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
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
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
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

Band Current Spin and Charge TransportThe band current is the major electron transport mechanism in the bulk metal. It occurs because of movent of electrons between scatterings. It is very efficient transport mechanism. A small injection conductivity σ_{injection}, zero detection conductivity σ_{detection}=0 and independence of the ordinary conductivity σ_{charge} on spin polarization of the electron gas are distinguished features of the band current.The same content can be found in this paper (V.Zayets JMMM 2018 )
Results in short The band current occurs because of the movement of electrons between scatterings. In electron gas the conduction electrons move between scatterings at a highspeed in all directions. In equilibrium, the numbers of electrons moving in any opposite directions are exactly the same. Therefore, there is no electron current. When a voltage is applied to the conductor, the numbers of electrons moving along and opposite to the electrical field become slightly different. Therefore, the band current flows along the electrical field and it transports the charge and the spin.
1 transport mechanism: The band current is the major transport mechanism in the bulk of a conductor. Except the case of the hopping conduction, when the scattering current is the major mechanism. The hoping conductivity occurs in an isolator with a large number of conductive defects.2 Ordinary charge conductivity σ_{charge}: (The Ohm law). The electrons at the Fermi energy are most effective for the transferring of the charge. Above and below the Fermi energy the ordinary conductivity sharply decreases. For example, the contribution to the ordinary conductivity of electrons at energies 5kT above/below the Fermi energy is near 50 times smaller than the contribution of electrons at the Fermi energy. It is the primary reason why the conductivity of a metal is much higher than the conductivity of a semiconductor.3. Spindiffusion conductivity σ_{spin}: (Spin diffusion). The σ_{spin} is always larger than the σ_{charge}. In a semiconductor the spindiffusion conductivity is nearly equal to the charge (ordinary) conductivity σ_{spin}≈σ_{charge};in a metal the spindiffusion conductivity σ_{spin} is 1060% larger than the charge (ordinary) conductivity σ_{charge}. It The difference is larger for a larger spin polarization sp of the electron gas.
4. Detection conductivity σ_{detection}: (Spin detection) (Charge accumulation along spin diffusion) In a semiconductor and a metal the detection conductivity is zero ; note: it becomes nonzero in a metal with defects or in the vicinity of an interface
5 Injection conductivity σ_{injection}: (Spin polarization of an electrical current) (Spin transfer by an electrical current) It is large in a semiconductor and it is small in a metal. . In a semiconductor the injection conductivity is equal to the charge (ordinary) conductivity . The sign of the injection conductivity is different for n and ptype semiconductors;In metal the injection conductivity is small and it is proportional to an energy derivation of the density of states at the Fermi energy
Origin of the Band Current
The band current occurs when the number of the electrons moving in one direction is not equal to the number of the electrons moving in opposite direction. This current occurs because of movent of electrons between scatterings. Q. Do all electrons participate in the band current?No. The localized delectrons do not participate. Also, the standingwave conduction electrons do not participate in the band current (See here). All conduction (delocalized) electrons in a solid can be either standingwave electrons or runningwave electrons. Only the runningwave electrons contributes to this current, the standingwave electrons does not contribute. (See details here)
Q. The band current. Where it occurs?A. It is the major spin/charge transport mechanism in a bulk of a metal a a semiconductor. Additionally, it is the major transport mechanism of a lowresistivity contact.
1. Band current is the major transport mechanism in a bulk of a metal and a semiconductor. Only another possible current is the scattering current. It is about 24 orders in magnitude less effective than the band current. In a vicinity of interface and contact between two metals, the runningelectronwave conductivity significantly decreases or even can become zero, because of the decrease of number of runningwave electrons. In contrary, the scattering conductivity significantly increases near an interface, because a significant increase of the asymmetry between scattering probabilities for a forward and backward scatterings.2. Band current is the major transport mechanism through a lowresistivity contact between two metals. Even though at a contact between two metals, the conductivity of the band current always become smaller, for a lowresistivity contact it is still larger than the conductivity of the scattering current. Due to the existence of the standingwave electrons at an interface between two metals, there is always a contact resistance between two metals even in the case when there is no energy barrier between them.For a higher resistivity contact the scattering current (ballistic current) becomes only a transport mechanism through the contact.Drift and diffusion band currentsDrift current is current drifted by an external applied electrical field. The drift current flows from a drain ("+" electrical potential) towards a source ("" electrical potential). Diffusion spin current is a diffusion of the spin and charge from a region of the larger spin accumulation towards a region of the smaller spin accumulation. The properties of diffusion and drift currents are drastically different.The drift current flows along gradient of the chemical potential ∇μ . The diffusion current current flows along gradient of the spin polarization∇sp . The drift and diffusion currents in the model of spinup/spindown bands is here.This page describes only the when there are no standingwave electrons and the band electrons are runningwave electrons (Ideal case) The standingwave electrons have a substantial influence on spin properties of the band current. As example See the current in the vicinity of interface and current in a conductor with defects. diffuseCurrent.m, driftCurrent.m Matlab functions, which calculates drift and diffusion band currentsDrift band current
The drift current flows along an applied electrical field. The drift current flows from a drain ("+" electrical potential) towards a source ("" electrical potential). 1. Charge drift current (ordinary current). The ordinary conductivity σ_{charge}. The ordinary conductivity significantly depends on electron energy (See Fig.20). The electrons at the Fermi energy are effective for the transfer of the charge. The state conductivity for them is highest. Above and below the Fermi energy the ordinary conductivity sharply decreases. The ratio of state conductivity at given energy to the state conductivity at the Fermi energy: 1) ± kT → 78.6 % 2) ± 2kT → 41.9 % 3) ± 3kT → 18.0 % 4) ± 4kT → 7.0 % 5) ± 5kT → 2.6 % The contribution to the ordinary conductivity of electrons at energies 5kT above/below the Fermi energy is near 50 times smaller than the contribution of electrons at the Fermi energy. It is the primary reason why the conductivity of a metal is much higher than the conductivity of a semiconductor.
2. Spin drift current. The injection conductivity σ_{injection}. Spin polarization of current. The injection conductivity σ_{injection} (See Fig.21) describes the drift of the spins along an electrical current and the spin polarization of an electrical current. When an electrical current flows from one region to another region, the spin accumulation in one region may increase and in another region it may decrease. When two regions belong to different metals, such effect is called the spin injection. The polarity of the spin injection is different for electrons, which energy is lower and higher than the Fermi energy. This means that electrons of different energies transform spins in opposite directions. In a metal the electron density is nearly constant at the Fermi energy and there is an almostequal amount of electrons, which transform the spins in the opposite directions. As result, the spin polarization of electrical current in a metal is near zero and an electrical current nearly does not transform the spin at all. Only in the case when in a metal there is a nonzero gradient of the density of state at the Fermi energy and there is a small difference of amounts of electrons bellow and above the Fermi energy, the value of spin injection becomes nonzero and an electrical current may transform the spins. However, in a metal it is still very inefficient. It means that in a metal the spin polarization of electrical current is always substantially smaller than the spin polarization of the electron gas (See Fig.22). In contrast, in a semiconductor the σ_{injection} is large and it is near equal to the ordinary charge conductivity σ_{charge}. It also means that in a semiconductor the spin polarization of an electrical current is near equal to the spin polarization of the electron gas. In a nondegenerate ntype semiconductor, the Fermi energy is below the conduction band and all conduction electrons are at energies above the Fermi energy. In this case the spins is transferred from a "" to a "+" electrode. It is similar to transfer of the spin by negativelycharged particles in vacuum, when the spin and the negative charge are carried in the same direction. In a nondegenerate ptype semiconductor, the Fermi energy is above the conduction band and all conduction electrons are at energies below the Fermi energy. In this case the spins are transferred from a "+" to a "" electrode. It is similar to transfer of the spin by positivelycharged particles in vacuum, when the spin and the positive charge are carried in the same direction.
Diffusion band current
Diffusion spin current is a current, which flows along a gradient of spin accumulation. 1. Spin diffusion current . The spin conductivity σ_{spin}. It describes the flow of the spins from a region of a higher spin accumulation to a region of a smaller spin accumulation.
The spin conductivity σ_{spin} is larger than the ordinary conductivity charge conductivity( See Fig. 23). Figure 25 shows the the ratio of spin conductivity σ_{spin} to ordinary charge conductivity σ_{charge}. Above and below the Fermi energy these conductivities are almost equal. The largest difference between them is at the Fermi energy and it is larger for a larger spin polarization sp of the electron gas. At spin polarization sp=75% , the σ_{spin} can be 2.5 times larger than σ_{charge}. The reason of the difference between σ_{spin} and σ_{charge}. is a higher density of spinpolarized electrons at the Fermi energy (See spin statistics).
2. The detection conductivity σ_{detection}. The charge accumulation along spin diffusion. The detection conductivity σ_{detection} describes the diffusion of the charge along a gradient of the spin accumulation. When spinpolarized electrons diffuse from a region of a higher spinaccumulation to a region a lower spin accumulation, additionally there is a current of spinunpolarized electrons, which flows exactly in the opposite direction. When the opposite currents of the spinpolarized and spinunpolarized electrons are not equal, the charge is accumulated along the spindiffusion (See here). This charge accumulation can be measured electrically. Since an amount of the charge accumulation is proportional to spin diffusion current, the value the spin diffusion current can be estimated from such electrical measurements. As can be seen from Fig.24, the σ_{detection} equals to zero for all energies. This means that in the bulk of a defectfree conductor there is always an exact balance between opposite currents of spinpolarized and spinunpolarized electrons.
Properties of the bulkMetals.The Fermi energy E_{Fermi} crosses one or several bands. As result, the density of state is nearly constant in region E_{Fermi}10kT<E<E_{Fermi}+10kT. (1) Only conduction electrons in energy range E_{Fermi}3kT<E<E_{Fermi}+3kT.mainly contribute to ordinary σ_{charge} and spin diffusion σ_{spin} conductivities. (2) Both the ordinary σ_{charge} and spin diffusion σ_{spin} conductivities are large. They do not dependent on charge accumulation. The σ_{charge} does depend on spinaccumulation. The σ_{spin} conductivities slightly increases for larger spin accumulation (max 50 % increase). (3) The spin polarization of an electrical current is significantly smaller than the spin polarization of the electron gas. The injection conductivity is much smaller than the ordinary conductivity σ_{injection}<<σ_{charge} (4) No charge accumulation along spin diffusion. σ_{detection}=0. SemiconductorsThe Fermi energy E_{Fermi} in the band gap. All states of conduction electrons are either substantially above or below the Fermi energy E_{Fermi}. (1) All conduction electrons contribute to ordinary σ_{charge} and spin σ_{spin} conductivities in case of a ntype semiconductor and only electrons of halffilled states contribute in the case of a ptype semiconductor. (2) Both the ordinary σ_{charge} and spin diffusion σ_{spin} conductivities are equal and relatively small. They both significantly depend on the charge accumulation and they do not depend on spin accumulation. σ_{charge} = σ_{spin} (3) The spin polarization of an electrical current is equal to the spin polarization of the electron gas. The injection conductivity is equal to the ordinary conductivity σ_{injection}=σ_{charge}. (4) No charge accumulation along spin diffusion. σ_{detection}=0.
Solution of the Boltzmann equation for the band currentWhy do we need the Boltzmann Transport Equations?????
From the Boltzmann Transport Equations, the ordinary charge conductivity σ_{charge}, spindiffusion conductivity σ_{spin}, detection conductivity σ_{detection} and injection conductivity σ_{injection} can be found
The Boltzmann transport equations are described here. In order to describe the feature of the band current, only two terms of the general Boltzmann transport equations have to be used. The simplified Boltzmann transport equations are :
where F_{i } are distribution functions and τ_{k} is the momentum relaxation time. The subscript "i" labels the distribution functions for the group of spinpolarized electrons as "TIA", group of spinunpolarized electrons as "TIS" and electrons of fullfilled states as "full". The solution of Eqs. (1.1) (See Appendix 1) is: where is the band current of spin polarized electrons , is the band current of spinunpolarized electrons and is the band current of electrons occupying the fullfilled states.
are are equilibrium distribution functions of groups of spinpolarized electrons, spinunpolarized electrons and electrons occupying the fullfilled states, correspondingly. They are calculated here. click here to see how to derive Eqs. (1.2) from the Boltzmann equations (Eqs.(1.1))
The band current occurs, because of the movement of band electrons (delocalized conduction electrons) in space. The movement of an electron literally means that if at time t the electron is at point x, at time t+dt the electron will be at point , where v_{x} is the electron speed along the direction of the xaxis. We assume that all electrons in electron gas are the runningwave electrons. (To see how to do calculations when it is not the case. Click here or here.)The change of electron distribution F(x) at point x can be described as where describes the movement of electrons to point x from point and describes movement of electrons from point x to point In the case of a short time interval dt, Eq. (14.1) can be simplified as or Taking into the account that the electron can move not only in the xdirection, but in any direction gives Using the he relaxationtime approximation and substituting of Eq. (14.4 ) into the Boltzmann transport equations Eqs. (1.1) gives The solution of the Boltzmann transport equations for the band current is The current due to movement of one electron in volume V is where q is the charge of an electron. Integrating Eq. (14.8a) over all quantum states and using Eq. (14.7), the band current can be calculated as In equilibrium there should be no current: The condition (A1.8) was used to simplify Eq. (A1.7). Similarly the currents flowing along j_{i,} and perpendicularly j_{i,⊥}to are calculated as Since j_{i,⊥}=0, the current flows only along and first equation of Eqs. (A1.10), the band current can be calculated as The explicit expressions for the band current of spin polarized electrons , the band current of spinunpolarized electrons and the band current of electrons occupying the fullfilled states are It is important! Integration in Eq (14.8) is only over states of runningwave electrons. The standing wave electrons should not be included into the integration.
How to include the standingwave electrons in the integration of Eqs. (14.9) is shown here and here.
All three currents , and transport the charge, but the spin is only transported by the current of the spinpolarized electrons . Therefore, the charge current and spin current can be calculated as
There are only two independent variables: the chemical potential and the gradient the spin polarization sp, which describe special variation of the distribution function. Therefore, the gradients of the distribution functions can be calculated as: Using Eqs (1.4) and (1.2), the charge, injection, spin and detection conductivities are calculated from Eqs. (1.3) as The conductivities at the right side of Eqs (1.5) are calculated as where the script j denotes "μ,TIA", "μ,TIS", "μ,full", "sp,TIA", "sp,TIS", "sp,full". The σ_{j}(E) are defined as the state conductivities and are calculated as
click here to see how to derive Eqs. (1.51.7) and to see description of properties of spinpolarized and spinunpolarized currents
In the following we calculate the charge , spindiffusion , detection and injection conductivities from Eqs. (1.3) using Eqs. (1.4), and (1.2). In order to simplify the solution and the analysis, two cases are calculated separately. At first, the drift current flowing along an electrical field is calculated. Next, the diffusion current flowing along a gradient of spin polarization is calculated. case 1. The drift current. In the case when there is a spatial gradient of the chemical potential μ, the gradient of the distribution function can be calculated as Substituting Eqs. (4.1) into Eqs. (1.2) gives the drift current as Similar as above, the subscript "i" labels the distribution functions for the group of spinpolarized electrons as "TIA", group of spinunpolarized electrons as "TIS" and electrons of fullfilled states as "full". The definition of the drift current is Comparison of Eq. (4.2) and Eq.(4.3) gives the conductivities of spin polarized electrons σ_{µ,TIA}, spinunpolarized electrons σ_{µ,TIS}, and electrons of fullfilled states σ_{µ,full}, for the drift current as where the state conductivities σ_{µ,TIA}(E) , σ_{µ,TIS}(E) , σ_{µ,full}(E) are defined as All conductivities of Eq.(4.5), which describe a drift current, are proportional to derivative of the energy distribution in respect to the electron energy. Figures 41 shows the calculated state conductivities σ_{µ,TIA}(E) , σ_{µ,TIS}(E) , σ_{µ,full}(E). The state conductivity σ_{µ,full}(E) for a band current of electrons of fullfilled states is positive for all energies. This means that the fullfilled states are drifted from a “–“ source toward a “+” drain. It is similar to the movement of a negativelycharged particle in an electrical field in vacuum. The state conductivities σ_{µ,TIA}(E) , σ_{µ,TIS}(E) for a currents of electrons of halffilled states are positive for energies above the Fermi energy and they are negative for energies below the Fermi energy. This means that the drift direction of these states depends on their energy. The halffilled states of energies above the Fermi energy are drifted from a “–“ source toward a “+” drain. It is similar to the movement of a negativelycharged particle in an electrical field in vacuum. However, the halffilled states of energies below the Fermi energy are drifted in the opposite direction from a “+“ source toward a “” drain. The drift becomes similar to the movement of a positivelycharged particle in an electrical field in vacuum.
case 2. The diffusion current. . In this case there is a spatial gradient of the spin polarization sp and the gradient of the distribution function can be calculated as Substituting Eqs. (4.1) into Eqs. (1.2) gives the diffusion current as The definition of the diffusion current is Comparison of Eq. (4.7) and Eq.(4.8) gives the conductivities of spin polarized electrons σ_{sp,TIA}, spinunpolarized electrons σ_{sp,TIS}, and electrons of fullfilled states σ_{sp,full}, for the diffusion current as where the state conductivities σ_{sp,TIA}(E) , σ_{sp,TIS}(E) , σ_{sp,full}(E) are defined as All conductivities of Eq.(4.10), which describe a drift current, are proportional to derivative of the energy distribution in respect to the electron energy. Figures 42 show the calculated state conductivities σ_{sp,TIA}(E) , σ_{sp,TIS}(E) , σ_{sp,full}(E). The conductivity of the spinpolarized electrons σ_{sp,TIA}(E) is positive. It describes the simple fact that the spinpolarized electrons diffuse from a region of a higher spin accumulation into a region of a smaller spin accumulations. The conductivities of spinunpolarized electrons σ_{sp,TIS}(E) and σ_{sp,full}(E) are negative. It means that the spinunpolarized electrons diffuse in the opposite direction. Notice the metal conductivity is defined as , where e is the charge of the electron.
How the calculations have been done? In the bulk of a conductor, the equilibrium distribution functions are described by the spin statistics. It can be assumed that in a conductor with a low density of dislocations, there are no standingwave electrons . All delocalized electrons are the runningwave electrons and all of them contribute to the runningwave electron currents. Therefore, the distribution functions of the spin statistics are used in Eqs. (14.16), (14.20) in order to calculate the state conductivities shown below.
The state conductivity, what is it?In order to calculate the metal conductivities it is necessary to know the density of state D(E) of the metal (Eqs.(14.15) . The calculation of the unitless state conductivities does not require the density of state. Therefore, even without knowing the density of states of the metal, some conduction properties of the metal can be calculated and analyzed from the state conductivities. It is an advantage of using the state conductivities. The normal conductivity σ_{i} can be calculated by integrating the state conductivity with the density of states D(E):
Charge or ordinary conductivity σ_{charge} The charge or ordinary conductivity σ_{charge} significantly depends on electron energy. The electrons at the Fermi energy are most effective for the transport of the charge. The ordinary charge conductivity for them is largest. Above and below the Fermi energy the ordinary conductivity sharply decreases. For example, the contribution to the ordinary conductivity of electrons at energies 5kT above/below the Fermi energy is near 50 times smaller than the contribution of electrons at the Fermi energy. It is the primary reason why the conductivity of a metal is much higher than the conductivity of a semiconductor. The ordinary conductivity σ_{charge} does not depend on the spin polarization sp of the electron gas. Spindiffusion conductivity σ_{spin} The spindiffusion conductivity σ_{spin} describes the diffusion of the spins along a gradient of spin accumulation. The spin conductivity σ_{spin} is larger than the ordinary charge conductivity σ_{charge}. Above and below the Fermi energy, these conductivities become near equal. The largest difference between them is at the Fermi energy and it is larger for a larger spin polarization sp of the electron gas. At spin polarization sp=75% , the σ_{spin} can be 2.5 times larger than σ_{charge} . The reason of the difference between σ_{spin} and σ_{charge} is a higher density of spinpolarized electrons at the Fermi energy injection conductivity σ_{injection} The injection conductivity σ_{injection} describes the drift of the spins along an electrical current. It also defines the spin polarization of an electrical current. When an electrical current flows from one region to another region, the spin accumulation in one region may increase and in another region it may decrease. When two regions belong to different metals, such effect is called the spin injection. The polarity of the σ_{injection} is different for electrons, energy of which is lower and higher than the Fermi energy. This means that electrons of different energies carry the spins in opposite directions. In a metal the electron density is nearly constant at the Fermi energy. Therefore, amounts of electrons, which carry the spin along an electrical current and in the opposite direction, are nearly equal. As result, the spin polarization of electrical current in a metal is near zero and an electrical current nearly does not transport the spin at all. Only in the case when in a metal there is a nonzero gradient of the density of state at the Fermi energy and there is a small difference of amounts of electrons bellow and above the Fermi energy, the value of σ_{injection} becomes nonzero and an electrical current may transport the spins. However, in a metal it is still very inefficient. It means that in a metal the spin polarization of electrical current is always substantially smaller than the spin polarization of the electron gas. In contrast, in a semiconductor the σ_{injection} is large and it is near equal to the ordinary charge conductivity σ_{charge} . It also means that in a semiconductor the spin polarization of an electrical current is near equal to the spin polarization of the electron gas. In a nondegenerate ntype semiconductor, the Fermi energy is below the conduction band and all conduction electrons are at energies above the Fermi energy. In this case the spins is transported from a "" to a "+" electrode. It is similar to the transport of the spin by negativelycharged particles in vacuum, when the spin and the negative charge are carried in the same direction. In a nondegenerate ptype semiconductor, the Fermi energy is above the conduction band and all conduction electrons are at energies below the Fermi energy. In this case the spins are transported from a "+" to a "" electrode. It is similar to the transport of the spin by positivelycharged particles in vacuum, when the spin and the positive charge are carried in the same direction. detection conductivity σ_{detection} The detection conductivity σ_{detection} describes the diffusion of the charge along a gradient of the spin accumulation. When spinpolarized electrons diffuse from a region with a higher spinaccumulation to a region with a lower spin accumulation, additionally there is a current of spinunpolarized electrons, which flows exactly in the opposite direction. When the opposite currents of the spinpolarized and spinunpolarized electrons are not equal, the charge is accumulated along the spindiffusion. This charge accumulation can be measured electrically. Since an amount of the charge accumulation is proportional to spin diffusion current, the value the spin diffusion current can be estimated from such electrical measurements. The effect is called the spin detection. As can be seen from Fig.44, the σ_{detection} equals to zero for all energies. This means that in the bulk of a defectfree conductor there is always an exact balance between opposite currents of spinpolarized and spinunpolarized electrons
Spinpolarization of electrical current
It describes the efficiency of the spin transfer by an electrical current. It equals to the the ratio of the injection conductivity to the ordinary conductivity σ_{injection}_{}/ σ_{charge}Semiconductors: It equals to the spinpolarization of the electron gas The reason: There is only one type of carriers. It is either the electrons or the holes
Metals: It is significantly smaller than the spinpolarization of the electron gas. The reason: There two types of carriers: the electrons and the holes. The electrons and the holes transport the spin in the opposite directions.
Conductivity of spin diffusion σ_{spin}The spindiffusion conductivity σ_{spin} describes transport of the spin in a diffusion spin current.
semiconductors: It equals to the ordinary conductivity σ_{charge}
Metals: It is larger than the ordinary conductivity σ_{charge} The reason: The larger amount of spinpolarized electrons at the Fermi energy (See here)
What is difference between a drift current and a diffusion current? A drift current is the electron current flowing along an electrical field or a gradient of the chemical potential. A drift needs both a source and a drain. A diffusion current is the electron current flowing along an a gradient of the spin polarization sp. A diffusion current needs only a source and it does not needs a drain. For more details about diffusion and drift currents click hereNotes:1) the ordinary charge conductivity σ_{charge} does not depend on the spin polarization . That means in this case there is no Magnetoresistance effect in the case of bulk current. There are no GMR, AMR and TMR effects.2) The injection conductivity σ_{injection} is equal to the ordinary conductivity σ_{charge} for energy more than (13)kT above the Fermi energy. The injection conductivity σ_{injection }is equal but of opposite sign to the charge conductivity for energy less than (13)kT below the Fermi energy. This means that in the bulk of nondegenerated semiconductors the transport of the spin by a drift current is the most effective. In p and nsemiconductors the injection conductivity is of opposite sign. This means that the holes and electrons transport the spin in the opposite directions.3) The injection conductivity σ_{injection} in bulk of a metal is small, because of equal but opposite contributions of electrons with energies below and above the Fermi energies (opposite contributions of holes and electrons).4) At higher spin polarization sp, the the injection conductivity in bulk of a metal decreases5) In both the semiconductors and the metals the detection conductivity σ_{detection} is zero.note: the classical model of the spinup/spindown bands incorrectly predicts that the detection conductivity is always equal to the injection conductivity (See here)6) The spindiffusion conductivity σ_{spin} is larger than the charge conductivity.
Electrons and Holesfor more details about the holes and electrons click hereBoth the “electrons” and the “holes” are absolutely identical particles. They both are halffilled states, in which one place is occupied by an electron and another place is empty. The only difference between “electrons” and the “holes” is their electron energy. The energy of an “electron” is above the Fermi energy E_{F }and the energy of a “hole” is below E_{F}. Depending on the electron energy, the properties of a halffilled state are substantially different. In an electrical current the "electrons", whose energy > E_{F} , transport the charge and the spin from a "" potential to a "+" potential similar to a current of negativecharged particles in vacuum. In contrast, the "holes", whose energy < E_{F}, transfer the charge and the spin in the opposite direction from a "+" potential to a "" potential similar to a current of positivelycharged particles in vacuum. In a metal there are almost equal amounts of "holes" and the "electrons" and they transport spin in the opposite directions. As result, the transferring of the spin accumulation by an electric current is ineffective in the bulk of a metal.
In diffusion current the properties of the electrons and the holes are absolutely identical (See Fig.12 right).
In drift current the properties of the electrons are very different. Below explanation is for the drift current (See Fig.12 left) In a nondegenerate ntype semiconductor, the Fermi energy is below the conduction band and all states available for transport are at energies above the Fermi energy. In this case the conductivity σ_{μ,full} of the fullfilled states is small and the major transport mechanisms is the transport of the halffilled states. A halffilled state is negatively charged and its spin is ½. Therefore, the halffilled states carry simultaneously the spin and the negative charge. It is similar to the movement an electron in vacuum, when the spin and the negative charge are carried in the same direction. For this reason, the conductivity in an ntype semiconductor is called the electron conductivity. In a nondegenerate ptype semiconductor, the Fermi energy is above the valence band and all states available for transport are at energies below the Fermi energy. In this case both the fullfilled and halffilled states substantially contribute to the transport. Since σ_{μ,full} is positive and σ_{μ,TIE},σ_{μ,TIS} are negative, the fullfilled and halffilled states transport a negative charge in opposite directions. The σ_{μ,full} is larger than σ_{μ,TIE}+σ_{μ,TIS} .Therefore, the charge conductivity σ_{charge} is positive for all energies (Fig.11, blue curve) and in total the negative charge is transported from ““ to “+”. The spin of a fullfilled state is zero and only halffilled states carry the spin. The spin is drifted in the opposite direction to the drift direction of the negative charge. It is from “+” to”“. It is similar to the movement a positive particle in vacuum, when the spin and the positive charge are carried in the same direction. The conductivity in a ptype semiconductor is called the hole conductivity. In a metal the electron and hole conductivities are almost equal. The drift direction of the charge is the same for the electron and hole currents. It is a drift of a negative charge from ““ to “+” or a drift of a positive charge from “+“ to “”. In contrast, the drift of spin is in opposite directions for the hole and electron currents and the opposite spin currents nearly compensate each other. Therefore, in a metal the spin drift current is significantly smaller than the drift charge current and the spin polarization of the drift current is always significantly smaller than the spin polarization of the electron gas. It is different from the case of a semiconductor, where the spin polarization of the drift current is equal to the spin polarization of the electron gas.
Charge, spindiffusion, injection and detection conductivities in a metal
In case when the Fermi energy is far from any band critical point, the density of states is nearly constant and the Fermi surface consists of only one band, the following approximation can be used where and D_F are the electron speed and the density of states at the Fermi energy, respectively. Substituting Eq. (15.1) into Eqs. (1.6) and integrating gives the conductivities in the bulk of a metal with a low density of defects as
From Eqs. (15.2) the following ratios between the conductivities in the bulk of a metal are obtained In comparison in the case of a nondegenerate semiconductor the conductivity ratios are
The spindiffusion length
Substituting Eq. (15.3) into Eq. (4.8) gives the spindiffusion length in a nonmagnetic metal as
The conductivities in a conductor with defects and multilayers
Runningwave electrons and standingwave electrons
It is important to emphasize that not all conduction electrons contribute to the band current. All conduction electrons can be divided into two different types: runningwave and standingwave electrons. The effective length of the conduction electrons is rather long (See here). In a semiconductor it can be as long as a hundred of nanometers. In the case when an average distance between defects in a conductor is comparable with the effective length of a conduction electron, the electron may bounce back and forwards between defects. It is similar to the case of a photon, which are bouncing between walls of a resonator. When bouncing between defects, one electron, which moves forward, is firmly fixed to the electron, which moves backward. These coupled electrons do not move along crystal. They are fixed at position of defects. Importunately, when an electrical field is applied, still there is one electron moving forward and one electron moving backward. The electrical field does not change the ratio of electrons moving in opposite directions for this type of electrons. Therefore, these electrons do not contribute to the band current. These coupled electrons are defined as the standingwave electrons. The electrons, which can move freely along crystal and which are not fixed to one position, are defined as the runningwave electrons. Only they contribute to the band current.
Do the standingwave electrons are described by the spinstatistics?Yes. Both the standingwave electron and the runningwave electrons are described by the spinstatistics, Why the standingwave are described by the spinstatistics, but the localized electron do not? They both do not move?It is because of a very large difference between scattering probabilities between a runningwave electron to a standingwave electron and a runningwave electron to a localized electron. The sizes of a runningwave electron are a standingwave electron are nearly the same. In contrast, the size of a localized electron is much smaller. The scattering probability between a runningwave electron to a standingwave electron is nearly the same as scattering probability between two runningwave electrons. The scattering probability between a runningwave electron to a localized electron is very small. It is important that the scattering probability between a standingwave electron to a localized electron is higher than between a runningwave electron to a localized electron. This is the reason why existence of the standingwave electron enhances the spd exchange interactions
How do we know how many standingwave electrons and how many runningwave electrons??The meanfree path is an important parameter, which determines the number of the standingwave electrons in a conductor. The longer the effective length of electron is, the higher the probability is for the electron to bounce between defects or interfaces to form a standingwave electron. Figure 3 shows the mean free path of the conduction electrons in the bulk of a conductor with defects, which is calculated here. The λ_{mean} significantly increases for a fullfilled states at energies E_{F}2kT and lower. Near the Fermi energy, all states have the same and rather short λ_{mean} The case of a large spinpolarization sp, when the λ_{mean} of spinpolarized electrons is longer, is an exception. Intuitively, the data of Fig. 3 can be understood as follows. Below the Fermi energy almost all states are fullfilled states, which have no unoccupied places where an electron can be scattered into. There are only a few of halffilled states, which have one unoccupied place. The probability of an electron to be scattered from a fullfilled states is very low, because most of time a fullfilled state is surrounded by other fullfilled states, into which an electron can not be scattered. Rarely a halffilled state is nearby and a scattering event may happen. Therefore, the life time of fullfilled state is long and the λ_{mean} is long as well. In contrast, a halfstate is always surrounded by fullfilled states, from which an electron can be scattered into the halffilled state with a high probability. Therefore, both the life time and λ_{mean} are short for a halffilled state. Near the Fermi energy, all states have enough possibilities for scatterings and λ_{mean} is short for all states, except the case of a high spin polarization sp of electron gas. In this case almost all states near the Fermi energy are occupied by spinpolarized electrons. As was mentioned above, electrons can not be scattered between states occupied by spinpolarized electrons. It makes the life time and λ_{mean} of the spinpolarized electrons longer. Influence of the standingwave electrons on conductivities: detection conductivity σ_{detection}
injection conductivity, σ_{injection} scattering current
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