Adiabatic quantum pumping through surface states in 3D topological insulators (2024)

Recently, surface states in topological insulators have attracted a lot of attention in the condensed-matter community[1]. Both in two-dimensional (e.g. HgTe) and in three-dimensional (e.g. Bi₂Se₃) compounds with strong spin–orbit interaction the topological phase has been demonstrated experimentally[2–6].² Although these compounds are insulating in the bulk (since they have an energy gap between the conduction band and the valence band), their surface states support gapless topologically protected states. In the simplest case these low-energy states of a strong three-dimensional topological insulator can be described by a single Dirac cone at the center of the three-dimensional Brillouin zone (Γ point)[2, 4, 7–10]. The corresponding Hamiltonian is given by[11]

Here represents a vector whose three components are the three Pauli spin matrices, Irepresents a 2×2 identity matrix in spin space, v_F is the Fermi velocity, and μ is the chemical potential. The low-energy states of (equation(1)) are topologically protected against perturbations[12]. This has prompted recent research on the transport properties of surface Dirac fermions. For example, the conductance and magnetotransport of Dirac fermions have been studied in normal metal–ferromagnet (NF) junctions, normal metal–ferromagnetic–normal metal (NFN) junctions and arrays of NF junctions on the surface of a topological insulator[13–15], suggesting the possibility of an engineered magnetic switch. An anomalous magnetoresistance effect has been predicted in ferromagnetic–ferromagnetic junctions[16]. Also, electron tunneling and magnetoresistance have been studied in ferromagnetic–normal metal–ferromagnetic (FNF) junctions[17, 18], for which it has been predicted that the conductance can be larger in the anti-parallel configuration of the magnetizations of the two ferromagnetic regions than in the parallel configuration. In addition, a large research effort has been devoted to studying models which predict the existence of Majorana fermion edge states at the interface between superconductors and ferromagnets deposited on a topological insulator[11, 19, 20].

In this article we investigate adiabatic quantum pumping of Dirac fermions through surface states of a strong three-dimensional topological insulator in the ballistic regime. Quantum pumping refers to a transport mechanism in meso- and nanoscale devices by which a finite dc current is generated in the absence of an applied bias by periodic modulations of at least two system parameters in the adiabatic regime (typically gate voltages or magnetic fields)[21–23]. In order for electrical transport to be adiabatic, the period T of the oscillatory driving signals has to be much longer than the dwell time τ_dwell of the electrons in the system, T=2πω⁻¹≫τ_dwell. In the last decade, many different aspects of quantum pumping have been theoretically investigated in a diverse range of nanodevices, for example charge and spin pumping in quantum dots and quantum Hall systems[24–28], the role of electron–electron interactions[29–33], non-adiabatic driving pumps[34–36], quantum pumping in graphene mono- and bilayers[37–43] and, recently, in a two-dimensional topological insulator[44]. On the experimental side, Giazotto et al[45] have recently reported an experimental demonstration of charge pumping in an InAs nanowire embedded in a superconducting quantum interference device.

In this paper we study quantum pumping induced by periodic modulations of gate voltages or exchange fields (induced by a ferromagnetic strip) in two topological insulator devices: a NFN and a FNF-junction, see figures1 and2. Using a scattering matrix approach, we obtain analytic expressions for the angle-dependent pumped current in both types of junctions and show that the adiabatically pumped current reaches a maximum each time a new resonant mode appears in the junctions. By analyzing in detail the spectrum of resonant modes, we find an analytic explanation for the position of these maxima. We also predict a characteristic difference between the pumped currents in the NFN- and FNF-junctions: whereas the former vanishes for carriers at normal incidence, the latter is finite due to the different nature of wavefunction interference in the junctions. To end we highlight an experimentally distinguishable feature between the pumped current and the conductance.

The remainder of the paper is organized as follows. In section2, we describe the NFN and FNF-junctions and use a scattering matrix model to calculate the reflection and transmission coefficients of both junctions. In section3, we analyze the condition for resonant transmission. In section4, we calculate and analyze the adiabatically pumped current for the two different pumps. Finally, in section5 we summarize our main results and propose possibilities for experimental observation of our predictions. In theappendix, we review the conductance of the NFN junction and present a detailed analysis of the plateau-like steps that appear in the conductance. We also analyze and compare the conductance of the FNF-junction with parallel and anti-parallel configuration of the magnetization.

We first describe the NFN-junction, see figure1. The junction is divided into three regions: region N_l (for x<0), region N_r (for x>d) and the ferromagnetic region F in the middle. We assume the interface between N and F regions to be ideal, i.e. without a potential barrier. The left- and right-hand side of the junction represent the bare topological insulator. The charge carriers (surface Dirac fermions) in these regions are described by the Hamiltonian (equation(1)) whose eigenstates are given by

Here +(−) labels the wavefunctions traveling from the left (right) to the right (left) of the junction. The angle of incidence α and the momentum k_n in the x-direction are given by

Here represents the energy of the Dirac fermions measured from the Fermi energy _F,³q denotes the momentum in the y-direction and v_F the Fermi velocity. In the normal (non-magnetic) regions N_l and N_r a dc gate voltage can be applied to tune the chemical potential μ in the left and right contacts and thereby control the number of charge carriers incident on the junction. We assume gate voltages to be small compared to the band-gap for bulk states (eV_i≪E_g∼1eV, i=l,r), so that transport is well described by surface Dirac states[10]. In this case, the eigenstates are given by

where the index i=l,r labels the left and right sides of the junction.

In the middle region of the junction (0<x<d), the presence of the ferromagnetic strip modifies the Hamiltonian by providing an exchange field. The Hamiltonian that describes the surface states in this region is then given by , with the induced exchange Hamiltonian[14, 16]

and magnetization . The magnitude M of the magnetization depends on the strength of the exchange coupling of the ferromagnetic film and can be tuned for soft ferromagnetic films by applying an external magnetic field[16]. The eigenstates of the full Hamiltonian are then given by

with

and

From equation(12) we see that for a given energy there exists a critical magnetization

beyond which the wavefunction changes from propagating to spatially decaying (evanescent) along the x-direction for all transverse (q) modes[13, 14].

We now describe the FNF-junction, see figure2. Region F_l (x<0) and region F_r (x>d) are ferromagnetic with different magnetizations M_l, M_r along the y-axis and corresponding wavefunctions ψ_Fl and ψ_Fr (equation(10)). The Dirac fermions in the middle region N (0<x<d) are described by the wavefunctions ψ_N (equation(2)). When calculating transport properties of the FNF-junction below, we focus on two different alignments of the magnetizations of the ferromagnetic regions: the parallel configuration, in which the magnetizations in the ferromagnetic regions point in the same direction (M_l∥M_r), and the anti-parallel configuration, in which the magnetizations are in opposite directions (M_l∥−M_r).

Using equations(2)–(12) we now calculate for both the NFN- and the FNF-junctions the reflection and transmission coefficients for a Dirac fermion with energy and transverse momentum q incident from the left. To this end, we consider a general F_lF_mF_r junction, where the wavefunctions in each of the three regions left (l), middle (m) and right (r) are given by:

Here ψ^±_j (j=l,m,r) are the wavefunctions(5),(6) or(10) (depending on the junction considered) and r_ll and t_rl denote the corresponding reflection and transmission coefficients. By requiring continuity of the wavefunction at the interfaces x=0 and d, such that ψ_l|_x=0=ψ_m|_x=0 and ψ_m|_x=d=ψ_r|_x=d, we obtain the reflection and transmission coefficients:

and

Here α_j denotes the polar angle of the wavevector in region j=l,m,r (equations(7) and(11)). The reflection and transmission coefficients r_rr and t_lr for Dirac fermions incident from the right are obtained in an analogous way. The expressions(15) and (16) for the reflection and transmission coefficients form the basis of our calculations of the pumped current in section4.

Before calculating the pumped current, it is useful to first analyze the resonance conditions for reflection and transmission in the junctions. After setting α_l=α_r≡α in equation(15), we begin by finding the conditions when the reflection coefficient is zero, i.e. r_ll=0. The first, trivial, condition α=α_m+2πn corresponds to the situation of an entirely normal junction (i.e. no ferromagnetic region). The second and more interesting condition is sin(k_md)=0. This is the case when transmission occurs via a resonant mode of the junction and can be written as (using equations(3) and (12)):

Equation(17) indicates that for a given M and there are certain privileged angles α_c for which the barrier becomes transparent:

with the dimensionless barrier length and the effective magnetization. We refer to the modes satisfying equation(18) as resonant modes.

Now we address the question why a mode becomes resonant in the barrier. Figure3 shows the angle of incidence αc (equation(18)) as a function of energy /μ for different values of n and for typical values of the barrier width d and magnetization M. For a given n, the resonant mode does not contribute to transmission for energies c<<αnc, with c≡ℏvFM/2−μ, because the imaginary part of the momentum km is non-zero and thus the mode decays along the x-direction. The critical energy αnc for each mode is given by

When >αnc, km is real and the mode becomes resonant. For large energies (not shown in the figure) all the modes asymptotically reach their saturation angle αsat=π/2.

The angle-dependent transmission probability of the NFN-junction T^NFN(α)≡|t_rl(α)|² is given by[13, 14]

Here α_l=α_r=α and α_m denotes the polar angle of the wavevector in the middle region as defined in equation(11). This angle can be expressed in terms of α, using the fact that momentum is conserved along the y-axis, as

Figure4 shows the transmission probability (equation(20)) as a function of the angle of incidence α for different values of energy /μ. The dashed (red) vertical lines correspond to the angles satisfying equation(18) for different n. For low-energy excitations only negative angles α (i.e. q-momenta anti-parallel to M) contribute to the transport, see figures4(a)–(d). As the energy increases, the resonant modes move from the left to the right and also positive angles α (i.e. q-momenta parallel to M) begin to contribute, see figures4(e) and (f). Notice that the modes are not sharply peaked, except for energies at which only one mode is present (n=1, see figure4(a)); in general, several angles contribute to each new mode.

The behavior of the resonant modes, as shown in figures3 and4, forms the basis for understanding the behavior of the pumped current in the next section.

We now investigate the adiabatically pumped current through NFN and FNF topological insulator junctions⁴. In general, a pumped current is generated by slow variations of two system parameters X₁ and X₂ in the absence of a bias voltage[21, 22]. For periodic modulations X₁(t)=X_1,0+δX₁cos(ωt) and X₂(t)=X_2,0+δX₂cos(ωt+ϕ), the pumped current I_p into the left lead of the junction can be expressed in terms of the area A enclosed by the contour that is traced out in (X₁,X₂)-parameter space during one pumping cycle[22]:

with

Here r_ll and t_lr represent the reflection and transmission coefficients into the left lead and the index m sums over all modes (a similar expression can be obtained for the pumped current into the right lead). Equation(22b) is valid in the bilinear response regime where δX₁≪X_1,0 and δX₂≪X_2,0 and the integral in equation(22a) becomes independent of the pumping contour.

First we analyze the NFN pump, where the pumped current is generated by adiabatic variation of the gate voltages V_l and V_r which change the chemical potential in the left and right leads of the junction[46],⁵ respectively (see figure1). The periodic modulation of the gate voltages is given by V_l(t)=V_l,0+δV_lcos(ωt) and V_r(t)=V_r,0+δV_rcos(ωt+ϕ). Calculating the derivatives of the reflection and transmission coefficients r_ll and t_rl with respect to α_l and α_r, substituting these derivatives into equation(23) and using ∂α_j/(e∂V_j)=tan(α_j)/|+μ−eV_j| (j=l,r), the angle-dependent pumped current for V_l,0=V_r,0≡V_g⁶ is given by

Here I^NFN₀≡ωe/(8π)sin(ϕ)(eδV_l/μ)(eδV_r/μ), Γ≡−(1+/μ−eV_g/μ)² and α_m is given by equation(21). Figure5 shows the angle-resolved pumped current I^NFN_p(α) (equation(24)) as a function of α for different values of energy /μ. By comparing figures3 and5 we see that whenever a new mode appears in the transport window, the pumped current exhibits a maximum and changes sign. As the energy increases, more and more angles of incidence contribute to I^NFN(α), which eventually becomes a smooth oscillatory function of α (see panels (e) and (f)).

The total pumped current I^NFN_p is obtained from equation(24) by integrating over all angles of incidence α:

In general, this integral cannot be evaluated analytically and we have obtained our results numerically. Figure6 shows I^NFN_p (in units of I^NFN₀) as a function of energy /μ. For low energies, the pumped current I^NFN_p is zero as no traveling modes are allowed in the junction. As we increase the energy, each time a resonant mode appears (see figure3) also the total pumped current exhibits a maximum and changes sign⁷. It is interesting to compare these features of the pumped current—vanishing current and change of sign at energies where a new resonant mode appears—to the behavior of the conductance in the same junction (see for more details theappendix). The conductance exhibits plateaus and, for large energies, oscillatory behavior as a function of /μ (see figuresA.1 andA.2). The position of the plateaus in the conductance corresponds to the position of vanishing pumped current. A characteristic difference between both currents is that while the conductance generally increases as a function of voltage, the pumped current changes sign. This difference can be used to distinguish pumped currents from conductance in these junctions.

Now we analyze the pumped current in the FNF-junction with parallel orientation of the magnetizations. In this system, the driving parameters are the magnetizations M_l and M_r in the left and right contacts, respectively, see figure2, where the periodic modulation is given by M_l(t)=M_l,0+δM_l cos(ωt) and M_r(t)=M_r,0+δM_r cos(ωt+ϕ). After calculating the derivatives of the reflection and transmission coefficients, substituting into equation(23), and using ∂α_j/∂M_j=ℏv_F/(|+μ|cos(α_j)) (j=l,r), the angle-dependent pumped current I^FNF_p(α) for M_l,0=M_r,0=M is found to be

Here I^FNF₀≡ωe/(8π)sin(ϕ)(ℏv_FδM_l/μ)(ℏv_FδM_r/μ), Γ'≡−(1+/μ)² and the polar angle of the wavevector in the middle region α_m is given by sin(α_m)=sin(α)−ℏv_FM/(|+μ|).

The total pumped current I^FNF_p is again obtained by integrating over all angles of incidence α. The behavior of I^FNF_p is similar to that of the pumped current in a NFN-junction (shown in figure6): also I^FNF_p exhibits maxima and subsequently changes sign at the same energies as I^NFN_p, where a new mode enters the transport window. There is, however, an interesting difference between both currents for normally incident carriers: whereas the pumped current in the NFN-junction vanishes for carriers at normal incidence, I^NFN_p(α=0)=0, the pumped current in a FNF-junction is given by I^FNF_p(α=0)=cos³(α_m)sin(2k_md)/(Γ'[1−sin²(α_m)cos²(k_md)]²). This difference arises because the two pumps are driven by different parameters (voltages in the NFN pump and magnetizations in the FNF pump) which leads to different interference patterns in the two junctions. In the voltage-driven NFN-pump destructive interference of normally incident Dirac fermions occurs (similar effects have also been predicted for graphene[37, 40]), while the magnetization-driven FNF-pump exhibits constructive interference of Dirac fermions at normal incidence.

Finally, we briefly analyze the behavior of the pumped current as a function of the width d of the middle region in the junctions. For energies below _c, the pumped current of the NFN-junction decays to zero as the width d increases (there are no resonant modes in the system). For energies larger than >_c, the pumped current oscillates as a function of width . Figure7 shows I^NFN_p as a function of for >_c. As expected, the pumped current I^NFN_p switches direction at the values of given by equation(18), where a new resonant mode enters the junction. This analysis also holds for a FNF-junction.

The conductance G^NFN of a topological insulator NFN-junction has been studied in earlier work by Mondal et al[13, 14], who predicted oscillatory behavior of G^NFN as a function of the applied bias voltage (see also figureA.1). In thisappendix we briefly review their results and then add a quantitative explanation for the oscillations of the conductance in terms of the resonant modes (discussed in section3). This explanation is crucial for understanding both the plateau-like behavior of the conductance and for comparing this with the behavior of the pumped current in section4. We also calculate and analyze the conductance in topological insulator FNF-junctions.

The general expression for the conductance G^NFN across the NFN-junction in terms of the transmission probability T^NFN(α) (see equation(20)) is given by

Here, ρ(eV )=|μ+eV |/(2π(ℏv_F)²) denotes the density of states, W is the sample width, and the integration is over the angles of incidence α.

FigureA.1 shows the conductance of the NFN junction as a function of the applied bias voltage eV (which plays the role of the energy here) for different values of the effective magnetization . For a given magnetization M, the conductance is zero for eV <_c, where the critical energy _c≡ℏv_FM/2−μ. Below this energy there are no traveling modes inside the barrier. Our results agree with previous results in the literature[13, 14]. We see from figureA.1 that the conductance changes from plateau-like to oscillatory as eV/μ increases. In order to provide an explanation for this behavior we first analyze the plateau-like regime in detail. The first plateau corresponds to the situation in which the first transmission mode appears at α=−π/2 (see also figure3 in the main text). As the energy increases, subsequent resonant modes appear and the conductance increases in a step-like manner. The plateaus are not sharp, since the modes are not sharply peaked, but have a distribution around a particular angle of incidence, see figure4. Once the bias voltage is large enough for there to be contributions from both positive and negative angles of incidence, the conductance becomes oscillatory. For very large bias voltages (eV ≫_c, not shown in the figure), the effect of the magnetic barrier disappears and the conductance G^NFN/G₀→1.

FigureA.2 shows the conductance as a function of eV for several values of applied gate voltages V_l=V_r≡V_g. As expected, the features of the conductance remain the same as V_gincreases, but the critical energy _c=ℏv_FM/2+eV_g/2−μ for the onset of the conductance increases and the spacing between consecutive resonant modes decreases. As a result the plateaus become narrower.

In the remaining part of thisappendix we study the conductance in a topological insulator FNF-junction, as shown in figure2. We consider both the junction with parallel and with anti-parallel magnetizations in the ferromagnetic regions. In the parallel configuration (using α_l=α_r≡α=sin⁻¹(ℏv_F(q+M)/|+μ|) and α_m=sin⁻¹(ℏv_Fq/|+μ|) in equation(16)), we find that the conductance is similar to the conductance of a NFN-junction, as displayed in figureA.1. A minor difference between the two is that in the FNF-junction the first resonant mode occurs for positive α (i.e. transverse momentum parallel to M) and as the energy increases, the resonances move towards negative values of the angle α. This, however, does not affect the total conductance as we sum over all possible angles of incidence, and the same analysis as presented above for the NFN-junction can be applied to understand the FNF-junction with parallel magnetization.

In the case of anti-parallel alignment of the magnetizations in the two ferromagnetic regions, the transmission probability T^FNF,AP(α_l,α_r)≡|t_rl(α_l,α_r)|² is given by[17, 18]

The conductance of the FNF-junction in the anti-parallel configuration is obtained from equation(A.2) by multiplying T^FNF,AP(α_l,α_r) with cos(α_r)/cos(α_l) and integrating over the allowed angles of incidence⁸, i.e. from α_c1=sin⁻¹(2ℏv_FM/(|+μ|)−1) to α_c2=sin⁻¹(2ℏv_FM/(|+μ|)+1):

with

FigureA.3 shows the conductance G^FNF,AP of the FNF-junction in the anti-parallel configuration (equation(A.3)). From the horizontal axis we see that the critical energy _c for the onset of the conductance is larger than in the corresponding parallel configuration (figureA.1). Moreover, as the bias voltage increases the conductance does not exhibit plateaus, but instead increases in an oscillatory manner. This oscillatory behavior can be understood from figureA.4, which shows G^FNF,AP(α_l,α_r) for four different values of eV (note that all the angles (α_l, α_m and α_r) can be expressed in terms of one angle, which we choose to be α_l). From figureA.4 we observe that the area under the curve oscillates as a function of the applied bias voltage, which leads to oscillations in the conductance.