more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionScatteringsSpinpolarized/ unpolarized electronsSpin statisticselectron gas in Magnetic FieldFerromagnetic metalsSpin TorqueSpinTorque CurrentSpinTransfer TorqueQuantum Nature of SpinQuestions & Answersmore Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11
more Chapters on this topic:IntroductionTransport Eqs.Spin Proximity/ Spin InjectionSpin DetectionBoltzmann Eqs.Band currentScattering currentMeanfree pathCurrent near InterfaceOrdinary Hall effectAnomalous Hall effect, AMR effectSpinOrbit interactionSpin Hall effectNonlocal Spin DetectionLandau Lifshitz equationExchange interactionspd exchange interactionCoercive fieldPerpendicular magnetic anisotropy (PMA)Voltage controlled magnetism (VCMA effect)Allmetal transistorSpinorbit torque (SO torque)What is a hole?spin polarizationCharge accumulationMgObased MTJMagnetoopticsSpin vs Orbital momentWhat is the Spin?model comparisonQuestions & AnswersEB nanotechnologyReticle 11

Ordinary Hall Effect
Spin and Charge TransportThe ordinary Hall effect occurs when a magnetic field is applied perpendicularly to the direction of a drift current. In this case both the electrons and the holes turn out from the flow direction and they are accumulated at same side of the wire. The charge accumulation at sides of the wire can be measured and the transport properties of the electron gas can be evaluated. Since there is a near equal amount of electrons and holes in a metal, they are of opposite charge and they both are accumulated at same side wire, the Hall voltage in a metal is not large.Due to the ordinary Hall effect, the spin accumulation may be significantly enlarged at one edge of a ferromagnetic wire. It is because both the spinpolarized electrons and the spinpolarized holes are accumulated at one edge of the wire.The same content can be found in this paper (http://arxiv.org/abs/1410.7511 or this site for a more upgraded version) .Chapter 12, pp. 3536Possible confusion!!: from 2014 to 2017 I have used names TIA and TIS for groups of spinpolarized and spinunpolarized electrons, respectively. The reasons are explained here.Main ResultSpin accumulation at a side of sample due to the Hall effect
Relativistic origin of the Hall effect (the Lorentz force)
When an electron moves in a magnetic field, it experiences the Lorentz force. The Lorentz force has a relativistic origin. The Theory of Relativity states that a particle moving in a magnetic field experiences an effective electrical field, which is directed perpendicularly to the magnetic field and perpendicularly to the particle movement direction. It is important to emphasize that the direction and magnitude of the effective electrical field does not depend either on the particle charge or on the particle spin.
According to the Theory of The Relativity the electric and magnetic field mutually transformed into each other depending on the speed of an observer. For example, if in a coordinate system of static observer there is only a magnetic field, a movable observer will experience this field as both an electrical field and a magnetic field. A particle moving in a static magnetic field experiences an effective electric field. The effective electrical field acts on the particle charge (the Lorentz force, Hall effect) and forces the particle to move along this field. A particle moving in a static electrical field experiences an effective magnetic field. The effective magnetic field acts on the particle magnetic moment (spinorbit interaction) and causes the precession of the magnetic moment around the direction of the effective magnetic field.
The Hall Effect and the SpinOrbit interaction are close cousinsthe Hall effect ==== results in ====> an effective electrical field
the SpinOrbit interaction ===results in=====> an effective magnetic field
The effective electrical field due to the Hall effect does not depend either on the particle charge or the particle spin. it is only depend on the particle movement direction and the direction of the magnetic field !!!!
Results in short
The Hall effect from the Boltzmann EquationsThe Hall effect can be described by the force term in the Boltzmann transport equations where In an equilibrium in the electron gas the electrons move in all direction. Since the Lorentz force only changes the electron movement direction, the Lorentz force does not change the equilibrium electron distribution F_i,0 Eq. (1.3) can be verified as follows Only if there is a drift current, the Lorentz force may change the electron distribution. In cases of the current of the runningwave electrons, substituting the solution for the current of the runningwave electrons Eq. (14.7) into Eq.(1.1) gives Using the relaxationtime approximation Eqs. (9),(12), ignoring the term of order H^2 and substituting Eq. (1.5) into the Boltzmann equations Eqs. (18) gives From Eq. (1.6) the Hall current can be calculated as (See Eq.(11.3) here) or Hall current can be calculated as
In the case when a magnetic field is applied perpendicularly to the flow direction of a drift current, the charge carriers experience the Lorentz force in the direction perpendicular to both the magnetic field and the current. Because of this force, the carriers are accumulated at the edges of the sample and a voltage transverse to the drift current is built up. This voltage is called the Hall voltage. In n and ptype semiconductors the Hall voltage is of opposite sign, because the holes and electrons are accumulated at the same side of sample, but they have opposite charge (Figs.1415). Because of a nearly equal number of holes and electrons in a metal, the Hall voltage in the metal is small and it is proportional to the gradient of the density of states at the Fermi level. When the electron gas in a metal is spinpolarized, both the electrons and holes are spinpolarized. Because of the Hall effect, the spinpolarized electrons and holes are accumulated at the same side of the sample. Therefore, at this side of sample a significant spin accumulation occurs. (Fig.16)
The Hall effect in ntype semiconductors.
In a ntype semiconductor there is only one type of carries. It is the electrons or "spin" states.
The electrons diffuse from a "" source toward a "+" drain. As they diffuse in the magnetic field they turn to the left. Therefore, the spin and the negative charge are accumulated at the right side of the sample.
The Hall effect in ptype semiconductors.
In a ptype semiconductor the holes are the carriers for the Spin and the Charge. In fact the hole current consists of two currents: the current of "spin" states and the current of "full" states.
The holes diffuse from a "+" source toward a "" drain. As they diffuse in the magnetic field they turn to the right. Therefore, the spin and the positive charge are accumulated at the right side of the sample. (see left figure)
Look closer!!!The hole current consists of the currents of "spin" and "full" states In a ptype semiconductor the "spin" states diffuses from a "+" source toward a "" drain. Therefore, the "spin" states are accumulated at the right side of the sample. (see right figure). The "full" states moves in the opposite direction and they are accumulated at the right side of the sample. Therefore, the spin and the positive charge are accumulated at the right side of the sample. (see right figure)
Remember!!!!The Hall effect is the relativistic effect. It does not matter for the Hall effect the charge or the spin of the "spin" states. It is only matter their movement direction. This is the reason why the "spin" states in the cases of the n and ptype semiconductors turn in the opposite directions.(Compare Fig.14 and Fig.15(right))
note about the current of "full" states (click to expand)
The current of "full" states is the normal current. This means that the negatively charged "full" states diffuses from a "" source toward a "+" drain. Due to the Hall effect they should turn to the left (similar to the electrons shown in Fig. 14). However, the most of the electrons, which occupies the "full" states, are the standingwave electrons. (See Fig.9 here) The standingwave electrons do not move, therefore they do not contribute to the Hall current. In contrast, the electrons, which occupies the "spin" states, are the runningwave electrons. For this reason, mostly the "spin" states contribute to the Hall effect in the case of the hole current.
The Ordinary Hall effect in metals.
The drift current in a metal consists of the electron and hole currents, which flow in the opposite direction. Since the charge of an electron and a hole is opposite, the direction of the charge transport is the same for the electron and hole current. This is the transport of the "" charge from "" to "+". Or what is the same, the transport of the"+" charge from "+" to "". Since the electrons and holes move in the opposite directions, in the magnetic field the electrons turn to the left and holes turn to the right. As result, both the electrons and the holes are accumulated at the same side of the sample. About the same amounts of the electrons and holes are accumulated
Since the charge of an electron and a hole is opposite, the charge is not accumulated. The Hall voltage in a metal is small.
In contrast, the spin direction of electrons and holes is the same, therefore there is a significant spin accumulation due to the Hall effect in a metal
Look closer!!!Both the electron and hole currents are currents of "spin" states The "spin" states of energies higher than the Fermi energy moves from "" to "+" similar to positive particlesThe "spin" states of energies lower than the Fermi energy moves from "+" to "" similar to positive particles (See here)Since the "spin" states of the different energies move in the opposite directions, they turn in the opposite directions due to the Hall effect. The "spin" states of higher energies turn to the left and the "spin" states of lower energies turn to the right. As result, the "spin" states of all energies are accumulated at the same side of the sample. There is a significant spin accumulation at this side of the sample. Notes1. The Hall voltage in a metal is small, but it is not zero. It can be positive and negative. It depends on whether the electron or hole current is larger in a metal. The Hall voltage in a metal is proportional to the energy derivative of the density of states at the Fermi energy. (See here)
2. From the Hall voltage in a metal the injection conductivity can be estimated. The injection conductivity is proportional to the Hall voltage. 3. For a larger the spin torque or a better spin injection efficiency, a metal having largest the Hall voltage should be used. The largest spin torque is required for efficient a MRAM cell The spin torque is proportional to (See here). The spin injection efficiency is proportional to (see here)
Q. How to choose a good metal for the spin injection or to achieve a largest spintransfer torque???
A1. Use a metal, which shows a largest Hall voltage. A2. If you want to inject the spin from one metal to another, use metals, which show the Hall voltage of the same sign.
Note 1. The conductivity in the bulk of a metal and in the vicinity of the interface is different. The spin injection and the spintransfer torque are the effects of an interface. The Hall effect is often measured in the bulk of a metal. Hall coefficient and Hall angle in metalsfrom Colin M. Hurd "THE HALL EFFECT IN METALS AND ALLOYS" , PLENUM PRESS· 1972Summary note m3 /A/s =Ohm*m/TMetals with holedominated transport (spin is drifted from "+" to "")
Metals with electrondominated transport (spin is drifted from "" to "+")
Iron FeThe transport is electrondominated. Hall coefficient at 6T is 1E9 B* kG/Ohm/cm nonordinary Hall coefficient is Rs=3040E11 Ohm*m/T at RT (Hall angle=34E3 1/T), it decreases almost to zero at lower T ordinary Hall coefficient is 2E11 Ohm*m/T at RT (Hall angle=0.2E3 1/T), it weakly decreases at lower T. Conductivity : 1.01E7(S/m) bcc structure Cobalt CoTransport is ???dominated. (maybe holedominated)
nonordinary Hall coefficient is small positive Rs=2E11 Ohm*m/T at RT(Hall angle=0.32E3 1/T), it decreases almost to zero at lower T, it becomes becomes negative below 250 C ordinary Hall coefficient is negative and large 8E11 Ohm*m/T(Hall angle=1.28E3 1/T), it weakly increases to 12E11at lower temparature; Conductivity : 1.602E7(S/m) hcp structure Nickel Ni Transport is holedominated
nonordinary Hall coefficient is big negative Rs=70E11 Ohm*m/T (Hall angle=10.1E3 1/T), decrease almost to zero at lower T. ordinary Hall coefficient is negative 4E11 Ohm*m/T (Hall angle=0.572E3 1/T);
Conductivity : 1.43E7(S/m) IronCobalt FeCo (bcc) nonordinary Hall coefficient is positive and very large Rs=400E11 Ohm*m/T, at Co=15% ordinary Hall coefficient is negative and largest 25E11 Ohm*m/T, at Co =35 % Properties of FeCo the spin polarization is negative for Co and positive for Fe ?? some calculated spin polarizations for FeCo , near 16 % of Co sp polarization changes its signs Fe=56 %; Co(25%) sp=26%; Co(50%) sp=67%; Co(75%) sp=73% Co sp=80% from book "Materials Design and Synthesis for Desirable Magnetic and Optical Properties By Hao Zhu" magnetic moment of BCC Co maybe the same as BCC Fe lattice constant =2.87 A x=0, it is near constant till 20 %, next it reduces to 2.83 A for BCC Co Cromium At 290 K the Hall coefficient is independent on magnetic field, but at 4.2 and 77 K, R decreased by a factor of three as the field was increased to 147 kG Titanium Ti has cphstructure Hall coefficient is of opposite sign for directions parrall and perpendicular to caxis R_parallel=4.211 Ohm*m/T (electrondominated) R_perpendicular= 7.711 Ohm*m/T note: All metals of cph structures (Ti,Y,Gd) have significant difference of Hall resistance along and perpendiculary to the caxis indicating very strong spinorbit interaction. Terbium In the paramagnetic region the values obtained for Ro and Rs are, respectively, 44E11Ohm*m/T and 420E11 Ohm*m/T (very large) Hall angle in semiconductors

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