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: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

Spin and Charge Transport near an Interface
Spin and Charge TransportThe spin and charge transport in the vicinity of a contact between two metals are substantially different from the transport in the bulk of metals. In the vicinity of the contact, The probability of electron back reflection becomes larger. As a result, the number of the runningwave electrons decreases and the number of the standingwave electrons increases compared to the bulk. In the bulk the major transport mechanism is the current of the runningwave electrons and the scattering current is often significantly smaller. In contrast, in the vicinity of an interface the current of the runningwave electrons decreases or even becomes negligible, but the scattering current increases and often became the dominant transport mechanism.The spin properties of the conduction are substantially different in the bulk and in the vicinity of an interface. If in the bulk the injection conductivity in metals and the detection conductivity in metals and semiconductors are neglegibally small. In the vicinity of the interface the injection and detection conductivities may be substantial.The conductivity at the interface becomes anisotropic. The conductivities for a current flows along and across a contact becomes different.The same content can be found in this paper (http://arxiv.org/abs/1410.7511 or this site for a more upgraded version) .Chapter 11, pp.2935Possible 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.
(Note): The conventional charge conductivity and spindiffusion, detection and injection conductivities are not fixed and unchangeable material parameters. They are changing significantly depending on the density and the distribution of spatial defects in a conductor. They are different in the bulk of a conductor and in the vicinity of an interface. For example, the detection conductivity is zero in the bulk of a metal or a semiconductor, but it can be substantial in the vicinity of a contact. Even the transport mechanism may change in the vicinity of a contact. If the major transport mechanism in a metal is the current of the runningwave electrons, the major transport mechanism becomes the scattering current, for example, in the vicinity of a tunnel junction.
Conduction near interface. Classical model vs presented model
In the classical model it is assumed that the conductivity at contact is steplike. Till the contact the conductivity is constant and it equals to the conductivity in the bulk of the metal. On the other side of the contact, the conductivity equals to the bulk conductivity of the second metal. Also, it is assumed that at an interface between metals there is a deltafunctionlike sharp increase of the resistance, which is called the contact resistance.
According to the presented model the conductivity through the contact does not have any steplike or deltafunctionlike features. The conductivity monolithically changes over a distances ~14 meanfree paths from the bulk conductivity of one metal to the bulk conductivity of the other metal. The runningwave conductivity decreases from the bulk conductivity of the first metal, reaches a minimum at the interface between the metals and again increases to the bulk conductivity of the second metal. The scattering conductivity changes from being negligibly small in the bulk of the metals to becoming a substantial at the interface. The scattering conductivity may even overcome the runningwave conductivity.
Q1. What happens at interface???Q2. Why there is a difference of electron transport in the bulk and in the vicinity of the interface???Two effect influence the conductivity in the vicinity of an interface:(1) Reduction of the number of runningwave electrons and increasing of the number of standingwave electrons. This causes the reduction of the current of the runningwave electrons.
(2) The electron scattering probability in the direction toward the interface becomes smaller that the scattering probability outward the interface. This causes significant increase of the scattering current.
Comparison of conductivity near an interface or a contact and conductivity in bulk of bulk of a metalBulk of metal:(1) Only the runningwave conduction. The scattering conductivity is near zero. (2) The detection conductivity is near zero. (3) The injection conductivity is near zero in a metal and it equals to the charge conductivity for a nondegenerate semiconductor. At interfaceRunningwave electron conductivity: (1) It becomes smaller , because of a reduction of the number of the running wave electrons (2) The detection conductivity may become substantial. (3) The injection conductivity of a metal may become substantial. (4) The charge conductivity may become dependent on the spin accumulation sp in a metal. (5) The conductivity at the interface becomes anisotropic. The conductivity for a current flows along and across a contact becomes different.
The scattering conductivity If in bulk of a metal the scattering conductivity is near zero, it becomes substantial near an interface. It is because near an interface the scattering probability of an electron to be scattered toward the interface and in opposite direction are substantially different. There are two types of the scattering conductivity at an interface. (1) The first type is the scattering conductivity by the runningwave electrons. It is the conductivity of a lowresistance contact (for example, a contact between two metal) (2) The second type is the scattering conductivity of the standingwave electrons. It is the conductivity of a high resistance contact (for example, a tunnel junction or a contact between a metal and a semiconductor) .
Runningwave electron conductivity at an interface or contact.
A contact or an interface between two metals can be considered is an obstacle for the movement of an electron. The electrons can be compared with photons of light. When light passes through the boundary between two materials, only a part of light passes through the boundary and a part of light is reflected back. Similarly for an electron, there is a finite probability that the electron is reflected from an interface and there is a finite probability the electron passes through the interface as a runningwave electron. Obviously, the electrons, which are reflected from the interface, do not transport the spin and charge through the contact. They can be considered as bounded by the interface and they become the standingwave electrons. In the case of a contact between similar metals, the electron reflection at the contact is a small and in vicinity of contact there is a substantial number of the runningwave electrons. In contrast, the reflection is nearly full at a contact between different materials, like a metalsemiconductor or a tunnel barrier, and almost all the electrons in the vicinity of this contact are the standingwave electrons. The electron meanfree path determines an effective length where the conductivity changes from a bulk type to an interface type.
Near interface the number of the runningwave electrons increase and the number of standingwave electrons increases. Amount of decrease of running wave electrons at a contact depends on materials of the contact. For example, in case of a contact between two similar metals the amount of the standingwave electrons at contact is not large. In contrast, in the case of a tunnel junction all electrons at the contact are expected to be standingwave electrons. Therefore, an interface can be characterized by a parameter, which describes a ratio of standingwave electrons to runningwave electrons. We define this parameter as the contact reflectivity R.
The contact reflectivity R may depend (1) Electron energy. For example in the case of a Schottky barrier, the width of the barrier is narrower for electrons of higher energy and they experience small reflectivity from the contact (2) Temperature. The meanfreepath independent scatterings (for example, scattering on phonons) may reduce the number of the standingwave electrons and increase the number of the runningwave electrons. It is similar to the the case of the electron transport in the bulk of metal with defects. This effect may only substantially influence the number of the runningwave electrons only in the case of a small interface reflectivity R.
The approximation of the squareshape wavefunction
Within the approximation of the squareshape wavefunction the special distribution of the wave function of an electron moving along the xdirection at point x=x0 is approximated as Main idea: It is assumed that when the wave function of an electron (Eq. (1.1)) touches the interface, a runningwave electron becomes a standingwave electron with the probability R, where R is defined as the reflectivity of the interface. The number of the standingwave electrons depends on the angle theta between the electron movement direction and the contact normal. In the case when the contact is in the yzplane at x=0, the electron at position x=x1 touches the contact if or In an equilibrium the electron movement directions are distributed equally in all directions. This means that the probability that an electron moves in the direction between theta and theta+ d_theta is
and From Eqs (1.3,1.4) the condition that an electron at position x=x1 touches the contact is As was mentioned above, the probability that a runningwave electron, which touches the interface, becomes a standingwave electron is R. Therefore, the number of the standingwave electrons at point x=x1 is calculated as It should be noted that in the case of a more realistic Gaussianshape wave function (Gaussianshape envelop function), the probability of electron to be a standingwave electron depends on the amount of overlap of the electron wave function with the regions of the first and second metals
When an electron is passing through a contact, it can be reflected or scattered to become a standingwave electron. Otherwise, it is passing the contact as a runningwave electron. In the vicinity of a contact, the electron wave function spreads into both metals. As a runningwave electron approaches the contact interface, the more of its wave function penetrates into the second metal making the probability of the electron reflection higher. This is the reason of the continuous decrease of the number of the runningwave electron when the distance to interface decreases. As a runningwave electron moves through the contact interface, the overlap of its wave function with the regions of the first and second metals changes continuously without a step. This literally means that at both sides of the contact the number of the runningwave electrons should be the same or in other words the distribution of the runningwave electrons throughout the contact should be without steps. This condition gives the ratio between contact reflectivity R1 of metal 1 and contact reflectivity R2 of metal 2 as where indexes 1 and 2 correspond to metal 1 and metal 2 from sides of the contact. From Eq. (1.9) the contact refractivity of one side of the contact can be calculated from the refractivity of another side of the contact as note: Since the contact refractivity R can not be negative, in case of n2 is greater than n1, the contact refractivity of the metal 2 can be only in the interval for example at a contact between a semiconductor and a metal, the contact refractivity in the metal is very close to 1 and almost all electrons near contact are the standingwave electrons. In contrast, the contact refractivity in the semiconductor may be from 0 to 1.
The number of runningwave electrons through the contact can be calculated as
The number of the standingwave electrons is calculated as
Figure 18 shows the distribution of runningwave and standingwave electrons in the vicinity of a contact between two metals with different numbers of the conduction electrons. At a distances longer than /2 from the interface all electrons are runningwave electrons. For shorter distances the number of the runningwave electrons decreases becoming smallest at the interface. In contrast, the number of the standingwave electrons increases becoming largest at the interface.
For calculation of Fig. 18 it is assumed that in each metal the meanfree path is the same for all electrons. However, the meanfree path depends significantly on the electron energy and the type of the state the electron occupies (See here). Figure 19 shows the calculated numbers of spin polarized and spin unpolarized conduction electrons at different distances from the interface. The energy dependence of was included into the calculations.
The "full" states are the most affected in the proximity of the contact interface. Even at the long distance of 4 um from the contact there is a significant number of the standingwave electrons occupying the "full" states at low energy. For a shorter distance of 0.5 um there is a significant number the standingwave electrons of all types and all energies.
The conductivity is different for currents flow along and across an interface!!!!
Calculation of the currentCurrents flow along and across interfaces calculated below. For both cases the current is calculated from Eq. (11.8).
As can be seen from Eq. (1.2) the number of the running wave electrons depends on the electron moving direction. The closer the moving direction is to the interface normal, the higher the probability for the electron to be the standingwave type. This makes the conductivity of the runningwave electrons to be anisotropic in the vicinity of a contact. This means that near an interface the conductivity to become different for the currents flowing along and across the interface.
Current flows across the interfaceThe current that flows perpendicularly to the interface can be calculated from Eq. (11.8) here., where theta is both the angle between the electron movement direction and the normal to the interface and the angle between the electron movement direction and the current direction. From Eq. (1.3) the probability that an electron is of the runningwave type is Substituting Eq. (1.20) into Eq. (11.8) here. and integrating gives the current flowing perpendicularly to the interface as
or
Current flows along the interface
In order to calculate the current flowing along the interface, we use the cylindrical coordinates, where theta is defined as the angle between the electron movement direction and the direction of current flow. Phi is the angle between the normal to the interface and the projection of the electron movement direction into the plane perpendicular to the current flow. Than, the condition that the electron of the running wavetype is The range of angles, where condition (1.22) can be satisfied is In the region in the vicinity of the interface, where , the current along the interface can be calculated from Eq. (11.8) here, and Eqs. (1.23), (1.22) as or The integrals of Eqs. (1.21b), (1.24b) can be evaluated numerically.
Figure 20 shows the conductivity of the current of the runningwave electrons in the vicinity of a contact between two metals and a contact between a metal and a dielectric. Through the contact the conductivity decreases from the value of the bulk conductivity of metal 1, reaches a minimum at the contact interface and then increases to the value of the bulk conductivity of metal 2. In the case when the reflectivity R of the contact is 1 (for example, a metaldielectric contact (Fig.20 (b))), at the contact interface the conductivity of the current of the runningwave electrons becomes zero and the scattering current becomes only the remaining transport mechanism. It is the case, for example, of transport through a tunnel junction. The reduction of the conductivity occurs in each metal within /2. The conductivity for a current across the interface decreases sharper than for a current along the interface.
For the calculation of Fig. 20 it is assumed that in each metal the meanfree path is the same for all electrons. As was explained above, it is necessary to include in calculation the dependence of on the electron energy and on the type of state, which the electron occupies. Figure 21 shows the charge, injection, spindiffusion and detection conductivities at different distance from the contact. Since the density of states may be different for the metals at the sides of the contact, the state conductivities can not describe the change of the conductivity through contact. Instead the derivation of the conductivities has been used in Fig.21. The conductivities can be calculated as
Conclusions(1) As closer position of an electron to interface, the higher possibility for the electron to be a standingwave electron (Fig.3) (2) The largest number of standingwave electrons occupies the "full" state with low energy and "spin" TIA states near the Fermi energy, because of the long meanfree path of these states. (Fig.3) note: in calculation of Figs.3,4 the meanfreepath independent scattering (scatterings on phonon) are ignored. This gives unrealistically large number of standingwave "full" states at lower energy, because of unrealistically long meanfree path. (3) Charge conductivity decreases. Therefore in the vicinity of an interface the conductivity is always lower than that in the bulk of a metal (the contact resistance). The most affected is the runningwave conductivity conductivity of the "full" states. (4) The injection state conductivity is still asymmetric as the function of the energy. Therefore, in the case of a metal the injection conductivity is not large. However, it is not fully asymmetric and even for the metal with constant density of the state, the injection conductivity can be nonzero (Fig.4(center)). (6) The spin conductivity decreases in the vicinity of the interface (7) The detection conductivity becomes nonzero in the vicinity of the contact. Note1: the detection conductivity is zero in the bulk of semiconductors and metals
Scattering conductivity at an interface.In contrast to the bulk, where there is no scattering conductivity or the scattering conductivity is small, near an interface the scattering conductivity becomes substantial. The tunneling through a tunnel barrier or a Schottky barrier are examples of the scattering conductivity through the interface.
CalculationsThe scattering conductivity is due to scatterings from one quantum state to another quantum state. The scattering probability from one quantum state to another quantum state is linearly proportional to the overlap integral of the wave functions of two states. Below the overlap integral is calculated.
Calculations of the scattering conductivity in the vicinity of an interface. Click here to expand.
(1) The approximation of the squareshape wave function is used. Note: Of cause, the approximation of the squareshape wave function is oversimplification. However, it gives a simple and easytounderstand solution, which is very close to real one. The usage is more realistic shape of wave function (for example, the Gaussian shape) is straightforward.(2) Both runningwave electrons and standingwave electrons contribute to the scattering conductivity. Below it is assumed that a standingwave and a runningway electron contributes equally to the scattering conductivity. (3) A contact between two metals, which are only different by their different meanfree paths, is calculated. In a realistic contact the densities of the states in metals are different as well. The inclusion this difference into the calculations is straightforward.
Scattering in the bulk Within the approximation of squareshape wave function, in bulk the wave function of an electron, which moves along the xaxis and which center at time t=to is at point x0, is calculated as From Eq. (3.1) the overlap integral between wave functions of two electrons, which centers are at points x0 and x0+delta_x (delta_x>0), is calculated as and the overlap integral for electrons at points x0 and x0delta_x (delta_x>0), is calculated as Eqs. (3.2) and (3.3) can be combined for any sign of delta_x as From Eq. (3.4) in the bulk the probabilities for electron to be scattered backward and forward are the same. This a reason there is no scattering current in the bulk for the running wave electrons.. Note: In the bulk the hoping conduction can exist. The hoping conduction is the scattering conduction between the standingwave electrons. In this case the electron wave function is different for a neighbor electrons in the forward and backward directions. Therefore, the overlap integral is different in the forward and the backward directions.
Scattering in the vicinity of an interface.
The wave function is continuous through the interface. It is assumed that even an wave function extended over interface keeps its square shape. A simplest contact between two is calculated, where only difference between the metals is different value of the meanfree path. Near an interface an electron can be scattered on imperfections of the interface. Therefore. the electron meanfree path is shortest near interface. Depending on the electron position x0, the meanfree path can be calculated as
Overlap IntegralThe overlap integral between the wave functions (Eq.(3.1) of two electrons at positions x0=x_a and x0=x_b (x_b> x_a) is calculated as
in the case when x_a> x_b Combining Eqs.(3.7) and (3.8), we have
.

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