Pseudospin-valve effect on transport in junctions of three-dimensional topological insulator surfaces

We show that the surface states of pristine 3D topological insulators (TIs) are analogs of ferromagnetic half metals due to complete polarization of an emergent momentum independent pseudospin (SU(2)) degree of freedom on the surface. To put this claim on firm footing, we present results for TI surfaces perpendicular to the crystal growth axis, which clearly show that the tunneling conductance between two such TI surfaces of the same TI material is dominated by this half metallic behavior leading to physics reminiscent of a spin-valve. Further using the generalized tunnel magnetoresistance derived in this work we also study the tunneling current between arbitrary TI surfaces. We also perform a comprehensive study of the effect of all possible surface potentials allowed by time reversal symmetry on this spin-valve effect and show that it is robust against most of such potentials.

Introduction: Three dimensional topological insulators (3-D TI) 1-4 represent a distinct class of 3-D band insulators which have topologically protected metallic surface states. Nontrivial topology of band structures of these materials leads to the immunity of these surface states against a variety of disorder and interaction potentials which has made them a topic of central focus since thier discovery. Possibility of surface magnetization and related spin textures of Fermi surface in these materials due to external doping 5-10 has also been a topic of great interest. Here we argue that, even the pristine TI surface can act like a ferromagnetic half metal 11 , though not due to spin but due to a pseudo-spin D.O.F. A natural way to confirm half metallic behaviour would be an observation of perfect spin blockade (complete suppression of tunneling due to orthogonality of the states in the spin sector) in tunnel conductance between two half metals with opposite directions of polarisation and then a lifting of the spin blockade by tilting the polarisation direction of one half metal with respect to another. We present analytic results which show appearance of such (pseudo) spin blockade physics and its subsequent lifting due to the rotation of pseudo-spin polarisations of surface states which is the central result of this letter. The low energy physics of popular TI materials like Bi 2 Se 3 can be described by a four band model arising from spin (σ) and orbital parity (τ ) quantum numbers of electrons belonging to a unit cell of TI crystal. Zhang et.al. 12,13 showed that a planar boundary of TI like Bi 2 Se 3 has surface states whose spectrum is described by two sets of independent SU(2) D.O.F which can be expressed as non-trivial combinations of σ and τ . Of these two SU(2) D.O.F, one of them is dispersing 14 while the other one is non-dispersing. This fact is natural as the surface state itself comes into being by freezing (making it momentum independent) two out of four bulk D.O.F (four band model) on the planar boundary. This makes the frozen SU(2) D.O.F fully polarised and hence surface state is endowed with a half metal like character. To proceed further with the discussion an angle θ is defined between the vector normal to a given planar crystal surface and the crystal growth axis of Bi 2 Se 3 which helps A and B of TI surfaces with an insulating barrier between them, with the crystal growth axis of both the samples along the z-axis. Sample 1 in both the junctions has the surface with τx = −1 exposed, shown by the blue color, where as sample 2 has the surface with τx = ±1 exposed in Junction-A(B) shown by the red (blue) color. (b): The two new surfaces exposed on cleaving a TI crystal perpendicular to the crystal growth axis have opposite orbital polarisation analogous to a bar-magnet as shown in (c).
in distinguishing different surfaces. For the simple case of θ = 0 (top surface) and θ = π (bottom surface), the dispersing and frozen D.O.F are given by σ and τ operators respectively. The direction of spin, σ, is determined via the spin-momentum locking angle while the τ points solely along the positive x (negative x) direction for the top (bottom) surface 12,15 for all momentum states hence defining a τ polarised half metal. 16 . It is interesting to note that the τ -polarisation is exactly opposite on the opposite surfaces (θ = 0 and θ = π surfaces respectively) 12,15 hence leading to distinction between the top and bottom surfaces which is very similar to the distinction of north and south poles of a bar magnet. It can be understood from the observation that, when a bar magnet is broken into two halves, it exposes two new ends with opposite polarity such that each new bar magnet has opposite polarity at its two ends. In an identical fashion when a film of TI with exposed θ = 0(top surface) and θ = π(bottom surface) is sliced parallel to θ = 0 plane creating two new TI films, the newly exposed surfaces will have exactly opposite τ x -polarisations consistent with the scenario of a bar magnet (see Fig. 1). The fact that the different surfaces of the TI are very distinct from one another due arXiv:1502.07871v1 [cond-mat.mes-hall] 27 Feb 2015 to this frozen D.O.F has been mostly underestimated and ignored. One of the most important outcome of our analysis is, though frozen, these D.O.F can have strong influence on transport across junctions of TI via physics analogous to the spin blockade physics. Surface states: In a coordinate system, where the exposed TI surface lies on the x-y plane with its crystal growth axis at an angle θ to the z-axis, the Hamiltonian governing the surface states can be written in the form where S is the dispersing SU(2) D.O.F given by and sin θ. Now, the exact wavefunction for the surface state of θ and π + θ surface can be obtained form the two fold degenerate eigenspace of H s (θ) by demanding the surface state to be an eigenstates of T x θ 12,15 with eigenvalue ±1 respectively where T θ represents the pseudospin D.O.F given by Hence the very construction for surface state for any θ naturally involves imposing a complete T θ polarisation on all the surface electron states independent of their momentum, making it a perfect analog of an ferromagnetic half metal where spins are fully polarised independent of momentum. Tunneling current: The pseudo-spin half metal described above manifests itself in the tunneling current in a junction between two TI surfaces. We consider a translation invariant tunnel Hamiltonian describing tunnelling between two planar surfaces of TI with arbitrary θ. Translation invariance leads to a momentum resolved tunnel-Hamiltonian density given by where c 1/2,k is the fermionic annihilation operator for the two TI surfaces and z k is the overlap between the two surface state wave functions at momentum k. The expectation value of the current density operator, defined 17 , calculated perturbatively to leading orders in J takes the form where ∆n F (k, µ 1 , µ 2 ) = n F (E 1 (k), µ 1 ) − n F (E 2 (k), µ 2 ) is the difference in the Fermi functions of the two surfaces, E 1/2 (k) being the respective energy dispersions. Here the expression for |z k | 2 is given by This decomposition is both nontrivial and instructive as it allows for interpretation of the total tunneling current in terms of individual responses form σ, τ sectors and a correlation term between them. The surface states being eigenstates of T x θ = α θ τ x + β θ τ y ⊗ σ y lead to a momentum independent part in |z k | 2 given by |z f | 2 ≡ τ x 1 τ x 2 + τ y ⊗ σ y 1 τ y ⊗ σ y 2 , which is representative of the overlap between the frozen sectors of the respective surfaces, hence it leads to the identification of the contribution to the total tunneling current from the pseudo-spin blockade physics. Junctions of θ = 0, π surfaces: For the θ = 0 and π surfaces, S θ and T θ reduce to the spin σ and the orbital pseudo-spin τ respectively. Consequently the expression for |z k | 2 in Eq.(6) simplifies to (7) Clearly the first term in Eq. (7) is a representative of the half metallic behaviour of the surface state and hence is common to all momentum modes. The second term represents the momentum dependent overlap of spin orientations of the spin-momentum locked states on the two surfaces. Now, we know that τ points along positive and negative x-axis for θ = 0 and θ = π surfaces for a given TI crystal. Hence for a junction between θ = 0 surface of one TI with the θ = π surface of another TI, the first term in Eq.(7) vanishes identically ( τ 1 · τ 2 = −1) leading to a perfect pseudo-spin blockade which directly stems from the half metallic behaviour discussed above (see Junction-A in Fig.(1) ). As the frozen pseudo-spin for θ = 0, π surface is give by τ which originates form orbital symmetries of unit cell, we call this situation an orbital blockade. Next we note that, if both the participating surfaces at the junction are θ = 0 surfaces for their respective crystals (see Junction-B in Fig.(1)), the orbital blockade is lifted ( τ 1 · τ 2 = 1). But, even if we start with two identical crystals with their crystal growth axis being parallel (taken as z-axis) to each other, still we have to rotate the crystal growth axis of one of the crystals with respect to the other by π so that the θ = 0 surfaces of the two crystals could face each other hence forming the junction. A careful observation immediately reveals that the π rotation of the crystal growth axis leads to a perfect orthogonality between the spins (σ) of the electron states on the two surfaces for a given momentum. Hence, though the orbital blockade gets lifted in the Junction-B but a spin blockade ( σ 1 (k) · σ 2 (k) = −1) arises leading to zero tunnelling current. So, we have zero tunnel To contrast the presence and absence of the orbital blockade in Junction-A and -B respectively, the spin (σ) blockade has to be lifted in Junction-B. This can be achieved in two ways (a) by applying an in-plane magnetic field which leads to an opposite shift of the Fermi surface of the two surface states or (b) by strong doping which is large enough to push the Fermi surface into the hexagonally warped regime of the energy spectrum 18 . B-field induced spin blockade lifting: In presence of magnetic field B = B/2ŷ parallel to the plane of junction (x-y plane), the spin-momentum locked massless Dirac Hamiltonian of θ = 0 surface given by v (σ × k) z gets modified to v k y σ x − (v k x +B)σ y whereB = gµ B B/2. Hence in effect the k = 0 point (Dirac point) of the spectrum gets translated along the k x -axis by an amountB/2v . However, since the other planar surface of the junction has its spin texture opposite to the first surface, the Dirac cone for the other surface shifts along k x by the same amount but in the opposite direction. For a given energy where is measured form the neutrality point of the Dirac spectrum, the two Fermi surfaces intersect at two points given by {0, ±(( 2 −B 2 /4)/v ) 1/2 }. The spin textures on the Fermi surface at these two points are clearly not anti-parallel (inset of Fig.(2)) and hence have a finite overlap, thus lifting the spin blockade allowing a finite current in Junction-B, however the presence of orbital blockade still keeps the current identically zero in Junction-A. In the linear response regime the expression of the total current in Junction-B is given by and the corresponding differential conductance (G) is plotted as function of the magnetic field B in Fig.(2).
Here µ represents the average chemical potential of the 3. The current,Ĩw, (normalized by the current for |z k | 2 = 1 ∀k) for Junction-B and |z k * | 2 (θ k * = 0) is plotted against the average chemical potential µ. The plot shows that it is linearly proportional to each other. The inset shows the behaviour of the spins on the θ k * = 0 line for the two surfaces (shown by the two color of the vectors) on the Bloch sphere for three different µ.
two surface forming the junction. The differential conductance shows a monotonically increasing behaviour with the magnetic field. This can be understood from the inset of Fig.(2), as B is increased, the relative shift between the two Fermi surface increases, and so does 1 + σ 1 · σ 2 at the points of their intersection. In the limiting case when the two Fermi surfaces touch at a single point, the two spins at the point of touching are exactly parallel to each other leading to a maximal current. This limiting value of B is given by 2µ/gµ B . Any further increase in B causes the two Fermi surfaces to completely move away from each other leading to zero current. Spin blockade lifting by warping: Another alternative to lifting of the spin blockade is to dope both the samples well enough such that the hexagonal warping effects get pronounced 18 . The warping term, which couples to the out-of-plane spin polarisation, induces a 3-fold symmetric pattern in the z-spin (out-of-plane ) polarisation with the Fermi surface breaking up into six symmetric regions of alternating positive and negative z-polarisation. On the top surface, the momenta with polar angle θ k * = nπ/3 where n goes form zero to five have maximal out-of-plane spin polarisations with its sign given by (−1) n . Note that, even in presence of warping the orthogonality of spin textures for the two participating surface states for a given momentum direction leads to zero current. Here we have assumed that the doping is same for both the surfaces. But, if the second sample is rotated by an angle π/3 about the z-axis, then the z-component of spin polarisation for a state from either surface with the same in-plane momentum direction have the same magnitude and sign. Thus this rotation maximally lifts the spin blockade leading to finite current in Junction-B. Under the π/3 rotated situation, an increased doping leads to a increased warping effect which leads to an increase of overlap of the spin polarisation of the surface states on the two surfaces with same momenta. Hence the tunnelling current grows as a function of doping(see Fig.(3).
For obtaining a geometric interpretation of the effect, maximally overlapping spins of the two surfaces state at momentums corresponding to states with maximal out of plane polarisation (which contribute maximally to the tunnelling current) is schematically shown on the Bloch sphere as a function of doping in inset of Fig.(3). Orbital blockade lifting by θ = 0 surface: Lets us consider a junction for which the surface-2 corresponds to θ = 0 while surface-1 corresponds to θ = θ ( = 0, π)(see Fig.(1)), then the corresponding half-metallic polarisations correspond to T x θ=0 and T x θ=θ respectively. As mentioned before, the half-metallic contribution to the tunneling current is given by |z f | 2 which turns out to be τ x 1 τ x 2 as τ y ⊗ σ y θ=0 = 0. Since the eigenstates of T x θ=0 and T x θ=θ operators are neither parallel nor orthogonal, the pseudo-spin blockade naturally gets lifted. Note that, for having finite current in momentum conserved tunnelling, the Fermi surfaces of the two TIs must have intersections in the momentum space within the bias window. Failure of the Fermi surfaces to meet this condition leads to a zero current which we call the Fermi surface blockade. Further note that the Fermi surface for a surface states corresponding to an arbitrary value of θ is elliptical with its eccentricity depending on θ. In fact, the presence or absence of the Fermi surface blockade depends on θ and also the relative doping between the two samples. For a given energy E measured from the neutrality point of the spectrum of the surface-1, the points of intersection of the Fermi surfaces depend on θ and doping D of the spectrum of surface-2 relative to the first (see Fig.(4)). Such intersections exist only if |ξ| ≤ 1 3 )], with |ξ| = 1 corresponds to the limiting cases when the semimajor/minor axes of the elliptic Fermi surface equals the radius of the circular one. For |ξ| < 1 there are four intersections in the momentum space parametrized in polar coordinates as k * = (k * , θ * k,i ); i ∈ Z : 1 ≤ i ≤ 4, where k * = (E − D)/ v , θ * k,1(2) = cos −1 ( ±(1 + f )/2) and θ * k,3(4) = θ * k,1(2) + π. These solutions are substituted in Eq. (6), and the expression in Eq.(5) is evaluated to calculate the current which is normalized by the current that would have been present had the two spinors been paral- FIG. 5. The normalized current (Ĩ) is plotted as a function of θ for four different doping. To lift the Fermi surface blockade at higher negative dopings of the circular Fermi surface, one needs to deviate more from θ = 0 as eccentricity of the elliptical Fermi surface increases with θ till π/2 and then goes down. The asymmetry in the current about θ = π/2 is understood by looking at the contribution of the different sectors to the current lel in the SU(4) space to define a dimensionless quantity (Ĩ) plotted in Fig.(5). The current is always zero for D > 0, due to the aforementioned Fermi surface blockade, which is also responsible for the extended regions of zero current observed for negative doping. The contribution to the current from the three different terms in Eq. (6) is also plotted, which shows that the contribution of the spin sector is symmetric about θ = π/2, in contrast to the orbital sector and the correlation term, which are antisymmetric. This leads to a current variation which is neither symmetric nor antisymmetric about θ = π/2. Choosing one of junction surfaces to be the θ = 0 surface results in overlap of frozen sector |z k | 2 , reducing to τ 1 · τ 2 which indeed varies form −1(orbital blockade) to +1 (complete lifting of orbital blockade) as a function of θ (see Fig.(5)). Hence the asymmetry of current about θ = π/2 can be taken as manifestation of pseudo-spin blockade physics arising form tunnelling between pseudo-spin half metals. Conclusion: A tremendous amount of effort has gone into understanding and exploring the surface states of 3D TIs but most of these studies primarily focuses on the dispersing D.O.F. The frozen D.O.F discussed in the letter has been completely ignored as they were believed to be of no observable consequence. On the contrary we have show that electrical transport in junctions of TIs are actually dominated by the frozen D.O.F. We have shown that the presence of the frozen D.O.F leads to a half metallic behaviour of the surface which manifests itself via spin-valve type electrical transport properties. We have discussed three different possibilities of probing this half metallic behaviour of surface states in electrical transport set up.