersion relation of the mass term in a Dirac-type Hamil-tonian which is valid for materials like Bi2Se3 [2, 3] or HgTe/CdTe semiconductor quantum wells [1, 6], and the wave functions of the surface states are formed from the states of bulk bands with the same spin directions. Therefore, the electron surface states in TI can be characterized by spin oriented perpendicular to the momentum. While in Refs. [4, 5] the mass dispersion is not needed to derive the surface states, and the surface states are formed from the bulk band states of the opposite spin directions.
Experimental study of transport properties related to the surface states is typically complicated by residual electrons and holes in the bulk. In this respect it would be interesting to study the possibility to induce the current in the surface state by absorption of circularly polarized light. As the left/right circular polarization of
e-mail: art@cplire.ru
the light is associated with the angular momentum projection ±1, the absorption of polarized light can vary the number of electrons with definite spin. Since the directions of the spin and momentum of electrons in the chiral states are related, variation of the spin distribution may affect the symmetry of the momentum distribution, and, hence, excite the current. As these effects are related only to the chiral surface state, the bulk contribution to the current should be absent.
Below we set e, ft, and kB to unity, restoring dimensional units in final expressions when necessary.
Semiconductors with few energy bands near the Fermi energy are usually described by an approach developed in seminal paper by Kohn and Luttinger [7]. In the simplest case when only two bands are taken into account the Hamiltonian has a form of the effective Dirac equation for the envelope functions. It has a form of the effective Dirac equation with the effective velocity of light c* proportional to the matrix element of the momentum for the interband transitions, and the mass term proportional to the energy gap EG = 2M,
H = c* (rx x ak ,pk) + tz M, (1)
where Pauli matrices ak operate in the spin state, and Tk refer to the conduction and valence band states. Based on such an approach, the chiral interface states with Weil spectrum were suggested long ago [4] at the surface of semiconductors, and at the interface of het-erostructures consisting of two semiconductors with inverted energy bands [5]. In the former paper the bulk Hamiltonian was supplemented with phenomenological boundary conditions at the surface of the semiconductor derived from condition that the Hamiltonian is Hermi-tian in restricted area and obeys time reversal symme-
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try. This phenomenological boundary conditions were shown [8] to be consistent with the results for het-erostructures and yield the surface chiral states that are similar to the interface states in heterojunctions.
Consider first the chiral surface states described by the simplest model [4] that describes the chiral surface states depicted in Ref. [4] as the Tamm states. The boundary condition links the spinors related to the conduction and valence bands = ia0(a, n)$c, where n is the normal to the surface of the semiconductor, or to the boundary of the heterocontact. In case of a free surface of a semiconductor a0 is a phenomenological constant the value of which is determined by the properties of the surface. In inverted heterocontacts the value of a0 was shown [8] to depend on work functions of semiconductors forming the heterocontact, and a0 = 1 for the symmetric case.
Eigenstates and energy spectrum ea(p) = e0 — avp, v = c* ■ 2a0/(1 + a2), describing the two-dimensional chiral surface states at plane z = 0 are characterized [4] by momentum p = (px,py) = p(cos p, sin p) and by the cone number a = ±1, and have a form
^(t) «
I |c) + ia0 ( ) |v)
—iaelv I \ iaelLp /
ipr — \z
(2)
Here |c) and |v) are periodic Bloch functions of the extrema states in the conduction and the valence bands, and (x,y) are the coordinates along the surface state. Decay of the wave functions in the direction perpendicular to the surface is described by single exponential dependence with the exponent A ~ v/Eg.
Three-dimensional Bi2Se3-type TI is described by the effective Hamiltonian [9-11] similar to (1) but with momentum depending mass term, M ^ M — Bp2. In contrast to the former case, the boundary of the TI with vacuum was usually described by model open boundary conditions. The wave-functions of the chiral states with the spectrum ea(p) = —avp are
^(t)<x
(|P 1+) + IP2—))eipr (e
\\z _ e — \2z\
(3)
Here, IP 1+) and IP2Z) refer to orbitals related of the conductivity and valence bands [9, 10]. Decay of the surface state in z-direction is described by two exponents
A
1,2
1 (v±Vv2 - 4MB).
2b
In both models the exactly backward scattering by non-magnetic defects in the chiral surface states is prohibited, the probability of elastic scattering with variation of momentum directions from the angle to being proportional to cos2 12V2. However, the elastic
scattering time is not suppressed much because of the sizeable scattering for a finite angle.
An important difference between chiral Tamm (2) and TI states (3) is that in the former case the surface states are formed from the bulk band states of the opposite spin directions, while in the latter case spin directions are the same. The finite expectation value of the electron spin in the Tamm states (2) is non-zero only due to asymmetry of the contributions from the conduction and the valence bands at a0 = 1, s = a(a2 — 1)/[2(a2 + + 1)] (— sin p, cos p, 0). While in the TI chiral states case (3) the spin expectation value is s = f(— sin cos 0).
The difference in the structure of the chiral states results in a difference of their optical properties. Calculation of matrix elements of vertical optical electric dipole transitions from the lower (a = 1) to the upper (a = —1) cone of the spectrum for the Tamm and TI chiral states yields
2ia0
d
Tamm —
2'(c|t%), cLti = 0, 1 + ai
where (c|d|w) is the dipole matrix element for the bulk interband transitions. Thus the dipole optical transitions between the lower and the upper cones are allowed in the Tamm states and are forbidden in the TI states.
As the electric dipole transitions in the TI states are forbidden it is worth to discuss the magnetic dipole transitions though they are much weaker. Matrix elements of the magnetic dipole transitions between the states (3) depend on direction of the momentum,
mTi « ^b (iHx cos ^>,iHy sin y,HZ),
(4)
where H is the magnetic field of the light. If Hz = 0 then for an electron moving at an angle $ to the magnetic field, according to (4), |mTi |2 « sin2 Therefore, for linearly polarized light propagating in the direction perpendicular to the surface of TI the optical absorption in the edge states is anisotropic. This must result in anisotropic photoconductivity, but the photoconductivity is not expected to be large as optical absorption is caused by magnetic dipole transitions.
Consider now the right (left) circularly polarized electromagnetic wave propagating along the y-axis parallel to the surface and characterized by its polarization. In this case the probability of the transitions from the state in the lower cone a =1 to the upper one, a = — 1, is proportional to (/i,bHo)2 cos4 ^ for the left polarized and to (pbHo)2 sin4 j for the right polarized light. So there is a preferential direction of the momentum of electrons excited by circularly polarized radiation. The left polarized wave preferentially creates electrons moving in positive x-direction and holes moving in the opposite direction. Hence, optical absorption must induce an elec-
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trie current in the direction perpendicular to the wave vector of the incident electromagnetic wave. However, we expect that this effect is difficult to observe since the circulating current is greatly suppressed by elastic scattering. There are more chances of experimental observation of
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