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by: Ashley Kunze


Ashley Kunze
GPA 3.61


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This 3 page Class Notes was uploaded by Ashley Kunze on Thursday October 22, 2015. The Class Notes belongs to CHEM 223 at University of California Santa Barbara taught by Staff in Fall. Since its upload, it has received 11 views. For similar materials see /class/226957/chem-223-university-of-california-santa-barbara in Chemistry and Biochemistry at University of California Santa Barbara.




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Date Created: 10/22/15
VOLUME 83 NUMBER 9 PHYSICAL REVIEW LETTERS 30 AUGUST 1999 Spin Hall Effect J E Hirsch Department of Physics University of California San Diego La Jolla California 920930319 Received 24 February 1999 It is proposed that when a charge current circulates in a paramagnetic metal a transverse spin imbalance will be generated giving rise to a spin Hall voltage Similarly it is proposed that when a spin current circulates a transverse charge imbalance will be generated giving rise to a Hall voltage in the absence of charge current and magnetic eld Based on these principles we propose an experiment to generate and detect a spin current in a paramagnetic metal PACS numbers 7215Gd 7361At Consider the spontaneous or anomalous Hall effect 1 In ferromagnetic metals the Hall resistivity trans verse electric eld per unit longitudinal current density is found to be empirically tted by the formula pH ROB 47TRSM l in cgs units with B the applied magnetic eld and M the magnetization per unit volume R0 is the ordinary Hall coef cient and Rs the anomalous Hall coef cient experimentally found to be generally substantially larger than the ordinary Hall coef cient as well as strongly temperature dependent Within models that assume that the electrons giving rise to magnetism in ferromagnetic metals are itinerant a variety of mechanisms have been proposed to explain the origin of the coef cient R 3 These include skew scattering by impurities and phonons and the side jump mechanism 1 In early work it was also proposed that the effect will arise in the absence of periodicitybreaking perturbations 2 but this is generally believed not to be correct 1 In this paper we will not discuss the origin of the anomalous Hall effect 3 Rather we take the exis tence of the effect in ferromagnetic metals as experimental proof that electrons carrying a spin and associated mag netic moment experience a transverse force when they are moving in a longitudinal electric eld for any of the rea sons listed above or others If there is a net magnetization in the system there will be a magnetization current asso ciated with the ow of electric current and the transverse force will give rise to a charge imbalance in a direction perpendicular to the current ow and hence to an anoma lous Hall effect Consider then the situation where no magnetization ex ists that is a paramagnetic metal or doped semiconductor or a ferromagnetic metal above its Curie point carrying a charge current in the x direction The electrons still carry a spin and the same scattering mechanisms that gave rise to the anomalous Hall effect in the magnetic case will scatter electrons with spin up preferentially in one direc tion perpendicular to the ow of current and spin down electrons preferentially in the opposite direction Here we have in mind a slab geometry as usually used in Hall effect experiments and spin up and spin down directions are de 1834 0031 9007 99 839 183441500 ned perpendicular to the plane of the slab Because there is an equal number of spin up and spin down electrons no charge imbalance will result but we argue that a spin im balance will there will be an excess of up spins on one side of the sample and of down spins on the opposite side The situation is depicted schematically in Fig 1 Although it may appear that if there is spin rotational invariance the spin up and down directions are not well de ned we argue that the slab geometry naturally de nes such directions The effect can be simply understood as arising from spinorbit scattering Consider 4 a beam of unpolarized electrons incident on a spinless scatterer with potential v Vcr Vsr539 Z 2 with 539 and L the electron s spin and orbital angular momentum respectively The term Vsr is the usual spinorbit scattering potential 4 proportional to the gradient of the scattering potential The scattered beam Hall effect I o L y EF VSH o L Y FIG 1 The charge carriers are assumed to be electronlike In the Hall effect the Lorentz force on the moving charges causes charge imbalance in the spin Hall effect skew scattering of the moving magnetic moments causes spin imbalance in a direction perpendicular to the current ow In the Hall effect the Fermi levels for up and down electrons are the same and the difference in the Fermi levels at both edges of the sample is the Hall voltage VH In the spin Hall effect the difference in the Fermi levels for each spin at both edges of the sample is VSH but it is of opposite sign for spin up and down electrons 1999 The American Physical Society VOLUME 83 NUMBER 9 PHYSICAL REVIEW LETTERS 30 AUGUST 1999 will be spin polarized with polarization vector 4 e fgfgA P n 3 f f2g2 0 where 13 is a unit vector perpendicular to the scattering plane in direction ki gtlt kf with kikf incident and scattered wave vectors f and g are respectively spin independent and spindependent parts of the scattering amplitude 4 13 has opposite signs for particles scattered to the right and left of the scatterer hence there is a left right asymmetry to the spin polarization of the scattered beam whose sign depends on the sign of Vsr In the geometry considered here the scattering plane is de ned by the plane of the slab since there is considerably more phase space for scattering in that plane than perpendicular to it Furthermore in a crystal preferred spin directions may arise from crystalline anisotropy and it may be useful to consider a single crystal sample where one such direction is perpendicular to the slab Finally a preferred spin direction is also de ned by the magnetic eld generated by the current ow which in the slab geometry will point predominantly in the z direction on half of the slab along the y direction and in the z direction on the other half This magnetic eld will contribute an additional spin imbalance which may add or subtract to the one discussed here depending on the sign of the skew scattering mechanism We will not be interested in this component of the spin imbalance for reasons discussed below Note also that no spin ip scattering can occur when the spin of the incident particle is perpendicular to the scattering plane 4 so that in the planar geometry multiple scattering events with a scattering potential of a given sign will simply enhance the leftright asymmetry In the case of the ordinary Hall effect the charge imbalance results in a difference in the Fermi levels of both sides of the sample and hence a voltage VH which can be measured with a voltmeter In the case under discussion here the Fermi levels for each spin electrons will also be different on both sides of the sample but the difference will be of opposite sign for both spins How can one detect this spin voltage VSH or equivalently the associated spin imbalance One possible way would be to measure the difference in magnetization at both edges of the slab This may per haps be achieved by using a superconducting quantum interference device microscope 5 with high spatial reso lution that can measure local magnetic elds However it would be necessary to separate the contributions from the effect discussed here and the magnetic eld generated by the current ow which is likely to be dif cult because the latter one should be much larger A more interesting way follows from the analogy with the ordinary Hall effect In that case if the two edges of the sample are connected by a conductor a charge current will circulate since the electrons in the connecting conductor do not experience the Lorentz force felt by the electrons in the longitudinal current Similarly in our case we argue that when the edges of the sample are connected a spin current will circulate This spin current will be driven solely by the spin imbalance generated by the skew scattering mechanisms affecting the longitudinal current and not by the component of the spin imbalance which is due to the magnetic eld originating in the current ow How does one detect such a spin current We may use the same principle that allowed the spin imbalance to be created in the rst place When the two edges of the sample are connected and a spin current circulates a transverse voltage will be generated that can be measured by a voltmeter The situation is schematically depicted in Fig 2 Let us consider some experimental parameters First the width of the sample L needs to be smaller than the spin diffusion length 55 55 is the length over which spin coherence is lost due to scattering processes that do not conserve spin We will rely on the seminal work of Johnson and Silsbee 6 JS who studied spin current ow between a ferromagnet and a paramagnet aluminum JS estimated 55 450 um at T 43 K and 55 170 um at T 366 K in their Al sample which had residual resistivity ratio of about 1000 We will assume for de niteness a sample of Al as in the JS experiment of width L 100 um with resistivity of order p 27 X 103 u cm at low temperatures The magnetization associated with the spin up elec trons in the sample is M mug with In the density of spin up electrons and M3 the Bohr magneton If only up electrons were present when a longitudinal current den sity jx ows an anomalous Hall voltage VH 4RsijnTLB would be generated with L the width of the sample and Rs the anomalous Hall coef cient Equation 4 gives also the spin Hall voltage for spin up electrons that will be generated and an equal one with opposite sign will result for the spin down electrons in the paramagnetic FIG 2 A transverse strip of width 1 connects both edges of the slab A spin current will ow and skew scattering will cause negative charge to accumulate on the left edge upstream from the primary current jx A charge imbalance will result and an electric potential that can be measured with a voltmeter 1835 VOLUME 83 NUMBER 9 PHYSICAL REVIEW LETTERS 30 AUGUST 1999 case Hence we obtain for the spin Hall voltage VSH ZWRSijn LB with n the total conduction electron concentration To obtain an estimate of the magnitude of the effect we will simply assume that Rs is the same as the free electron ordinary Hall coef cient of Al R0 1nec 345 X 10 11 m3C As mentioned above values of the anomalous Hall coef cient tend to be larger than those of the ordinary one For a current density jx 6 X 106 Am2 as used in the JS experiment Eq 5 yields a spin Hall voltage VSH 22 nV When we connect the two edges of the sample by a transverse metal strip a spin current will ow in that strip Assuming that the resistivity for the spin current is the same as that for the charge current we have for the current for each spin jg VSHpL Assuming the same skew scattering mechanism operating on the transverse sample the resulting spin Hall voltage due to this spin current is VSUIH 477Rslj039n039MB with Z the width of the transverse strip Now however because spin up and down currents circulate in opposite directions the spin voltages add giving rise to a real voltage due to the spin current VSC 2V TC which can be detected by an ordinary voltmeter The voltage due to the spin current is then 2 VSC 8772R z 136 7 Note that the transverse width L has dropped out in Eq 7 because even though it gives larger spin voltage VSH it also increases the resistance to the spin current in the transverse direction Still a dependence on L is im plicit in Eq 7 since when L becomes comparable to or larger than the spin diffusion length 53 Vsc will decrease Neither does the thickness of the transverse layer enter in Eq 7 a thicker layer would increase the spin cur rent but not the current density For the parameters under consideration here assuming for example Z 100 um Eq 7 yields VSC 58 nV easily measurable In the more general case where the transverse strip is of differ ent composition andor purity than the longitudinal strip Eq 7 becomes 2 Vsc SszsiRszz 8 where indices 1 and 2 refer to longitudinal and transverse strips respectively Figure 3 shows top and side views of the sample envisaged A thin insulating layer should be deposited on top of the sample longitudinal strip of width L and small contact areas should be etched to expose the sample surface and allow for metallic contact between the longitudinal and transverse strips Then a thin transverse strip of width I should be deposited on the insulator such that it also covers the contact areas The length of the 1836 Top view I contact transverse insulator quot9 insulator L 5 gt contact I Side view transverse strip insulating layer FIG 3 Top view along the z direction and side view along the y direction of the sample envisaged for detection of the spin Hall effect The voltage V measured by the voltmeter will be the spin current voltage Eq 7 in the absence of applied magnetic eld or the voltage VB Eq 10 in the presence of a magnetic eld B in the z direction contacts along the x direction should be suf ciently small that no signi cant voltage drop should occur on them due to the longitudinal current which would be transmitted to the transverse strip The voltage drop along a contact of length 16 Vd cpjx 7 should be substantially smaller than the signal VSC For the parameters used as an example here Vd 02 nV for a contact width 16 1 um Also a spurious voltage may arise if the two contacts are not perfectly aligned Again if the contacts are offset by Ax the magnitude of the spurious voltage will be at most Eq 9 with Ax replacing 16 so for our parameters Vd 1 nV if Ax 5 um Note also that a smaller resistivity p both increases the signal voltage Eq 7 and decreases the spurious voltage Eq 9 Finally the resistance of the contacts should be much smaller than the resistance of the transverse strip in order for Eq 7 to remain valid This argues for a thin transverse strip large resistance and a thin insulating layer smaller contact resistance along the thickness of the layer It would appear to be simple to achieve a contact resistance at least 2 orders of magnitude smaller than the transverse strip resistance Note also that the sign of the expected signal VSC as indicated in Fig 2 is opposite to the voltage Vd that would arise from voltage drop across the contacts As long as the signs of the anomalous Hall coef cients R31 and R32 are the same so in particular for R51 R32 the sign of the spin current voltage Vsc will always be as indicated in Fig 2 that is Vsc drives a current in the direction opposite to the primary current jx Thus a


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