Optically tunable spin texture of the surface state for Bi2Se3 and SmB6 topological insulators

Y. F. Peng; C. M. Dai; H. Z. Shen; X. X. Yi

doi:10.1364/OE.26.018906

1. Introduction

Three-dimensional topological insulators (TIs) have attracted much attention in recent years due to spin-helical metallic edge or surface states protected by time reversal symmetry [1–3]. In TIs, electron spin of the surface states is locked into wave vector and winds by 2π around the fermi surface, which resulted from strong spin-orbit interaction and inversion symmetry breaking. The latter effect, known as the Rashba effect, occurs on the surfaces of many materials and produces spin degeneracy-lifted surface states and makes chiral spin textures. For the Bi₂Se₃ family, |P_z〉 is coupled to the left-handed helical spin textures for the upper Dirac cone [4–6]. Furthermore, the radial orbital textures |P_r〉 are always coupled to left-handed spin textures and a tangential orbital textures |P_t〉 is coupled to a right-handed spin textures. The situation is exactly opposite for the lower Dirac cone [7–12]. These characters of spin textures strongly restrain the backscattering [13–18]. Besides, such spin textures on Dirac cone result in anti-localization properties and play a central role for novel quantum phenomena [19, 20]. All these novel properties make the topological insulator a potential candidate as spintronic devices and quantum logic gates in quantum information processing.

Many experimental results of angle resolved photoemission spectroscopy (ARPES) show that the spin directions of photo-electrons behave differently in electromagnetic fields due to the polarization of the incident light [21–26]. Physically, spin is determined by classical field-matter interactions, and the related theoretical researches can be found in paper [27, 28], which incorporates $\vec{k} \cdot \vec{p}$ model and symmetry theory to investigate the spin textures. These results, however, are limited to classical external field. Note however, when the coupling with the external field is sufficiently strong and the field itself is weak, then it becomes necessary to treat the external field quantum mechanically as a collection of photons. On one hand, the total Hamiltonian including the photons has a larger dimension; On the other hand, it becomes time independent and our intuition about static systems can be useful again. As a result, several works such as the constructions of nontrivial topological, spin-1/2 geometric phase and stationary two-level atomic inversion induced by a quantum field have been done [29–33].

Based on current research progress, we study the spin polarization of topological surface state (TSS) driven by quantum field. Surface states coupled to quantized light polarized in three modes are analytically derived. New spin polarization characteristics are observed. The similarities and differences of the spin polarization for TSS between ours and [27] are discussed. To be specific, in this work, we try to answer the following questions: First, when the field is quantized, how does the spin behave. Second, how does the spin depend on the polarization of the quantized driving field?

The remainder of the paper is organized as follows: In Sec. 2, we introduce a model to describe the couplings of TIs to the quantized polarized light. In Sec. 3 and Sec. 4 we derive the spin polarization on the Bi₂Se₃ and SmB₆ surface and do some discussions on the characteristics of spin polarization. Conclusions are given in Sec. 5.

2. Model to describe the system coupled to quantum field

Generally speaking, the surface state of TIs irradiated by incident polarized light can be described by

{\hat{H}}_{o} = \vec{σ} \times \vec{d} {(\vec{k} - \frac{e}{c ℏ} \vec{A})}_{z} + ℏ ω {\hat{a}}^{†} \hat{a},

which is obtained via the Peierls substitution

\vec{P} \to \vec{P} - e \vec{A} / c

, where

\vec{A}

is the vector potential for the incident light.

\vec{σ}

are Pauli operators,

\vec{k}

are the wave vectors on the surface of TIs, ω is the frequency of the field and

{\hat{a}}^{†} (\hat{a})

is the bosonic creation (annihilation) operator. The relationships between the quantum-polarized lights and the material are schematically shown in Fig. 1.

Fig. 1 (a) Diagram of the experimental geometry. Linear polarization (σ-polarized and π-polarized) and circular polarization (righ tand left circular polarized) of photons can be continuously rotated by the θ and α angles. (b) The two-dimensional Brillouin zone for (111) surface with hihg-symmetry points Γ = (0, 0). (c) Corresponding two-dimensional Brillouin zone ofor the (001) surface with its hihg-symmetry points $\bar{Γ} = (0, 0)$ , $\bar{X} = (- π, 0)$ , $\bar{Y} = (0, - π)$ . Circles and ellipses around these hihg-symmetry points represent constant energy contour.

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3. Bi₂Se₃ surface state coupled to three kinds of single-mode quantum field

The spin-orbital interaction in Bi₂Se₃ preventing the bulk band from crossing and an inversion between $| p 1_{z}^{+} 〉$ and $| p 2_{z}^{-} 〉$ at Γ point drives the system into topological phase [34]. Previous work demonstrates that Bi₂Se₃ belongs to three-dimensional TIs with large bulk gap of 0.35 eV and one single Dirac cone on the (111) surface. Taking $\vec{d} (k)$ to be a simple linear function (close to the Dirac point), the structure of the electron state on the surface of Bi₂Se₃ irradiated by quantized field can be described by the Hamiltonian [35]

{\hat{ℋ}}_{d} = v_{0} {[\vec{σ} \times (ℏ \vec{k} - \frac{e}{c} \vec{A})]}_{z} + ℏ ω {\hat{a}}^{†} \hat{a},

where v = 5 × 10⁵ m/s is the electron velocity at the Dirac point. Formally, the Hamiltonian (2) can be rewritten as

{\hat{ℋ}}_{d} = {\hat{ℋ}}_{e} + {\hat{ℋ}}_{F} + {\hat{ℋ}}_{i}

, where

{\hat{ℋ}}_{e} = ℏ v_{0} (σ_{x} k_{y} - σ_{y} k_{x})

is the Hamiltonian of the bare electron in TIs,

{\hat{ℋ}}_{F} = ℏ ω {\hat{a}}^{†} \hat{a}

is the Hamiltonian of the field and

{\hat{ℋ}}_{i} = - e v_{0} / c (σ_{x} A_{y} - σ_{y} A_{x})

is the interaction Hamiltonian between the electron in and the field. With β is the angle between

\vec{k}

and the

+ {\vec{k}}_{x}

direction, an unitary transform

U_{R} = (\begin{matrix} e^{i β / 2} & i e^{- i β / 2} \\ e^{i β / 2} & - i e^{- i β / 2} \end{matrix})

brings the Hamiltonian (2) to the space spanned by the eigenstates |e〉 and |g〉 of

{\hat{ℋ}}_{e}

{\hat{ℋ}}_{N} = (ℏ v_{0} k - \frac{e v_{0}}{c} A_{y} \sin β - \frac{e v_{0}}{c} A_{x} \cos β) τ_{z} + ℏ ω {\hat{a}}^{†} \hat{a} + \frac{e v_{0}}{c} (A_{y} \cos β - A_{x} \sin β) τ_{y},

where τ_z = |e〉〈e| − |g〉〈g|, τ_x = |e〉〈g| + |g〉〈e|, τ_y = i |g〉〈e| − i |e〉〈g|. Considering the field is weak with respect to the free energy of the system, ħv₀k, we can get

\hat{H} = ℏ v_{0} k τ_{z} + ℏ ω {\hat{a}}^{†} \hat{a} + \frac{e v_{0}}{c} (A_{y} \cos β - A_{x} \sin β) τ_{y} .

The expression for

\vec{A}

have been discussed in [27] and [36]. Combing the results of the two papers, we obtain easily the expressions of field operator

\vec{A}

for σ-polarized light, π-polarized light and circular polarized light. The spin textures of Eq. (2) can be calculated for quantized light with different polarization modes.

3.1. Quantized σ-polarized light

Quantized σ-polarized light can be expressed as,

\vec{A} = \sqrt{\frac{2 π ℏ c^{2}}{ω V}} ({\hat{a}}^{†} + \hat{a}) (- s i n α, c o s α, 0),

where V = (c2π/ω)³ is the volume of the space with field valid for the whole paper. Substituting Eq. (5) into Eq. (4), we obtain

{\hat{H}}_{σ} = ℏ v_{0} k τ_{z} + ℏ ω {\hat{a}}^{†} \hat{a} + i G \cos (β - α) (τ_{-} - τ_{+}) ({\hat{a}}^{†} + \hat{a}),

where

G = e v_{0} \sqrt{2 π ℏ / ω V}

is valid through the article. In this paper, the problem is investigated by adopting the method in [37] to deal with the anti-rotating wave approximation. Here, if we set

{\hat{H}}_{0} = ℏ v_{0} k τ_{z} + ℏ ω {\hat{a}}^{†} \hat{a}

,

{\hat{H}}_{I} = i G \cos (β - α) (τ_{-} - τ_{+}) ({\hat{a}}^{†} + \hat{a})

.

\hat{s} = x τ_{-} \hat{a} + y τ_{+} {\hat{a}}^{†}

is anti-Hermitian operator, where x and y remain to be determined. Up to the second order in G, the effective Hamiltonian

{\hat{H}}_{e f f} = \exp (\hat{s}) {\hat{H}}_{σ} \exp (- \hat{s})

is given by

{\hat{H}}_{e f f} = {\hat{H}}_{0} + {\hat{H}}_{I} + [\hat{s}, {\hat{H}}_{0}] + [\hat{s}, {\hat{H}}_{I}] + \frac{1}{2} [\hat{s}, [\hat{s}, {\hat{H}}_{0}]] \dots .

Now we require

τ_{-} \hat{a} + H . c .

to be eliminated from the first order term, namely

{\hat{H}}_{I} + [\hat{s}, {\hat{H}}_{0}] = i G \cos (β - α) (τ_{-} {\hat{a}}^{†} - τ_{+} \hat{a})

, we restrict

[\hat{s}, {\hat{H}}_{0}] = [\hat{s}, ℏ v_{0} k τ_{z}] + [\hat{s}, ℏ ω {\hat{a}}^{†} \hat{a}] = - y (2 ℏ v_{0} k + ℏ ω) τ_{+} {\hat{a}}^{†} + x (2 ℏ v_{0} k + ℏ ω) τ_{-} \hat{a} .

The preceding equation gives the coefficients

x = y = - \frac{i G}{(2 ℏ v_{0} k + ℏ ω)} \cos (β - α) .

Using the result in Eq. (9), we have

\begin{array}{l} [\hat{s}, [\hat{s}, {\hat{H}}_{0}]] = \frac{- 2}{2 ℏ v_{0} k + ℏ ω} {[G \cos (β - α)]}^{2} τ_{z}, \\ [\hat{s}, {\hat{H}}_{I}] = [\hat{s}, i G \cos (β - α) (τ_{-} \hat{a} - τ_{+} \hat{a} + τ_{-} {\hat{a}}^{†} - τ_{+} {\hat{a}}^{†})] = \frac{2 n}{2 ℏ v_{0} k + ℏ ω} {[G \cos (β - α)]}^{2} τ_{z}, \end{array}

where we have omitted the high-frequency intercrossing terms such as

{\hat{a}}^{†} {\hat{a}}^{†}

and

\hat{a} \hat{a}

. The modified energy for the system is

Ω_{σ} = ℏ v_{0} k + \frac{n}{2 ℏ v_{0} k + ℏ ω} {[G \cos (β - α)]}^{2},

where n denotes the photon occupation number. The Hamiltonian then follows

{\hat{H}}_{e f f} = Ω_{σ} τ_{z} + ℏ ω {\hat{a}}^{†} \hat{a} + i G \cos (β - α) (τ_{-} {\hat{a}}^{†} - τ_{+} \hat{a}) .

Eq. (12) can be solved exactly. To solve the problem, let us introduce notations |e, n〉 and |g, n + 1〉 to describe the states of the system. Namely, the first denotes a state of the electron in the upper band with the electromagnetic field in fock state with photon number n, while the second for electron in lower band with n + 1 photons. With this notation, the eigenstates and eigenvalues of the Eq. (12) can be described by |ϕ〉 = a|e, n〉 + b|g, n + 1〉 and

E = ℏ ω (n + 1 / 2) \pm \sqrt{{(Ω_{σ} - ℏ ω / 2)}^{2} + (n + 1) \cos {(β - α)}^{2} G^{2}}

respectively. The probabilities of the two-level system on the corresponding bands are given by

\begin{array}{l} | a |^{2} = \frac{(n + 1) \cos {(β - α)}^{2} G^{2}}{(n + 1) \cos {(β - α)}^{2} G^{2} + {(Ω_{σ} + n ℏ ω - E)}^{2}}, \\ | b |^{2} = \frac{{(Ω_{σ} + n ℏ ω - E)}^{2}}{(n + 1) \cos {(β - α)}^{2} G^{2} + {(Ω_{σ} + n ℏ ω - E)}^{2}} . \end{array}

Considering the spin is always perpendicular to the momentum, we can define a helicity operator

\hat{h} = (1 / k) \hat{z} \cdot \vec{k} \times \vec{σ}

, which commutates with the Hamiltonian, to characterize the handness of the spin textures. For the upper Dirac cone of surface states, the helical operator

\hat{h} = - 1

leading to a left-handed spin textures in the momentum space while for the lower Dirac cone,

\hat{h} = 1

yields a right-handed spin textures [11]. Considering

exp (\hat{s})

in Eq. (7) up to the second order terms, we have

\begin{array}{l} {| ψ 〉}_{σ} = e^{\hat{s}} | ϕ 〉 \approx (1 - \hat{s} + \frac{1}{2} {\hat{s}}^{2}) | ϕ 〉 \\ = a | e, n 〉 + b | g, n + 1 〉 + i m \cos (β - α) [\sqrt{n + 2} b | e, n + 2 〉 + \sqrt{n} a | g, n - 1 〉] \\ - \frac{1}{2} m^{2} \cos^{2} (β - α) [{\sqrt{n}}^{2} a | e, n 〉 + {\sqrt{n + 2}}^{2} b | g, n + 1 〉], \end{array}

where m = G/2ħv₀k + ħω is valid for the whole paper. Up to now, we show that the effective Hamiltonian (12) and its eigenstate |φ〉 are obtained by applying an unitary transformation on Hamiltonian (6). Then we obtain the approximate eigenstate of Hamiltonian (6) through an inverse unitary transformation as Eq. (14). The eigenstates of TSS coupled to quantum field in the rest of the paper are obtained by this method, too. The pseudospin vector of the two-band system in the quantized σ-polarized light can be calculated as

\begin{array}{l} {〈 σ_{z} 〉}_{φ, σ} =_{σ} 〈 ψ | τ_{x} {| ψ 〉}_{σ}, \\ {〈 σ_{x} 〉}_{φ, σ} =_{σ} 〈 ψ | \sin β τ_{z} - \cos β τ_{y} {| ψ 〉}_{σ}, \\ {〈 σ_{y} 〉}_{φ, σ} =_{σ} 〈 ψ | - \cos β τ_{z} - \sin β τ_{y} {| ψ 〉}_{σ} . \end{array}

It is easy to find that

\begin{array}{l} _{σ} 〈 ψ | τ_{y} {| ψ 〉}_{σ} =_{σ} 〈 ψ | τ_{x} {| ψ 〉}_{σ} = 0, \\ _{σ} 〈 ψ | τ_{z} {| ψ 〉}_{σ} = | a |^{2} ({[1 - \frac{1}{2} m^{2} \cos^{2} (β - α) {\sqrt{n}}^{2}]}^{2} - m^{2} \cos^{2} (β - α) {\sqrt{n}}^{2}) \\ - | b |^{2} ({[1 - \frac{1}{2} m^{2} \cos^{2} (β - α) {\sqrt{n + 2}}^{2}]}^{2} - m^{2} \cos^{2} (β - α) {\sqrt{n + 2}}^{2}) . \end{array}

In the low energy region (±33meV) relative to the Dirac point where the Fermi surface is circular [22], we require the coupling strength

m \sqrt{n} = 0.1

is valid through the paper. The spin operator

\vec{S}

is related to the pseudospin operator

\vec{σ}

as (S_x, S_y, S_z) = (g_xxσ_x, g_yyσ_y, g_zzσ_z) [3, 27]. The real spin then reads

〈 \vec{S} 〉 \propto_{σ} 〈 ψ | τ_{z} {| ψ 〉}_{σ} (g_{x x} s i n β, - g_{y y} c o s β, 0),

where _σ〈ψ|τ_z|ψ〉_σ is given by Eq. (16). For Bi₂Se₃ case, g_xx = g_yy = 0.3068 are constants [27,34] under three kinds of polarized light. The real spin textures of Eq. (17) for the TIs coupled to quantized σ-polarized light on the upper Dirac cone are shown in Fig. 2.

Fig. 2 The real spin textures of surface states coupled to quantized σ-polarized light. Arrow indicates direction and relative length of the real spin in xy plane with different azimuth α.

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3.2. Quantized π-polarized light

For quantized π-polarized light,

{\vec{A}}_{π} = \sqrt{\frac{2 π ℏ c^{2}}{ω V}} ({\hat{a}}^{†} + \hat{a}) (\cos θ \cos α, \cos θ \sin α, - \sin α) .

Eq. (4) takes

{\hat{H}}_{π} = ℏ v_{0} k τ_{z} + ℏ ω {\hat{a}}^{†} \hat{a} - i G \cos θ \sin (β - α) (τ_{-} - τ_{+}) ({\hat{a}}^{†} + a) .

The spin polarization with different azimuth angle α and polar angle θ is shown in Fig. 3.

Fig. 3 The spin polarization of TSS in the xy plane of Bi₂Se₃ coupled to the quantized π-polarized light with different {α, θ}. The first and second rows indicate the spin textures on the upper and lower Dirac cone, respectively.

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3.3. Quantized circular polarized light

The quantized circular polarized light is described by

\begin{array}{l} {\vec{A}}_{η} = \sqrt{\frac{π ℏ c^{2}}{ω V}} [(\cos θ \cos α ({\hat{a}}^{†} + \hat{a}) - i η \sin α (\hat{a} - {\hat{a}}^{†})) {\vec{e}}_{x} \\ + (\cos θ \sin α ({\hat{a}}^{†} + \hat{a}) + i η \cos α (\hat{a} - {\hat{a}}^{†})) {\vec{e}}_{y} - \sin θ ({\hat{a}}^{†} + \hat{a}) {\vec{e}}_{z}], \end{array}

where η = ±1 denote the left/right handed quantized circular polarized light, respectively. Substituting A_η,x and A_η,y into Eq. (4), we get

{\hat{H}}_{η} = ℏ v_{0} k τ_{z} + ℏ ω {\hat{a}}^{†} \hat{a} + i G / \sqrt{2} (κ {\hat{a}}^{†} + κ * \hat{a}) (τ_{-} - τ_{+})

, where κ = cos θ sin(β − α) − iη cos(β − α). Applying the method in [37] to get the effective Hamiltonian

{\hat{H}}_{η}

{\hat{H}}_{η} = Ω_{η} τ_{z} + ℏ ω {\hat{a}}^{†} \hat{a} + i G / \sqrt{2} (κ {\hat{a}}^{†} τ_{-} - κ * τ_{+} \hat{a}),

where Ω_η=ħv₀k + κκ*G²(2n + 1)/(4ħv₀k+2ħω). Then, the expect value of τ_z on the eigenstate (given by the unitary transformation on the eigenstate of Eq. (21), which is similar with Eq. (14)) follows

〈 τ_{z} 〉 = \frac{(1 - 0.5 n m κ κ *) (n + 1) G^{2} κ κ * / 2}{{(Ω_{η} + ℏ ω n - E_{η})}^{2} (n + 1) G^{2} κ κ * / 2} - \frac{(1 - 0.5 (n + 2) m κ κ *) {(Ω_{η} + ℏ ω n - E_{η})}^{2}}{{(Ω_{η} + ℏ ω n - E_{η})}^{2} + (n + 1) G^{2} κ κ * / 2} .

The real spin vector takes

〈 \vec{S} 〉 \propto 〈 τ_{z} 〉 (g_{x x} s i n β, - g_{y y} c o s β, 0)

The results are shown in Fig. 4.

Fig. 4 The real spin of Bi₂Se₃ irradiated by quantized circular polarized light. Arrow indicates direction and relative length of the real spin in the xy plane with different {α, θ}. The first and second rows indicate the spin textures on the upper and lower Dirac cone, respectively.

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3.4. Discussions on the textures at the Bi₂Se₃ surface

Figures 2–4 show the orientation and relative length of the spin vector $〈 \vec{S} 〉$ of surface state near the high-symmetry point Γ along (111) direction in momentum space for Bi₂Se₃ coupled to single-mode quantum field. The surface normal corresponds to the z direction and the standard convention of the right-handed coordinate system applies are shown in Fig. 1(b). We find that the spin polarization vectors are aligned preferentially in the xy plane pointing $\vec{k} \times \vec{x}$ and the helicity is left (right) handed for the upper (lower) Dirac cone. The spin continuously curls around the z-axis to form a vortex such that the system possesses a persistent spin current but no net density current. The spin vortex state is analogous to the flux-closure magnetic states in micromagnetics. In the vicinity of the z-axis, to reduce the exchange energy, the magnitude of the spin gradually decreases and vanishes on the z-axis. This differs from the magnetic vortex observed in nanoscale ferromagnets [38] where, due to the conservation of local spin moment, the magnetization in the vortex core turns toward the z-axis.

All these salient features can be understood by the following illustrations. For electron in crystal free from external field case, its eigenstate can be denote as |ψ1〉 = v_k |↑〉+ u |↓〉, the expectation values of three pseudospin components: $〈 σ_{x} 〉 = v_{k}^{*} u_{k} + u_{k}^{*} v_{k}$ , $〈 σ_{y} 〉 = u_{k}^{*} v_{k} - i v_{k}^{*} u_{k}$ , $〈 σ_{z} 〉 = u_{k}^{*} v_{k} - v_{k}^{*} u_{k}$ . The magnitude of the pseudospin $| \vec{σ} | = | v_{k} |^{2} + | u_{k} |^{2}$ constant 1. Besides, “spin-momentum locking” results in a fixed spin direction for each $\vec{k}$ of the eigenstate. Therefore, in the curling configuration, with a constant length of spin vector, its direction changes gradually in-plane, and the amplitude of variation satisfies time reversal symmetry $[E (\vec{k}, ↓) = E (- \vec{k}, ↑)]$ . This is the electron spin behaviors presented in [27]. When surface state interacting with quantum field, considering the degree of freedom of photon, the eigenstate of Eq. (2) reads |ψ2〉 = ∑_n,j a_n |↑, n〉 + b_j |↓, j〉 (n, j indicates photon number). Based on |ψ2〉, we can derive the density matrice for the two-level system $ρ_{s} = \sum_{n = j} | a_{n} |^{2} | ↑ 〉 | ↓ 〉 + | b_{j} |^{2} | ↓ 〉〈 ↓ | + a_{n} b_{j}^{*} | ↑ 〉〈 ↓ | + b_{j} a_{n}^{*} | ↓ 〉 | ↓ 〉$ . The expectation value $〈 σ_{x} 〉 = \sum_{n = j} b_{j} a_{n}^{*} + a_{n} b_{j}^{*}, 〈 σ_{y} 〉 = \sum_{n = j} - i b_{j} a_{n}^{*} + i a_{n} b_{j}^{*}, 〈 σ_{z} 〉 = \sum_{n = j} | a_{n} |^{2} - | b_{j} |^{2}$ and the magnitude of pseudospin reads $| \vec{σ} | = \sum_{n = j} | a_{n} |^{2} + | b_{j} |^{2}$ which is always less than 1 and inversely proportional to the angle dependent coupling strength. Therefore, Figs. 2–4 show the length of arrow which represents the spin polarization deviates from unity and varies on the constant energy contour for all $\vec{k}$ . The coupling shows angle dependence cos(β − α) or sin(β − α), therefore, different coupling strengths under different azimuth angles α cause the variation of spin length for each $\vec{k}$ , whose performance in spin textures is the rotation around the Dirac point counterclockwise (clockwise) for the upper (lower) Dirac cone under increasing azimuth angle α. Considering quantized circular polarized light is the superimposition effect of σ-polarized light and π-polarized light, when θ=π/2, the in-plane circular polarized light degenerates into σ-polarized light, therefore, the second row of Fig. 4 is the true portrayal of Eq. (17) for the lower Dirac cone; Besides, the spin polarization of TSS is irrelevant to the helicity of external quantum field, which can be found from Eq. (22) where κ is paired with κ^∗, and therefore their product κκ^∗ is independent of η, which is different from the spin polarization in [27] where the photoemitted electron irradiated by classical field.

4. SmB₆ surface states under three kinds quantized field

In SmB₆, the hybridization between the conduction electrons occupying d orbital and pre-dominantly localized electrons residing on f orbital drives an insulating gap opening at low temperatures and significantly enhances the spin-orbit coupling, which is responsible for the inversion of the bands with opposite parity. What makes SmB₆ different from Bi₂Se₂ family is the presence of strong on-site Coulomb interaction between the samarium f electrons. As a result, the band inversion in the modified band structure happens between the 5d and 4f band(with total angular momentum j = 5/2) around three X points in the Brillouin zone (BZ) leading to three different Dirac points on the (001) surface [39–41]. In this section, we will investigate the spin textures at the aforementioned Dirac points $\bar{Γ}$ and $\bar{Y}$ .

4.1. Surface states at $\bar{Γ}$ point

The Hamiltonian for surface states near $\bar{Γ}$ point is given in [35]. Considering the topological surface state interacting with quantum field, we can get the following Hamiltonian

H_{\bar{Γ}} = - ℏ v_{0} [σ_{x} (k_{y} - \frac{e}{c ℏ} A_{y}) + σ_{y} (k_{x} - \frac{e}{c ℏ} A_{x})] + ℏ ω {\hat{a}}^{†} \hat{a} .

The crystalline symmetry at

\bar{Γ}

point is characterized by group C₄_v, which are j_z = ±1/2 and j_z = ±3/2 representation, respectively. The first-principles calculations show that the Dirac surface states at

\bar{Γ}

point belong to the representation of j_z = ±3/2 [27, 35]. Denoting

{\hat{H}}_{s \bar{Γ}} = - ℏ v_{0} (σ_{x} k_{y} + σ_{y} k_{x})

and preforming an unitary transformation

\hat{u} = (\begin{matrix} - i e^{- i β / 2} & e^{i β / 2} \\ i e^{- i β / 2} & e^{i β / 2} \end{matrix})

on Eq. (23) to transform

{\hat{H}}_{s \bar{Γ}}

into the space spanned by the eigenstates |E〉 and |G〉 of

{\hat{H}}_{s \bar{Γ}}

, we obtain

{\hat{H}}_{\bar{Γ}} = ℏ v_{0} k τ_{z}^{I} + ℏ ω {\hat{a}}^{†} \hat{a} + \frac{e v_{o}}{c} (A_{y} \cos β + A_{x} \sin β) τ_{y}^{I},

where

τ_{z}^{I} = | E 〉 〈 E | - | G 〉 〈 G |

,

τ_{x}^{I} = | E 〉 〈 G | + | G 〉 〈 E |

, and

τ_{y}^{I} = i | G 〉 〈 E | - i | E 〉 〈 G |

. Moreover,

\hat{u} σ_{x} {\hat{u}}^{- 1} = - \sin β τ_{z}^{I} + \cos β τ_{y}^{I}

,

\hat{u} σ_{y} {\hat{u}}^{- 1} = - \cos β τ_{z}^{I} - \sin β τ_{y}^{I}

and

\hat{u} σ_{z} {\hat{u}}^{- 1} = - τ_{x}^{I}

. Then, we can derive the real spin under three kinds of quantum field.

4.1.1. For quantized σ-polarized light

Substituting Eq. (5) into Eq. (24), we obtain

{\hat{H}}_{σ} = ℏ v_{0} k τ_{z}^{I} + ℏ ω {\hat{a}}^{†} \hat{a} + i G \cos (β + α) (τ_{-}^{I} - τ_{+}^{I}) ({\hat{a}}^{†} + \hat{a}) .

Denoting |Ψ〉_σ is the eigenstate of Eq. (25) for the upper Dirac cone, the real spin textures can be described by

{〈 \vec{S} 〉}_{φ, π} \propto σ 〈 Ψ | τ_{z}^{I} {| Ψ 〉}_{σ} (- g_{x x} s i n β, - g_{y y} c o s β, 0)

, where g_xx = 0.095 and g_yy = −0.095 are always valid near

\bar{Γ}

point. The real spin textures are shown in Fig. 5.

Fig. 5 The real spin textures with the quantized σ-polarized light in the vicinity of $\bar{Γ}$ point. Arrows indicate the spin directions in the xy plane with different {α}.

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4.1.2. For quantized π-polarized light

Considering Eq. (18) and Eq. (24), we can obtain a Hamiltonian

{\hat{H}}_{π} = ℏ v_{0} k τ_{z}^{I} + ℏ ω {\hat{a}}^{†} \hat{a} + i G \cos θ \sin (β + α) (τ_{-}^{I} - τ_{+}^{I}) ({\hat{a}}^{†} + \hat{a}) .

The spin polarization with different azimuth angles and polar angles are shown in Fig. 6.

Fig. 6 The spin polarization with the quantized π-polarized light in the vicinity of $\bar{Γ}$ point. Arrows indicate the spin directions in the xy plane with different {α, θ}. The first and second rows indicate the textures on the upper and lower Dirac cone, respectively.

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4.1.3. For quantized circular polarized light

Substituting Eq. (20) into Eq. (24), we can obtain the Hamiltonian as

{\hat{H}}_{η} = ℏ v_{0} k τ_{z}^{I} + ℏ ω {\hat{a}}^{†} \hat{a} + i G / \sqrt{2} (μ {\hat{a}}^{†} + μ^{*} \hat{a}) (τ_{-}^{I} - τ_{+}^{I}),

where μ = cos θ sin(β + α) − βη cos(β + α). These spin polarization are shown in Fig. 7.

Fig. 7 The real spin with quantized circular polarized light in the vicinity of $\bar{Γ}$ point. Arrows indicate the spin directions in the xy plane with different {α, θ}. The first row indicates the textures on the upper Dirac cone and the second row on the lower one.

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4.1.4. Discussions on the textures feature at $\bar{Γ}$ point

Figures. 5–7 show direction and relative length of spin polarization for TSS irradiated by quantized polarized light near high-symmetry point $\bar{Γ}$ lying in two-dimensional Brillouin region of SmB₆ for (001) surface which is shown in Fig. 1(c). There is no great difference in spin polarization configurations between Bi₂Se₃ and SmB₆ cases. In Sec. 3.4, we have detailly illustrated the spin polarization of TSS for Bi₂Se₃, here, we mainly emphasize their differences. Surface states near point $\bar{Γ}$ of SmB₆ obey double group of C₄_v and belong to the representation j_z = ±3/2, therefore the signs of σ_x and σ_y are all negative in Eq. (23) obtained from $\vec{k} \cdot \vec{p}$ model with linear approximation. While in Bi₂Se₃, surface states defer to C₃_v crystalline symmetry with an angular momentum j_z = ±1/2, signs of σ_x and σ_y are opposite in Eq. (2). Different symmetries of surface states compared with Bi₂Se₃ result in two main results: (i) spin and momentum rotate around $\bar{Γ}$ point in the same (opposite) direction for the upper (lower) Dirac cone; (ii) azimuth angle dependence sin(β + α) or cos β + α) derives the textures rotate clockwise (anticlockwise) for the upper (lower) Dirac cone for continuous increasing α.

4.2. Surface states at $\bar{Y}$ point

The crystalline symmetry at $\bar{Y}$ point is characterized by C₂_v group and time reversal symmetry. The first-principles calculations show that the Dirac surface states at $\bar{Γ}$ point belong to the representation of j_z = ±3/2. The Hamiltonian at $\bar{Y}$ point is given as ${\hat{H}}_{\bar{Y}} = ℏ (υ_{y} σ_{y} k_{x} - υ_{x} σ_{x} k_{y})$ , υ_y/υ_x = 5/4 [27] and υ_x = 5 × 10⁵m/s are valid for the spin near $\bar{Y}$ point. The eigenstates of ${\hat{H}}_{\bar{Y}}$ read $| \pm 〉 = {(e^{i ϕ / 2}, \mp e^{- i ϕ / 2})}^{T}$ with tan ϕ = υ_yk_x/υ_xk_y. Irradiated by the quantum field, the surface state can be described as

\hat{H} = ℏ υ_{y} σ_{y} (k_{x} - \frac{e}{c ℏ} A_{x}) - ℏ υ_{x} σ_{x} (k_{y} - \frac{e}{c ℏ} A_{y}) + ℏ ω {\hat{a}}^{†} \hat{a} .

It is convenient to transform Hamilitonian (28) into the Hilbert space spanned by |+〉 and |−〉 via an unitary transformation

U_{\bar{Y}} = (\begin{matrix} - e^{- i ϕ / 2} & e^{i ϕ / 2} \\ e^{- i ϕ / 2} & e^{i ϕ / 2} \end{matrix})

. We can get the following expression

{\hat{H}}_{\bar{Y}} = ℏ D (k) τ_{z}^{I I} + ℏ ω {\hat{a}}^{†} \hat{a} + \frac{e v_{x}}{c} (\frac{v_{y}}{v_{x}} \cos ϕ A_{x} - \sin ϕ A_{y}) τ_{y}^{I I},

where

D (k) = \sqrt{υ_{x}^{2} k_{y}^{2} + υ_{y}^{2} k_{x}^{2}}

. Moreover,

τ_{y}^{I I} = i (| - 〉 〈 + | - | + 〉 〈 - |)

and

τ_{z}^{I I} = | + 〉 〈 + | - | - 〉 〈 - |

.

4.2.1. Quantized σ-polarized light

The Hamiltonian describing surface state irradiated by quantized σ-polarized light reads

{\hat{H}}_{σ} = ℏ D (k) τ_{z}^{I I} + ℏ ω {\hat{a}}^{†} \hat{a} - i f (ϕ) (τ_{-}^{I I} - τ_{+}^{I I}) ({\hat{a}}^{†} + \hat{a}),

where

f (ϕ) = e υ_{x} \sqrt{2 π ℏ / ω V} (\cos α \sin ϕ + 5 / 4 \sin α \cos ϕ)

. With the eigenstate |Φ〉_σ for the upper Dirac cone of Eq. (30), the real spin at

\bar{Y}

point under quantized σ-polarized light can be given by

〈 \vec{S} 〉 \propto_{σ} {〈 Φ | τ_{z}^{I I} | Φ 〉}_{σ} (- g_{x x} \cos ϕ, g_{y y} \sin ϕ, 0)

. The coupling strength

e υ_{x} \sqrt{2 n π ℏ / ω V} / ℏ ω = 0.2

, g_xx = 0.0678 and gyy = −0.1223 are valid for the real spin at

\bar{Y}

point under three kinds of quantum field. Numerical results are shown in Fig. 8.

Fig. 8 The real spin for the system irradiated by the quantized σ-polarized light in the vicinity of $\bar{Y}$ point. Arrows indicate the spin directions in the xy plane with different α.

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4.2.2. Quantized π-polarized light

The surface state under quantized π-polarized light obeys the Hamiltonian

{\hat{H}}_{π} = ℏ D (k) τ_{z}^{I I} + ℏ ω {\hat{a}}^{†} \hat{a} + i F (ϕ) (τ_{-}^{I I} - τ_{+}^{I I}) ({\hat{a}}^{†} + \hat{a}),

where

F (ϕ) = e υ_{x} \sqrt{2 π ℏ / ω V} \cos θ (5 / 4 \cos α \cos ϕ - \sin α \sin ϕ)

. The spin polarization given by this equation is shown in Fig. 9.

Fig. 9 The spin polarization for the system irradiated by π-polarized light in the vicinity of $\bar{Y}$ point with different {α, θ}. The first and second rows indicate the textures on the upper and lower Dirac cone, reapectively.

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4.2.3. Quantized circular polarized light

For quantized circular polarized light, we obtain

\hat{H} = ℏ D (k) τ_{z}^{II} + ℏ ω {\hat{a}}^{†} \hat{a} + \frac{ie υ_{x}}{c} \sqrt{\frac{π ℏ c^{2}}{ω V}} (ζ {\hat{a}}^{†} + ζ^{*} \hat{a}) τ_{y}^{II},

where ζ = cos θ(5/4 cos ϕ cos α−sin ϕ sin α)+iη(5/4 cos ϕ sin α+sin ϕ cos α). The spin textures are shown in Fig. 10.

Fig. 10 The real spin in the vicinity of $\bar{Y}$ point for the upper Dirac cone under quantized circular polarized light with different {α, θ}. Arrows indicate the spin directions in the xy plane.

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4.2.4. Discussions on the textures feature at $\bar{Y}$ point

Figures. 8–10 show direction and relative length of spin polarization for TSS irradiated by quantized polarized light corresponding to high-symmetry point $\bar{Y}$ lying in two-dimensional Brillouin region of SmB₆ for (001) surface which is shown in Fig. 1(c). Here, we mainly focus on the difference of spin polarization between $\bar{Y}$ point and $\bar{Γ}$ point. Tight-binding Hamiltonian H_bulk(k_x, k_y, k_z) and spin operator for bulk band are expanded by d, f orbitals and pseudospin basis vectors. Replacing H_bulk(k_x, k_y, k_z) → H_bulk(0, 0, i∂z), which satisfies H_bulk(−i∂z)ψ(z) = Eψ(z) [41]. The g factor relating pseudospin to the real spin operator for surface state is obtained by projecting bulk spin operator into ψ(z). Thus obtained g_xx ≠ g_yy leads to spin polarization of TSS for $\bar{Y}$ point no longer in $\vec{k} \times \vec{z}$ direction. The coupling between TSS and quantized polarized light only tune the length of spin arrow for each $\vec{k}$ . Besides, spin textures under circular polarized light degenerate into σ-polarized light case as θ = π/2, therefore, the second row of Fig. 10 is consistent with Fig. 8.

5. Conclusion

To summarize, we have directly worked out the spin texture of TSS near high-symmetry points coupled to a single-mode quantum field polarized in three modes for Bi₂Se₃ and SmB₆. We find that the spin polarization of TSS is aligned preferentially in the xy plane and optically tunable. The azimuth angle can control the rotating mode of spin texture along the constant energy contour around the high-symmetry point and the increase of zenith angle results in the length decrease of spin arrow. Moreover, the spin texture is irrelevant to the helicity of circular polarized light. Our results have potential application in quantum optics and condensed matter physics.

Funding

National Natural Science Foundation of China (NSFC) (11534002, 61475033, 11775048, 11705025); China Postdoctoral Science Foundation (2016M600223, 2017T100192); Fundamental Research Funds for the Central Universities (2412017QD005).

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Optically tunable spin texture of the surface state for Bi₂Se₃ and SmB₆ topological insulators

Abstract

1. Introduction

2. Model to describe the system coupled to quantum field

3. Bi₂Se₃ surface state coupled to three kinds of single-mode quantum field

3.1. Quantized σ-polarized light

3.2. Quantized π-polarized light

3.3. Quantized circular polarized light

3.4. Discussions on the textures at the Bi₂Se₃ surface

4. SmB₆ surface states under three kinds quantized field

4.1. Surface states at $\bar{Γ}$ point

4.1.1. For quantized σ-polarized light

4.1.2. For quantized π-polarized light

4.1.3. For quantized circular polarized light

4.1.4. Discussions on the textures feature at $\bar{Γ}$ point

4.2. Surface states at $\bar{Y}$ point

4.2.1. Quantized σ-polarized light

4.2.2. Quantized π-polarized light

4.2.3. Quantized circular polarized light

4.2.4. Discussions on the textures feature at $\bar{Y}$ point

5. Conclusion

Funding

References and links

Cited By

Figures (10)

Equations (32)

Optics Express

Abstract

1. Introduction

2. Model to describe the system coupled to quantum field

3. Bi2Se3 surface state coupled to three kinds of single-mode quantum field

3.1. Quantized σ-polarized light

3.2. Quantized π-polarized light

3.3. Quantized circular polarized light

3.4. Discussions on the textures at the Bi2Se3 surface

4. SmB6 surface states under three kinds quantized field

4.1. Surface states at Γ¯ point

4.1.1. For quantized σ-polarized light

4.1.2. For quantized π-polarized light

4.1.3. For quantized circular polarized light

4.1.4. Discussions on the textures feature at Γ¯ point

4.2. Surface states at Y¯ point

4.2.1. Quantized σ-polarized light

4.2.2. Quantized π-polarized light

4.2.3. Quantized circular polarized light

4.2.4. Discussions on the textures feature at Y¯ point

5. Conclusion

Funding

References and links

Cited By

Figures (10)

Equations (32)

Optics Express

3. Bi₂Se₃ surface state coupled to three kinds of single-mode quantum field

3.4. Discussions on the textures at the Bi₂Se₃ surface

4. SmB₆ surface states under three kinds quantized field

4.1. Surface states at $\bar{Γ}$ point

4.1.4. Discussions on the textures feature at $\bar{Γ}$ point

4.2. Surface states at $\bar{Y}$ point

4.2.4. Discussions on the textures feature at $\bar{Y}$ point