SUPPLEMENTARY INFORMATION

Transcription

1 SUPPLEMENTARY INFORMATION doi: 1.138/nnano.9.4 All-electric quantum point contact spin polarizer P. Debray, J. Wan, S. M. S. Rahman, R. S. Newrock, M. Cahay, A. T. Ngo, S. E. Ulloa, S. T. Herbert, M. Muhammad and M. Johnson nature nanotechnology 1

2 supplementary information doi: 1.138/nnano.9.4 A. Conductance Measurement P D LG P S LG D S I RG P D RG P S (a) (b) Fig. S1. (a) Optical image of a Hall bar with a side-gated QPC at the centre (red circle). (b) Scanning electron micrograph of the x-y plane of a side-gated QPC. LG and RG are side gates created by cutting isolation trenches by wet etching. The arrow indicates the direction of the x-axis. The QPC devices were written by e-beam lithography at the centre of Hall bars made from InAs quantum-well structures. A constant voltage V DS at 17 Hz in the range 3 and 1 μv was applied between the drain (D) and source (S) contacts. The current I and the voltage drop between the Hall bar probes P D and P S were measured by two synchronized Lock-in amplifiers. All measurements were carried out in a screened room. Batteryoperated dc voltage sources were used to apply constant voltage to the side gates LG and RG for controlling the confining potential of the QPC. nature nanotechnology

3 doi: 1.138/nnano.9.4 supplementary information B. Influence of Perpendicular Magnetic Field Electron Density We first consider, for illustrative purposes, a transverse (y-axis), harmonic, electrostatic 1 confinement, U = m ω y. Electron propagation is along x-axis. In a perpendicular magnetic field B, the Hamiltonian H is given by ( py ) ( p + eby) 1 x H = + + m ω y. m m 1 h kx h kx E ( n, k x) = ( n + ) h ω + = E n +, m ( B ) m ( B ) ω = ω + ω c ω c = e B / m m ( B ) = m ω ω A magnetic field B causes an increase in the subband spacing as well as results in a magnetic effective mass m B ) > ( m. The 1D density of states per unit length including spin degeneracy is given by, ω m D ( E, ω) = π hω E E n 1 / Because of the increase in effective mass, the 1D density of states narrows and sharpens. The magnetic field thus flattens out the parabolic dispersion which acquires a dispersion less region for small k x. When the field is strong, ω C >> ω, Landau states localized well within the channel develop. The effect is most pronounced for the lowest subband. nature nanotechnology 3

4 supplementary information doi: 1.138/nnano.9.4 Assuming transport in the fundamental mode with the Fermi level E f in the lowest 1D subband, the 1D electron density n ( B ) = E f E1 D n ( B ) is given by, ω ( E,ω) d E = { m ( E f E )} π hω 1 1 /, 1 where E 1 = hω is the energy of the lowest 1D subband bottom. We are interested in the influence of magnetic field on the channel electron density when the.5 plateau occurs. This plateau results when the pinched off channel is gradually opened up by adjusting the gate voltages and the Fermi level crosses the lowest subband bottom from below and 3.5 n(b) (1 7 m - ) Harmonic Confinement B(T) Fig. S. Influence of magnetic field B on electron density n in the lowest 1D subband when the.5 plateau occurs. A harmonic confinement was considered. enters the first subband. Consider the point P of maximum curvature on the.5 plateau (Fig. 1c). This point, designated as the G.5 feature, results when the Fermi level has just entered the lowest 1D subband and is close to and above it. This is true irrespective of the value of E 1 and therefore of the magnitude of the applied perpendicular magnetic field. Let s say this happens when the Fermi level is ΔE = E f - E1 above the subband 4 nature nanotechnology

5 doi: 1.138/nnano.9.4 supplementary information bottom. For our devices at 4.K, a value of 1 mev for ΔE ( kt) seems reasonable. A plot of n(b) as a function of B is shown in Fig.S calculated with ω = 4 1 s, m =. 3m field.. The electron density is found to increase monotonically with magnetic The potential U(x,y) for a side-gated QPC can not be approximated by a harmonic confinement. A realistic model will be one with a flat bottom and sharply rising walls at the edges. The.5 plateau occurs close to pinch-off at fairly large negative bias voltages of the gates. The effective channel with is therefore much smaller than the lithographic channel width when it occurs. Because of the uncertainties involved, it is very difficult to estimate this value. We can use a guideline as follows. From our experimental results, we know that the confinement introduced by an applied magnetic field of 8T is not strong enough to create Landau levels and make the system behave like a DEG. This means the channel width is still smaller than the cyclotron diameter. From the measured electron density, we obtain a DEG Fermi level E f of 148 mev. Using the notion of classical turning points, we have E f ( 1 / ) m ω t =, c where ω c is the Larmor frequency for 8T and t = W/. W is the effective channel width. We can find an approximate value for W using the above relation. It turns out to be equal to 6 nm. This estimate corresponds to the maximum effective channel width when the.5 plateau occurs. A realistic U(x, y) was chosen to reflect as best as possible the rise at the transverse edges of the confining potential profile of a side-gated QPC. U(x, y) included the nature nanotechnology 5

6 supplementary information doi: 1.138/nnano.9.4 conduction band discontinuities at the InAs channel/vacuum interface. A square QPC of side 1 nm was considered with a transverse width of the flat bottom equal to 6 nm. A cosine function was chosen to represent a smooth transition from 4.5 ev (electron affinity for InAs) at the vacuum/channel interface to zero in the InAs channel over Δy nm. The potential is shown in Fig. S3. For such a potential an analytical solution is not possible. Numerical calculations of the areal electron density in the lowest 1D subband were carried out for magnetic fields, 4, 8, and 1T for E f -E 1 = 1 mev. The results are shown in Fig. 3d 4 U(eV) 1 5 y(nm) 5 x(nm) 1 Fig. S3. The potential U(x,y) of a model square QPC device of side 1 nm. The transverse confinement has a flat bottom of width 6 nm rising sharply at the edges. This potential is representative of the effective confinement of a real QPC when the.5 plateau appears. Effect of Switching Field Direction The spin polarization induced by LSOC is perpendicular to the device plane and its direction (up or down) depends on the direction of the current. For a fixed current direction it should be possible to align the perpendicular magnetic field in a direction 5 6 nature nanotechnology

7 doi: 1.138/nnano.9.4 supplementary information parallel or opposite to the spin polarization. An increase in the applied field is accompanied by an increase in the electron density and a corresponding decrease in the spontaneous spin polarization. If the applied field is strong enough to cause Zeeman spin polarization of its own and is opposed to the spontaneous spin polarization, there will be a critical value of the field when it will balance out the spontaneous spin polarization resulting in the disappearance of the.5 plateau. Further increase in the field will result in the reappearance of the.5 plateau with opposite spin polarization. If the applied field is parallel to the spontaneous spin polarization, there will be no balancing out. Any decrease in the spontaneous spin polarization due to increasing field will be compensated by a parallel and increasing Zeeman polarization and no disappearance of the.5 plateau is expected. In this case, the applied field may have no significant effect on the.5 plateau. If, however, the applied field is not strong enough to cause any Zeeman spin polarization of its own, it will have no direct influence on the spin polarization of the.5 plateau, but will do so indirectly because of its influence on the electron density. Since this influence is the same whether the applied perpendicular magnetic field is up or down with respect to the plane of the device, the effect on the device conductance will therefore be independent of the direction of the perpendicular magnetic field. Shubnikov de Haas oscillations (Fig. S4) measured at 4. K in a magnetic field up to 8T showed no evidence of any Zeeman spin splitting. This field strength is not therefore sufficient to cause any observable Zeeman spin polarization. The perpendicular magnetic field will have the same effect on the.5 plateau irrespective of its direction, consistent with our observations. nature nanotechnology 7

8 supplementary information doi: 1.138/nnano.9.4 C. Model potential U(x, y) The transverse confining potential energy of a side-gated QPC can be reasonably expected to be given by, ( y / ) U ( y ) = C F W for y > W / and zero otherwise. W is the width of the flat bottom and C is a constant that depends on the electron affinity at the channel/vacuum interface (trench wall). The function F represents the sharp rise of the potential at the transverse edges. The potential U(y) can be made asymmetric in y by applying asymmetric bias voltages to the two edges. The electrons are considered to propagate in the x-direction, along which the potential is assumed to be constant. The potential U(x, y) used for NEGF calculation of the spin density and conductance of a model QPC was chosen to reflect as best as possible the confining potential of the experimental side-gated QPC. The problem was kept tractable numerically by modelling only the central region of width (y-axis) 5 nm and length (x-axis) 5 nm of the experimental device. The details of the model potential U(x, y) are given below. The potential U(x, y) includes the conduction band discontinuities at the vacuum/inas channel interface. The cosine function is chosen to represent a smooth transition of the band discontinuity ( ΔE C ) from 4.5 ev at the vacuum region to in the InAs channel part, where 4.5eV is the electron affinity for InAs. The width of the transition region is around 8 nature nanotechnology

9 doi: 1.138/nnano.9.4 supplementary information 5nm. U(x, y) also includes the asymmetric side-gate voltage ( Δ V ) and drainsource voltage (V ds ). sg = V I V II x Δ E C q V DS + V I + V T, 5 ΔE C y x y cos π + 1 q VDS + ( V I V I I ) + VT, x y U ( x, y ) = q V DS + ( V I V I I ) + VT, ΔE C y x y cos π + 1 q V DS + ( V I V I I ) + VT, x Δ E C q V DS + V I I + V T, 5 y < 1.667nm 1.667nm y < 7.5nm 7.5nm y < 17.5nm 17.5nm y < 3.333nm y < 5nm D. Shubnikov de Haas Oscillations 5 InGaAs/InAs QW R xx (Ω) T = 4. K /B(T -1 ) Fig. S4. Shubnikov de Haas measurements on a Hall bar ma de from InGaAs/InAs QW structure used in this work. Zeeman spin splitting or any beating pattern is absent. nature nanotechnology 9

10 supplementary information doi: 1.138/nnano.9.4 E. The Free-Electron Hamiltonian The free-electron or single-particle Hamiltonian H of a QPC device can be written as, H = H + V ( y ) + H h kx H = + U ( y ) m 1 V ( y ) = m ω r H = β σ. SO ( y y ) r r ( k V ( y ) ) x SO (1) U(y) is a hard-wall potential, zero for W/ < y < W/, and infinity otherwise. W is the lithographic width of the QPC. The potential V(y) due to gate bias voltages is assumed to be harmonic. A numerical solution of the Schrödinger equation for H can be carried out by expanding the wave function Ψ ( x, y ) in terms of the eigenfunctions ψ ( y ) of the bare Hamiltonian H. Since we are concerned with transport only in the fundamental mode, we have two eigenvalues, one each for the up and down spin species with the ikx eigenfunctions Ψ σ ( x, y ) = ψ 1 ( y ) e χ σ. The eigenvalues are given by, n E E = E = E V ( y) 1 1 V ( y) 1 + βk βk x x 1 dv ( y) / dy1 1 dv ( y) / dy1, () where 1 E is the lowest eigenvalue of H. Equation () shows energy spin splitting for non-zero values of k x and 1 dv ( y) / dy1. The latter is non-zero only when the effective confining potential is asymmetric. A very small spin splitting is observed for non-zero k x for the extreme asymmetric case of y = W/ with W = 3 nm, ω = 4.x1 13 s -1, β = 1 nature nanotechnology

11 doi: 1.138/nnano.9.4 supplementary information 1.x1-18 m, and m =.3m. Though equation () shows a spin splitting for, it does not affect either the group velocity or the density of states of the two spin species. As a result, we do not expect any additional structure in the conductance of the device. Figure S5 shows the conductance calculated using scattering-matrix formalism. The conductance does not show the.5 plateau or any other feature indicative of spin k x 1. G T G(e /h).75.5 G = G VG(V) Fig. S5. Conductance G from a solution of the single-particle Hamiltonian showing the complete absence of spin splitting. V G is a common-mode potential that shifts the potential energy V(y). G T is the sum of spin-up and spin-down conductance. polarization. The single-particle Hamiltonian is not, therefore, adequate for explaining the occurrence of the.5 plateau. This is not surprising. The Hamiltonian H is invariant under time reversal. A consequence of this time-reversal invariance (TRI) for a system with spin ½ particles is that each single-particle energy level must have at least a two-fold Kramers degeneracy. TRI must be broken for spin polarization to be possible. One way of doing it would be to add by hand a term to the free-electron Hamiltonian that nature nanotechnology 11

12 supplementary information doi: 1.138/nnano.9.4 depends on spin polarization and takes into account the experimentally observed fact of LSOC induced spin polarization. F. Hamiltonian with e-e interaction The Hamiltonian H σ of the system for electrons with spin σ is a sum of the noninteracting Hamiltonian H and the total interaction self-energy ( r ) given by σ σ int H H σ σ σ int = H σ h = m r ( ) = γn ( r ) σ int ( k + k ) + x σ r ( ) r y + U ( y ) + H r The parameter γ indicates the interaction strength and n (r ) is the density of electrons with spin -σ. In the Thomas-Fermi (TF) approximation, γ for a DEG is directly proportional to the screening length and has an approximate value 1 of ( h / m ) σ SO π. Beyond the TF approximation, the screening length increases as electron density goes down; γ can be fairly large for a low-density 1D electron system. The interaction selfenergy acts like an additional local potential, different for the different spin directions. A spin-up electron encounters a potential proportional to the density of spin-down electrons and vice versa. Hence there is a repulsive interaction only between electrons with opposite spin directions. Any initial imbalance between the densities of spin-up and spindown electrons is enhanced by the e-e interaction. Such an imbalance or spin polarization can be induced either by a small external magnetic field or, in the absence of such a field, by LSOC when the confining potential of the QPC is asymmetric.. 1 nature nanotechnology

13 doi: 1.138/nnano.9.4 supplementary information G. The.7 Structure Soon after the discovery of the conductance quantization, 1, an additional plateau was observed at G.7G in the absence of any magnetic field in a AlGaAs/GaAs QPC. 3 Since then, this anomalous plateau, referred to as the.7 Structure by the scientific community, has been observed in both short (QPC) and relatively long 1D wires. The.7 structure shows the following distinct experimentally observed features: An anomalous temperature dependence showing increase of the conductance to the unitary value of G as the temperature is lowered. 3,4 Conversely, as the temperature increases the structure becomes more pronounced. In a parallel magnetic field the structure evolves smoothly into the Zeeman spinsplit plateau value of.5g. 4,5 A drain-source bias increases the conductance of the structure. 6 The formation of a zero-bias anomaly (ZBA) in the non-linear differential conductance. 5,7,8 Ever Since the observation of the.7 structure a number of theoretical models have been proposed and attempts have been made to explain its origin based on static 9 and dynamic 1,11 (Kondo effect) spin polarization, Wigner crystallization, 1 ferromagnetic spin coupling, 13 and very recently the Rashba spin-orbit coupling 14. Its origin is still debated even after a decade of research. 15. The zero-bias anomaly and the associated enhancement of the linear conductance as the temperature is lowered and its disappearance in applied parallel magnetic field are hallmarks of the Kondo effect in quantum dots 1,11. The Kondo model requires a local nature nanotechnology 13

14 supplementary information doi: 1.138/nnano.9.4 moment or localized spin. Though a density functional theory (DFT) has been used 16 to show that a dynamical local moment with a net of one electron spin forms in the vicinity of the QPC barrier, it is difficult to understand how a localized spin can form in an open QPC. Moreover, recent studies 8 show that the ZBA characteristics in quantum wires are inconsistent with the spin-one-half Kondo physics. The static spin-polarization models 9,17,18 deserve our special attention. A static spin polarization has been experimentally found to be associated with the.7 structure observed in a hole QPC. 5 References 1. D. A. Wharam et al., J. Phys. C. 1, L9 (1988).. B. J. Van Wees et al., Phys. Rev. Lett. 6, 848 (1988). 3. K. J. Thomas et al., Phys. Rev. B 58, 4846 (1998). 4. S. M. Cronenwett, et al., Phys. Rev. Lett. 88, 685 (). 5. L. P. Rokhinson et al., Phys. Rev. Lett. 96, 1566 (6). 6. A. Kristensen et al., Phys. Rev. B 6, 195 (). 7. R. Danneau et al. Phys. Rev. Lett. 1, 1643 (8). 8. T.-M. Chen et al., Phys. Rev. B 79, (9). 9. D. J. Reilly, Phys. Rev. B 7, 3339 (5). 1. L. Kouwenhoven and L. Glazman, Physics World, January R. M. Potok et al., Nature 446, 167 (7). 1. K. A. Matveev, Phys. Rev. Lett. 9, 1681 (4). 14 nature nanotechnology

15 doi: 1.138/nnano.9.4 supplementary information 13. K. Aryanpour and J. E. Han, Phys. Rev.Lett.1, 5685 (9) and references therein. 14. J. H. Hsiao et al., Phys. Rev. B 79, 3334 (9). 15. For a review, see J. Phys. Condens. Matter, 3 April K. Hirose et al. Phys. Rev. Lett. 9, 684 (3). 17. A.A. Starikov et al., Phys. Rev. B 67, (3). 18. P. Havu et al., Phys. Rev. B 7, 3338 (4). H. The Hanle Effect The observation of the.5 plateau in the ballistic conductance of a side-gated QPC is a signature of complete spin polarization by the QPC. This, however, is indirect evidence. A direct evidence of the spin polarization can be obtained by electrical measurement of the Hanle effect 1. A magnetic moment when placed in a uniform perpendicular magnetic field experiences a torque, which causes a precession of the moment about the field with the Larmor frequency, ω, where B is the magnetic field and m L = e B / m the electron effective mass. This is the Hanle effect. If there is no spin precession, a spin polarization simply decays at the spin relaxation rate and randomizes over the spin coherence length. The Hanle effect adds another relaxation mechanism. It has traditionally been measured using optical techniques,3. To electrically measure the Hanle effect and detect the spin polarization generated by a side-gated QPC, one needs a QPC spin polarizer and a QPC spin analyzer separated by a distance smaller than the spin coherence length (Fig. S6a). nature nanotechnology 15

16 supplementary information doi: 1.138/nnano.9.4 When the polarizer and the analyzer orientations are set parallel, the current through the device can be modulated by controlling the spin precession with the applied magnetic field. A 18 precession of the electron spin when it arrives at the analyzer will ideally give zero current or a low minimum in a real situation. Maximum current will result for 36 precession. An oscillating current through the device as a function of the applied magnetic field will be a direct evidence of electron spin polarization. There is no current oscillation if the electrons are not spin polarized. Figure S6b shows the scanning electron micrograph of a preliminary device incorporating two side-gated QPCs separated by a distance of few microns which can be a 1D Channel b SG SG SG SP L X B SA QPC 1 (SP) Central QPC channel (SA) SG SG SG Fig. S6. (a) Schematic representation of the principle of Hanle effect showing spin precession under the influence magnetic field B applied perpendicular to spin. SP and SA are, respectively, the spin polarizer and analyzer. The green and red arrows indicate spin orientation. (b) Scanning electron micrograph of a dual QPC device that can be used to electrically measure Hanle effect. The width of the central channel can be adjusted by side gates to ensure transport in 1D fundamental mode. 16 nature nanotechnology

17 doi: 1.138/nnano.9.4 supplementary information made shorter, if needed. QPC1 can be set to act as the spin polarizer and QPC as the analyzer of Fig. S6a. The width of the channel between the QPCs can be tuned by the side gates SGs to ensure 1D transport. Such a device can used to electrically measure the Hanle effect to obtain direct evidence spin polarization by the QPCs. References 1. M. I. Dyakonov and V. I. Perel, Optical Orientation in Modern Problems in Condensed Matter Sciences, edited by F Meier and B Zakharchenya (North-Holland, Amsterdam, 1984) and references therein.. O. Maksimov, X. Zhou, M. C. Tamargo, and N. Samarth, Physica E: Lowdimensional Systems and Nanostructures, 3, 399 (6). 3. M. Furis et al., New Journal of Physics 9, 347 (7). nature nanotechnology 17