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: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

MeanFree Path. Bulk conductivity of a metal with defects.
Spin and Charge TransportThe meanfree path is average distance an electron passes between scatterings. The meanfree path is also the effective length of the wavefunction of delocalized electrons.The influence of the meanfree path on transport becomes substantial in two cases. The first case is when the meanfree path becomes comparable to the average distance between defects. The second case is when an electron distance from an interface is comparable or less than the meanfree path. In both cases the number of runningwave electrons decreases and the number of standingwave electrons increases. As a consequence, the conductivity reduces and importantly the spin properties of the conductivity are significantly modified. The detection conductivity becomes nonzero and the injection conductivity becomes larger.
The same content can be found in this paper (http://arxiv.org/abs/1410.7511 or this site for a more upgraded version) .Chapter 10, pp. 2529
Results in short I. The length of the meanfree path significantly depends on the electron energy.
2.Because of defects, the number of standingwave electrons increases and the number of the runningwave electron decreases. 3. The probability for an electron to become a standingwave electrons become higher when a) distance between defects become shorter b) meanfree path becomes longer 4. The number of standingwave electrons depends on the electron energy and the type of the state the electron occupies. There are significant number of standing electrons occupying: a) "full" states at energies lower than the Fermi energy. b) "spin" states of the TIA assembly at energies near the Fermi energy. 5. The detection conductivity becomes nonzero.6. The charge conductivity becomes dependent on the spin polarization of the electron gas sp. There is a Magnetoresistance Effect in this case
7. It might be possible that the spin properties of a metal can be engineered by fabricating artificial defects inside the metal with an optimized distribution.
The meanfreepathThe meanfree path is the distance through which an electron moves between scatterings.
Since an electron is a wave package, the meanfree path can be considered the effective width of this electron wave package or an effective size of an electron or the effective length within which the electron interacts with defects, phonons and other inhomogeneities.
In case of a larger number of scatterings, the meanfree path becomes. Q1. What is the difference of electron transport in a conductor with defects and without defects?
A. There are two distinguished differences between conductivities in conductors with and without defects. The first difference is the value of the detection conductivity . It is zero in a conductor without defects for any electron energy and it is nonzero in a conductor with defects at energies near the Fermi energy. The second difference is the dependence of the charge (conventional) conductivity on the spin polarization sp of the electron gas. The charge conductivity is independent of the spin polarization in a conductor without defects. This means that there is no magnetoresistance effect in a conductor without defects. In contrast, in a conductor with defects, the charge conductivity is dependent on the spin polarization of the electron gas and there is a magnetoresistance effect. The conduction of electrons of only energies near the Fermi energy depends on the spin polarization. Therefore, this magnetoresistance effect may exists only in a metal, but not in a nondegenerate semiconductor. Q2. Why the charge is accumulated along the spin diffusion in a conductor with defects, but the charge is not accumulated in a conductor without defects?In the bulk of a conductor without defects a spin diffusion current flows without any charge diffusion current. Such spin current is possible only because in the bulk of a conductor without defects there is a balance. The numbers of spinpolarized and spinunpolarized electrons, which diffuse in the opposite directions along the gradient of the spin accumulation, are exactly equal. This results in a spin current without any charge accumulation and =0. In a conductor with defects the number of the runningwave electrons in “full” states of the TIS assembly (spinunpolarized electrons) becomes substantially smaller at lower energies. The number of the runningwave electrons in the “spin” states of the TIA assembly (spinpolarized electrons) becomes smaller at the Fermi energy. This unbalances the number of spinpolarized and spin–unpolarized electrons, which diffuse in opposite directions. As a result, the charge is accumulated along the spin diffusion and . Q3. How it is possible that the spinpolarized and spinunpolarized electrons have different conductivity in a conductor with defects?Q4. Why there is a magnetoresistance effect in a conductor with defects, but there is not in in a conductor without defects?In a conductor without defects, all electrons are runningwave electrons. When the spin polarization of the electron gas changes, the numbers of spinpolarized and spinunpolarized electrons change. However, the total number of the runningwave electrons remains constant and the charge conductivity remains unchangeable. In a conductor with defects the ratio of the standingwave to runningwave electrons is different for the TIS and TIA assemblies. This means that when the spin polarization of the electron gas changes, the total number of the runningwave electrons changes. As a consequence, the conductivity of the runningwave electrons become dependent on the spin polarization of the electron gas.
Calculations of the meanfree path in a conductor with defect.The mean free path is finite, because an electron experiences scatterings If there would be no scatterings, the electron wave function would extend from one edge of sample to other edge.
3 factors, which influences the electron scatterings:
1. quantum contribution,2. meanfreepathdependent scatterings3. meanfreepathindependent scatterings
The Quantum contribution describes the fact that an electron can be scattered only when there is an unoccupied neighbor state where the electron can be scattered to (See Fig.1). The meanfreepathdependent scattering is due a scattering centers which have a defined position in the space (for example, the scattering on defects and dislocations). The meanfreepathindependent scattering is due a scattering centers which do not have a defined position in the space (for example, the scatterings on phonons). Also, the scattering on defects contribute on this scattering, because an electron has nonzero speed and even if electron meanfreepath is very short, the electron can reach the defect and be scattered after some time.
Three mechanisms, which influence , can be distinguished. The first mechanism is the quantummechanical limitation for a scattering. An electron is a fermion and it can be scattered only if there is an unoccupied state where the electron can be scattered into. In The second mechanism is the meanfreepathdependent scattering. This is the scattering on defects and dislocations, which have some spatial distribution with some average distance between them. When mean is significantly smaller than this distance, the scattering probability is small, but when mean is comparable or longer than the average distance between defects, the scattering probability becomes larger. The third mechanism is the meanfreepathindependent scattering. This is the scattering on phonons and magnons, which does not depend on mean. the meanfree path mean. can be calculated as
Combining these three contributions, the meanfreepath can be calculated as where is the probability that there is an available unoccupied states into which an electron can be scattered, is the scattering probability, which is independent on the mean free path; is the scattering probability, which is independent on the mean free path.; is mean free path when there is no meanfreepath dependent scatterings and there is a sufficient number states in which an electron can be scattered into Meanfreepathdependent scattering he distribution of the defects in a metal depends on the metal fabrication method. The normal or Gaussian distribution of defects is the most probable. In this case the probability, that the distance from one defect to the closest neighbor defect equals d, can be calculated as Where d_0 is the average distance between defects and delta_d is the width of the distribution. Normalizing the distribution (1.3)
we obtain
We can assume that an electron can be scattered only if its wave function overlap at least one defect at any time. Therefore, an electron is scattered only if its meanfree path is shorter than the distance between defects. Then, the scattering probability is calculated as Integrating Eq. (1.6) the probability of the meanfreepathdependent scattering is obtained as
Quantum contribution to the scattering probability
The Quantum contribution describes the fact that an electron can be scattered only when there is a place where to be scattered to (See Fig.1) . The electron from a "full" state can be scattered either into "empty" state or "spin" state. The probability that near a "full" state there is "empty" state or "spin" states. Since such probabilities described by the distributions functions, the for "full" states is calculated as: where are distribution functions for the "spin" states of the TIA assembly, the "spin" states of the TIS assembly and "empty" states The electron from a "spin" state can be scattered either into "empty" state or another "spin" state. Also, an electron from a "full" state can be scattered into the "spin" state. The electron spin influences the scattering from one "spin" state to another "spin" state. For example, all electrons of the TIA assembly have the same spin direction and an electron can not be scattered from one "spin" state. In contrast, the electron can be scattered into a "spin" state of the TIS assembly, because spin direction of this "spin" state can any direction with equal probability. Therefore, he for "spin" states of TIA assembly is calculated as: An electron from a "spin" state of the TIS assembly can be scattered into a "empty" state or an electron from a "full" state can be scattered into the "spin" state or "spin" states of the TIA assembly or "spin" states of the TIS assembly. Therefore, he for "spin" states of TIS assembly is calculated as:
The meanfree path is calculated from Eq.(1.1) using Eqs. (1.71.10) Figs.47 shows calculated meanfree path as the function of an electron energy. In all figures, when it was not mentioned, the spin polarization is 50%, the average distance between defects d is 2 um, the width of the distribution of the distances between defects delta_d is 0.3 um, is 1 um (See Eq.(1.1) ) , the ratio of probabilities of dependent to independent scatterings is 80 %.
Main Properties of the meanfree path
1) The meanfree path significantly depends on the electron energy, spin polarization of a metal and the defect distribution in the metal.
2) Below the Fermi energy the meanfree path of "full" states is very long. For example, 3kT below the Fermi energy the meanfreepath of "full" states may be 15 times longer than that at the Fermi energy.
3) The length of the meanfree path of "spin" states of the TIA assembly significantly increase near the Fermi energy with increasing of the spin polarization sp of metal. It can be 10 times larger than the length of the meanfree path of "spin" states of the TIA assembly at spin polarization larger than 90 %.
Runningwave electrons & Standing wave electrons
Note: In bulk of a metal, the major electron transport mechanism is runningwave conduction. This conduction is proportional to the number of the runningwave electrons in the metal.. Note: As the number of defects, dislocations and phase boundaries in a metal increases, the number of the runningwave electrons decreases and the number of the standingwave electrons increases.
The depending scatterings are scatterings from defects, which have defined position in space or standing centers. Therefore, the number of standingwave in a metal should be proportional to the ratio of probability of dependent scattering to probability of all types of scatterings. It should be notice that it is not always the scattering on a defect causes forming of a standingwave electron. We the defect refractivity R_d as a probability that an electron will be a standingwave electron after a scattering on the defect. Therefore, the ratio of the standing wave (SW) electrons to all electrons (standingwave electrons(SW) + running wave (RW)) in metal can be calculated as is the scattering probability, which is independent on the mean free path; is the scattering probability, which is independent on the mean free path (See Eq.(1.1) )
Calculated example of runningwaveelectron conductivity.The case when along runningwave electrons, there is standingwave electrons.
For the case, when there are no standingwave electrons, click here
The conclusions from Fig. 11.
(1) There is a significant number of standingwave electrons for "full" states at energies lower than the Fermi energy.Reason: at these energies the electrons are mainly in the "full" states.
(2) There is a significant number of standingwave electrons for "spin" states of the TIA assembly at energies near the Fermi energy at the spin Reason: At the Fermi energy there is a small number of "full" and "empty"states. This number even further decreases when the spin polarization of a metal increases.
(3) The distribution functions become less monotonic with some local peaks. Note: The present calculation ignores inelastic scattering, where the electron energy does not conserved. The inelastic scattering may make the peaks less sharp.
Dependence on defect distributionclick to expand
Dependence on the distance between defects
Engineering the spin properties of a conductor
It should be noted that the values of the detection conductivity, the injection conductivity and the magnetoresistance depend on a defect distribution. Materials having the larger detection conductivity or/and the larger injection conductivity or/and the larger magnetoresistance are required for a variety of the practical applications. It might be possible that the spin properties of a metal can be engineered by fabricating artificial defects inside the metal with an optimized distribution. A similar technique was proved to be effective in the case of a photonic crystal, where the optical properties of the crystal can be engineered by inserting artificial defects into the crystal.
Normal current & Strange current
An electrical field forces an electron as a negativelycharged particle to move from "" potential to "+" potential.However, in an electron gas the electrons diffuses from regions of higher electron concentration to regions of lower electron concentration. It might not be the direction from "" potential to "+" potential Normal currentThe flow of electrons along an electrical field form "" potential to "+" potential. In this case the charge conductivity is positive.
Strange currentThe flow of electrons along an electrical field form "+" potential to "" potential. In this case the charge conductivity is negative.
Note: in total the electron current should be always normal. Only a current of some type of electrons or a current of electrons of some energy may be strange. For example, the current of "spin" states in the hole current of the runningwave electrons.
Examples of the strange currents1. The current of "spin" states with energies below the Fermi energy. Even though the "spin" states are negatively charged, they diffuse from "+" source to "" drain However, the total current (the hole current), which consists of the current of the "spin" states plus the current of the "full" states, is the normal current. It transport the negative charge from from "" source to "+" drain

I will try to answer your questions as soon as possible