Inter-surface coupling in thin-film topological insulators can reduce the surface state mobility by an order of magnitude in low-temperature transport measurements. The reduction is caused by a reduction in the group velocity and an increased *s _{z}* component of the surface-state spin which weakens the selection rule against large-angle scattering. An intersurface potential splits the degenerate bands into a Rashba-like bandstructure. This reduces the intersurface coupling, it largely restores the selection rule against large angle scattering, and the ring-shaped valence band further reduces backscattering by requiring, on average, larger momentum transfer for backscattering events. The effects of temperature, Fermi level, and intersurface potential on the Coulomb impurity scattering limited mobility are analyzed and discussed.

Topological insulators (TI) form a new class of quantum materials with an insulating band gap in the bulk and Dirac-cone surface states. Unlike normal materials, the surface states of TI materials are robust against disorder, inhomogeneities, and against time-reversal-invariant perturbations.^{1–4} A combination of the high velocity of the Dirac-cone surface states and their topological protection against back-scattering make TI materials appealing from the perspective of charge transport. 3D TIs such as Bi_{2}Te_{3} and Bi_{2}Se_{3} have demonstrated surface state mobilities on the order of ∼10^{4} cm^{2}V^{−1}s^{−1}.^{5,6} When the thickness of a 3D TI thin film is reduced to several quintuple layers (QLs), the high mobility is suppressed to 10^{2}–10^{3} cm^{2}V^{−1}s^{−1}.^{7–9} The suppression of the mobility in the low-temperature transport measurements has been attributed to the strong scattering from the p-type dopants that are required to move the Fermi level (*ϵ*_{F}) close to the Dirac point^{9–11} and charged surface adsorbates.^{7} Prior theoretical studies focused on the effect of inhomogeneities on the transport of TI surface states on a single surface.^{12,13}

The surface states of 3D TI thin films couple and hybridize the opposite spins of the top and the bottom surface states.^{14–16} As a result, a surface band gap is opened as illustrated in Fig. 1(a), the surface band-edge group velocity decreases, and the original momentum-spin (**k**–**s**) relation that prohibits back-scattering of surface states is broken.^{17} This last effect is illustrated in Fig. 1(c). These effects might further explain the reduction in the carrier mobility in TI thin films compared to the expected value of bulk surface states. The question we seek to answer is to what extent the mixing of the surface states affects the in-plane mobility of the TI surface states. To address this issue, we calculate the mobility of the coupled surface states on a TI thin film, and we explore the effects of several different externally controlled variables such as the Rashba-like splitting of the bands under interlayer bias illustrated in Fig. 1(b), temperature, and Fermi level.

The low-energy band-structure can be described by a 4 × 4 **k** ⋅ **p** Hamiltonian with 4 basis states $|\chi \alpha \u27e9$ corresponding to the spin up and spin down Bi and Se *p _{z}* orbitals.

^{1}We discretize in real space the

*k*component of this Hamiltonian into

_{z}*N*sites, so that each on-site block is a 4 × 4 matrix. Eigenstates $|k,p\u27e9$ are linear combinations of the 4 basis orbitals on the

*N*sites, $\u27e8r|k,p\u27e9=eik\xb7rA\u2211i=1N\u2211\alpha =14c\alpha ,ip(k)\u27e8r|\chi \alpha ,i\u27e9$, where

*α*is the orbital index,

*i*is the site index, and

*p*is the band index. The band index runs over the 4 bands shown in Figs. 1(a) and 1(b) corresponding the top and bottom gapped Dirac cones of the top and bottom surfaces. To distinguish the energetically degenerate states, an infinitesmal Zeeman term (10

^{−5 }eV) is applied to numerically split the opposite spins. Details of the discretization of this model can be found elsewhere.

^{16}

A typical band structure of a TI thin film is shown in Fig. 1(a). Hybridization of the top and bottom surface states generates a small band gap at the Dirac point. The gap decreases with increasing film thickness as shown in Fig. 1(d), and it is negligible beyond 6 quintuple layers.

In a thick film, opposite surfaces states have opposite **k**–**s** chiralities where **s** is the spin of state $|k\u27e9$. With decreasing film thickness, the opposite surface wave functions of the same momentum hybridize and mix their opposite spins, resulting in a non-zero *s _{z}* spin component. The magnitude of

*s*is a measure of the strength of the inter-surface coupling which is shown in Fig. 2(a) as a function of energy. The maximum inter-surface coupling occurs at the gapped Dirac point, and it decreases for energies away from the band edges. This trend agrees with prior results given by both

_{z}*ab inito*calculations

^{18}and analytical models.

^{14,17}

To quantitatively determine the mobility in TI thin films, a four-band semi-classical calculation is carried out. For a screened Coulomb scattering center at layer *j*, the Hamiltonian matrix element between states $|k,p\u27e9$ and $|k\u2032,p\u2032\u27e9$ is written as

where *A* is the unit area, *q _{e}* is the single-electron charge,

*κ*= 100

*ϵ*

_{0}is the static dielectric constant, $\beta =|k\u2212k\u2032|$, and $z=|i\u2212j|\Delta $ is the vertical distance between site

*i*and the layer of the Coulomb scattering center

*j*. The discretization length Δ = 0.3 nm. Equation (1) results from the 2D Fourier transform of the screened Coulomb potential $q2e\u2212q0r4\pi \kappa r$. The inverse screening length

*q*

_{0}is given by $q0=2\pi qe2\kappa d\rho d\u03f5F$, where

*ρ*is the 2-dimensional (2D) charge density and

*ϵ*

_{F}is the Fermi energy.

^{19}

The group-velocity scattering rate due to each impurity on layer *j* is

where **v**^{p}(**k**) and $vp\u2032(k\u2032)$ are the initial and final 2D group velocities. The corresponding group-velocity relaxation time is given by summing over the impurities and the final states to obtain $1\tau p(k)=\u2211jANj\u2211k\u2032,p\u2032Sk\u2032,p\u2032,k,p$, where *N _{j}* denotes the impurity density on layer

*j*. Following the estimates given by previous experimental investigations,

^{7,9}we fix the Coulomb impurity density at 10

^{13}cm

^{−2}. To calculate the carrier mobility, an ensemble average of the group velocity driven by an external electric field is calculated at finite temperature, $\u27e8vx\u27e9=\u2211k,\alpha vxp(k)fAp(k)\u2211k,if0p(k),$ where $fAp(k)$ is the asymmetric component of the non-equilibrium distribution function for band

*p*. From a relaxation time approximation, $fAp(k)=\u2212\tau p(k)qeEx\u210f\u2202f0p(k)\u2202kcos\u2009\theta $, where $f0p(k)$ is the Fermi function, $Ex$ is the electric field along the transport direction and

*θ*denotes the direction of

**k**with respect to the

*k*axis. The mobility is defined as $\mu =\u27e8vx\u27e9/Ex$.

_{x}The mobilities calculated for different film thicknesses are shown in Fig. 2(b) for three different temperatures. The Fermi level *ϵ*_{F} is aligned with the valence band edge, *ϵ*_{V}. The mobility decreases by an order of magnitude as the film thickness decreases from 9 QLs to 2 QLs.

The inter-surface hybridization resulting from the inter-surface coupling reduces the mobility for many reasons. Once a band gap is formed, the near band-edge group velocity decreases as shown in Fig. 2(c). Another factor that decreases the mobility is the increased *s _{z}* component of the spin in the states $|k,p\u27e9$.

On the top surface of a thick TI, the surface state can be written as a plane-wave times a spinor. For a non-magnetic scattering mechanism, the matrix element squared of the scattering potential between two states in the same band is $|Hk\u2032,k|2=|\u27e8k\u2032|H|k\u27e9|2|\u27e8s\u2032|s\u27e9|2=|\u27e8k\u2032|H|k\u27e9|2(1\u2212cos\u2009\theta k\u2032,k)$, where $\theta k\u2032,k$ is the angle between $k\u2032$ and **k**. For a bulk surface state, elastic, intra-surface backscattering is prohibited by the matrix element of the spinors.

In the case of thin films, mixing of opposite surface states introduces an *s _{z}* component to the spin so that $|\u27e8s\u2032|s\u27e9|2\u2260(1\u2212cos\u2009\theta k\u2032,k)$ and backscattering events are no longer prohibited. Fig. 2(d) demonstrates the normalized, angle-resolved scattering rate for an initial state along

*k*in a 4 QL film. Backscattering is significant for states close to

_{x}*ϵ*

_{V}where the inter-surface coupling gives

*s*≈ 1 as shown in Fig. 2(a).

_{z}There are several ways to increase the mobility of the thin film. One way is to increase the temperature in the low-temperature transport regime where the Coulomb scattering dominates. The mobility increases by almost an order of magnitude when the temperature increases from 10 K to 50 K as shown in Fig. 2(b). A similar trend was observed in a transport measurement in a 6 nm Bi_{2}Se_{3} thin film.^{8} The increase in temperature broadens the transport thermal window, so that higher energy states contribute to the carrier transport. Since the inter-surface coupling decreases away from the gapped Dirac point, states away from the band edge have a higher group velocity and better back-scattering protection due to the smaller *s _{z}* component of the spin. As shown in Fig. 2(d), the back-scattering protection is almost restored for the states 15 meV away from

*ϵ*

_{V}in a 4QL thin film. For the same reasons, shifting

*ϵ*

_{F}away from the band edges increases the mobility as shown in Fig. 2(e).

Another reason for this mobility increase for energies away from the band edges is the nature of the Coulomb scattering mechanism. Since the carrier concentration of a 2D thin film is significantly smaller than that of the bulk, screening is weak. Using the bare Coulomb potential, the scattering rate given by Eq. (2) can be simplified to $S\u0303k\u2032,k=2\pi \u210fqe44A2\kappa 2\beta 2Ik\u2032,k(1\u2212cos\u2009\theta )\delta (\u03f5\u2212\u03f5\u2032)$, where $Ik\u2032,k=|\u2211iN\u2211\alpha =14ci,\alpha p\u2032*(k\u2032)ci,\alpha p(k)|2$ is the overlap integral. For states close to the band edge, the two degenerate bands from oppisite surfaces have large but opposite *s _{z}* components, and, as a result, the inter-band overlap integral is close to zero. Thus, inter-surface scattering processes are neglected. States close to the band edge on the same surface have large and aligned

*s*components, so that $Ik\u2032,k\u22481$. For low-energy, intra-surface scattering, the group-velocity relaxation time can be calculated analytically as $1\tau \u0303(k)=AN\u2211k\u2032S\u0303k\u2032,k=Nqe48\u210f2\kappa 2kv$ where

_{z}*N*is the total surface density of the Coulomb scattering centers. The relaxation time is proportional to the group velocity,

*v*, and the radius of the contour of equal energy,

*k*. Thus, as

*ϵ*

_{F}moves away from the band edge,

*kv*at the Fermi level increases, so that states with greater relaxation times are included in the transport thermal window. Since

*μ*is proportional to

*τ*, increasing the

*kv*product increases the mobility at low temperatures.

The mobility is further enhanced by a vertical potential drop (*U*) through the thin film which creates a Rashba-like splitting of the bands as shown in Fig. 1(b). Experimentally, this can be achieved by creating different electrostatic environments on each surface such as applying a strongly coupled substrate or using gating mechanisms.^{20} The vertical potential drop leads to a structural inversion asymmetry (SIA) that can restore the topologically nontrivial surface states in a TI thin film.^{14,17,21} As demonstrated in Fig. 3(a), for a 4QL TI thin film, a vertical potential drop of 0.1 V enhances the mobility by up to an order of magnitude when *ϵ*_{F} is close to the valence band edge. The reduced inter-surface coupling is evident in the reduced *s _{z}* component of the spin close to the Rashba-split band edges shown in Fig. 4(a). The suppressed

*s*component restores the back-scattering protection. As demonstrated in Fig. 4(b), back-scattering is suppressed for states 5 meV away from

_{z}*ϵ*

_{V}when a vertical potential drop of 0.1 V is applied.

Besides the reduced inter-surface coupling, another reason for the mobility increase given by the Rashba-like splitting is the change in the band structure. As illustrated in Fig. 1(b), the surface-to-surface potential drop creates a valence band edge that is a ring in *k*-space. The band edge forms a circle of radius *k*_{0} that increases with *U*. The increase in *kv* linearly increases the relaxation time. Although the band-edge group velocity is almost unchanged due to the Rashba-splitting as shown in Fig. 3(b), the relaxation time increases by almost an order of magnitude when the potential drop increases from 0 to 0.1 V as demonstrated in Fig. 3(c). The corresponding mobility is enhanced from ∼10^{3} cm^{2}V^{−1}s^{−1} to ∼10^{4} cm^{2}V^{−1}s^{−1}.

In conclusion, the inter-surface coupling of TI thin films can reduce the surface state mobility by an order of magnitude in the low-temperature transport regime where the carrier transport is dominated by Coulomb impurity scattering. Hybridization of the surface states introduces a band gap and an *s _{z}* component to the spin. The presence of a band gap reduces the average group velocity, and the

*s*component of the spin reduces the protection against large angle scattering. Increasing the temperature or shifting the Fermi level away from the band edges can increase the mobility back to the level of an isolated bulk surface state. An inter surface potential resulting in a Rashba-like splitting reduces the inter-surface mixing and the associated

_{z}*s*component of the spin and restores the protection against large angle scattering. The Rashba-like splitting also creates a ring shaped band edge which increases the average momentum transfer required for a backscattering event.

_{z}This material is based upon work supported by the National Science Foundation under Grant Nos. NSF 1128304 and NSF 1124733. It was also supported in part by FAME, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA.