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Running-wave and standing-wave electrons.

The Boltzmann transport equations.

Spin and Charge Transport

All conduction electrons can be divided into two groups: running-wave electrons and standing-wave electrons. The standing-wave electrons are conduction electrons, which bounce between defects or interfaces. Therefore, they do not transport the spin and the charge during their movement. The running-wave electrons freely move in along the crystal. They transport the the spin and the charge during their movement. Both the running-wave electrons and standing-wave electrons transport the spin and the charge due to scatterings.

The the Boltzmann Transport Equations are used to calculate the charge σcharge, spin-diffusion σspin, detection σdetection and injection σinjection conductivities. These 4 conductivities are used in Spin/Charge Transport equations in order to describe the spin and charge transport in different materials and geometries.

Importance of the Boltzmann Transport Equations is that they can describe the spin/charge transport in a case when different transport mechanisms equally contribute. For example, in the case of the Spin Hall effect or the Anomalous Hall effect the band current, which flows along applied electrical field, induces the scattering current, which flows perpendicularly to the applied electrical field.


Possible confusion!!: from 2014 to 2017 I have used names TIA and TIS for groups of spin-polarized and spin-unpolarized electrons, respectively. The reasons are explained here.

 

It is important:

Comparison of spin/charge transport mechanisms

transport mechanism where? number of standing-wave electrons ordinary conductivity, σcharge spin-diffusion conductivity, σspin injection conductivity, σinjection detection conductivity, σdetection

magneto-resistance

dependence of σcharge on spin polarization

spin effects

Band current 1

All electrons are running-wave electrons
(See here)
bulk of a low-resistance conductor none highest highest

semiconductor: high

metal: low

zero none

weak

Band current 2

standing-wave +running-wave electrons
(See here and here)

(1) a conductor with defects

(2) a low- contact-resistance interface

 

moderate moderate moderate moderate moderate weak/ moderate

moderate

Scattering current

(See here)

(1) a conductor with many defects

(2) a high- contact-resistance interface

all lowest lowest high high high

good

 

 

1. in an electron gas there are two types of electrons

- running-wave electrons

- standing-wave electrons

electrons of different types contribute differently to the transport

2. There are two types of electron current:

- band current

- scattering current

-The standing-wave electrons do not contribute to the band current. Both the standing-wave electrons and the running-wave electrons contribute to the scattering current.

-Relative number standing-wave electrons significantly influence the spin transport

-There are more standing-wave electrons in the vicinity of an interface or in a conductor with defects

 

 

 

 


Running-wave electrons & standing-wave electrons

Running waves

Standing waves

The running-wave electrons are moving in the space. They can be scattered. Therefore, these electrons contributes to both conductivities The quantum mechanic does not require that an object with non-zero speed necessary moves in the space. One example is a standing wave, which has non-zero speed, does not move in the space. A standing wave can be decomposed as a sum of two waves running in opposite directions. The standing-wave electrons contribute only to the scattering conductivity.

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: running-wave and standing-wave electrons. The standing-wave electrons do not contribute to the band current. This fact is explained as follows. The effective length of the conduction electrons is rather long (See Fig. 3). 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 is moving forward, is firmly fixed to the electron, which is moving backward. These coupled electrons do not move along crystal. They are fixed at position of defects. Importunately, when an electrical field is applied, there is still one electron moving forward and one electron moving backward. The electrical field does not change the ratio of electrons moving in the opposite directions for this type of electrons. Therefore, these electrons do not contribute to the band current. These coupled electrons are defined as the standing-wave electrons. The electrons, which can move freely along crystal and which are not fixed to one position, are defined as the running-wave electrons.  Only they contribute to the band current.

 

The wave nature of electrons significantly influences the spin and charge transport in a solid. Since an electron is a wave, it is not necessary that an electron with non-zero speed moves and transports a spin and charge. A standing wave has a non-zero speed, but it does not move. For a calculation of the transport it is important to distinguish two classes of the electrons: the standing-wave electrons and the running-wave electrons. A standing-wave electron can be imagined as an electron bouncing back and forward between two defects. Also, a standing wave electron can be viewed as two electrons, which move in opposite directions. In order to build up a standing-wave electron two defects are not necessarily required, but one defect is often sufficient. Also, a standing-wave electron can form at an interface. The probability a standing-wave electron can be formed at a defect or an interface can be characterized by a parameter called the reflectivity R. The reflectivity R is defined as the ratio of the probability for an electron to be a standing-wave type to the probability to be a running wave type.

The standing-wave electrons or running-wave electrons, why it is important?

A standing-wave electron. An example, an electron bouncing between two defects.

Standing-wave photon in laser cavity

Note: An electron can bound to only one defect making a standing wave electron

Note: An electron can bound to an interface making a standing wave electron

Photon is reflecting back and forward between two mirrors forming a standing wave. When the photon passes through the semi-transparent mirror, it becomes the running-wave photon.

Similarly, a standing-wave electron is reflected between two defects (note, only one defect is sufficient to make a standing wave electron)

A. There are standing-wave electrons in conductors with defects, in multilayers and in vicinity of the interface between two conductors( See here and here). The existence of the standing-wave electrons influences significantly the spin and charge transport. For example, in the vicinity of an interface there are more standing-wave electrons and less running-wave electrons comparing to the bulk of the conductor. As result, the ordinary conductivity σcharge decreases near the interface. The detection σdetection  and injection σinjection conductivities experiences even larger changes. The values of both conductivities are near zero in the bulk, but in the vicinity of interface their values may increase substantially and their values may become comparable to the value of the ordinary conductivity σcharge. It is the reason why the spin detection, which is the effect of charge accumulation along spin diffusion, only has been observed at an interface, but not in the bulk of a conductor. It is also the reason why the spin transfer by an electrical current and the spin injection are more effective along or across an interface than in the bulk of a conductor.

 

In the bulk of a high-crystal-quality metal with a low density of defects and dislocations, most of the electrons are running-wave electrons. In a metal with a larger amount of defects or in the vicinity of an interface, there is a substantial amount of the standing-wave electrons. In the vicinity of a contact between two metals, there are both running-wave electrons and standing-wave electrons, because the reflectivity of such a contact is usually small. In contrast, in the vicinity of higher resistance contacts like a tunnel barrier contact or a semiconductor-metal contact almost all electrons are standing-wave electrons.

Do the standing-wave electrons are described by the spin-statistics?

Yes. Both the standing-wave electron and the running-wave electrons are described by the spin-statistics,

Why the standing-wave are described by the spin-statistics, but the localized electron do not? They both do not move?

It is because of a very large difference between scattering probabilities between a running-wave electron to a standing-wave electron and a running-wave electron to a localized electron. The sizes of a running-wave electron are a standing-wave electron are nearly the same. In contrast, the size of a localized electron is much smaller. The scattering probability between a running-wave electron to a standing-wave electron is nearly the same as scattering probability between two running-wave electrons. The scattering probability between a running-wave electron to a localized electron is very small.

It is important that the scattering probability between a standing-wave electron to a localized electron is higher than between a running-wave electron to a localized electron. This is the reason why existence of the standing-wave electron enhances the sp-d exchange interactions

 


 

Band Current & Scattering current

There are several known mechanisms of the electron transport in a solid such as diffusive, ballistic, hopping transport and so on. Here we have divided all transport mechanisms into two groups.  The transport related to the electron movement between scatterings is assigned to one group and the transport related to the electron movement due to the scatterings is assigned to another group. The first transport mechanism is defined as the band current. Only running-wave electrons can contribute to this current. The second transport mechanism is defined as the scattering current. All the localized and delocalized (conduction) electrons, running-wave and standing-wave electrons contribute to this current.

Simplified picture describing two conductivities in a solid. The running-wave electron transport corresponds to electron movement between two consequent scatterings over the length of electron mean free path (red line). The scattering transport occurs over a short distance during electron scattering (blue line)

 

 

There are two major mechanisms of electron transport in a solid:

1. Band current. This transport occurs because of movement of electrons between scatterings. Only running-wave electrons contributes to this transport.

2. Scattering current. This transport occurs because of movement of electrons due to scatterings. Both the running-wave and standing-wave electrons contributes to this current.

There are two major mechanisms of electron transport in an electron gas. A running-wave electrons transport the spin and charge, because they move along crystal. If there is a smaller amount of electrons moving in one direction than the amount of electrons moving in the opposite direction, there is a flow of charge and/or spin in this direction. Such a type of current is defined as the band current. This transport describes the transport of the charge and the spin by electrons between scatterings. It is important to emphasis that only the running-wave electrons, but not standing-wave electrons contribute to this current.

The band current occurs because of the movement of electrons between scatterings. In electron gas the conduction electrons move between scatterings at a high-speed 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.

In the electron gas the electrons are continuously scattered from one quantum states to another quantum state. When the probability for an electron to be scattered in one direction is higher than the probability to be scattered in the opposite direction, there is a flow of the spin and charge in this direction. This current is defined as the scattering current. Both the standing-wave and running-wave electrons may contribute to this current. Usually the scattering current is significantly less efficient than the running-wave-electron current for the charge and spin transport

Why do we need to divide the current into different kinds?

Reason 1: It is simpler to calculate each current separately.

Reason 2: The path of each electron in the phase space can be separated when it is scattered and when it moves between scatterings. Therefore, the contribution of each transport mechanisms should be treated in the Boltzmann transport equation individually and separately.

Reason 3: From a case to a case the contributions of the band and scattering currents are very different.

Reason 4: Spin properties

Two mechanisms of the Spin&Charge transport by Running-Wave Electrons.

(1) Band Current . Origin: Constant movement of electrons in a metal

(2) Scattering current. Origin: electron scattering from one state to another

Click on image to enlarge it. Blue balls represent defects. Brown ellipse shows distribution of wave function of the running-wave electrons. A scattering occurs when wave functions of two states are overlapped (shown in red color) Note: An electron current is shown. A hole current of the running wave electrons is shown here.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


 

 

Why do we need the Boltzmann Transport Equations ?????

Spin/Charge Transport Equations

 

From the Boltzmann Transport Equations, the charge conductivity , spin conductivity , detection conductivity and injection conductivity can be found

 

 

 

 

Derivation of the Boltzmann transport equations

 
Wikipedia article about the Boltzmann transport equation is here.

What we are looking for???

The Boltzmann transport equations are the equations to find the distribution function . Knowing the distribution function , it is straightforward to calculate spin and charge currents.

The probability to find an electron at a point r with pulse p at time t is described by the distribution function (often it is called the density function, but instead I will use the more-common quantum-mechanical name the distribution function).

How to include spin in the transport equations???

three distribution functions are used, which describes the probabilities for a quantum state to be filled by one, two or none electrons correspondingly. Also, the half-filled states can be filled either by an electron from group of spin-polarized or spin-unpolarized electrons. The condition that a quantum state can be only in one of these three states gives

Instead of one classical Boltzmann transport equation, it should be used a system of three transport equations for half-filled “full” and “empty” states.

Factors, which can change the distribution functions :

1) movement of electrons in the space (diffusion). The running-wave electrons are always moving. Additionally the electrons can move by scattering from a state to a state.

2) changing of electrons energy and momentum under an applied field.

3) The conversion of electrons from one assembly to another assembly.

4) The relaxation of a distribution function to an equilibrium distribution function.

 

The Boltzmann equation, which describes a temporal evolution of the distribution function, is given as as

where "i" labels the distribution functions for half-filled “full” and “empty” states; is the term, which describes a change of the distribution function due to the movement of the running-wave electrons; is the scattering term, which describes the changing of the distribution function due to movement of the electrons by scatterings from a quantum state to a quantum state;is the force term, which describes the changing of the distribution function due an external field; describes electron conversion between assemblies; is the relaxation terms, which describes relaxation of the distribution function to an equilibrium distribution function due to the electron scatterings.

 

 

 

Equilibrium

it is a case when there is no external field, there are no gradients of temperature or spin polarization. Therefore, there are no charge current and no spin current.

In equilibrium the distributions of spin-polarized electrons , the distributions of spin-unpolarized electrons, electrons in full-filled states and "empty" states are determined by the spin statistics. and the spin polarization sp of the electron gas. The spin polarization sp of the electron gas can be calculated as

.

Since usually the conversion rate between TIA and TIS assemblies are not energy dependent, from Eq (6) the spin polarization of electron gas is given as

Since a quantum state can be filled either by one, none or two conduction electrons, the equilibrium distribution functions are satisfied to the condition:

 

The Relaxation-Time Approximation

It allows to describe the relaxation term without specifying the mechanisms of the scatterings.

In this approximation two facts are used:

1) When all external perturbations are switched off, the distribution functions relax to the equilibrium distribution functions.

2) The speed of the relaxation is linearly proportional to a deviation of the distribution function from equilibrium

Except for some special cases it is safe to assume that an external perturbation (applied electrical and magnetic fields, a thermo gradient, a gradient of spin accumulation, a sp-d exchange effective field and a spin-orbit effective magnetic field) is sufficiently small so that under the perturbation the distribution function is only slightly different from the distribution function in an equilibrium. In this case the distribution function can be represented as

where is the distribution function in equilibrium and condition (8) is valid for any point of the phase space.

In the case when the applied field is switched off, the distribution function F_i relaxes to the equilibrium distribution function F_i,0 Then

where "i" corresponds to half-filled, full-filled and "empty" states" and is the momentum relaxation time.

Since electrons are constantly scattered between assemblies at a high rate, is the same for both assemblies and is the same for distributions of half-filled, full-filled and "empty" states.

 



 

Spin conversion

terms describes the conversion of electrons between assemblies

The electrons of the TIA assembly are converted into the TIS assembly, because of the different mechanisms of spin relaxation. The rate of the conversion due to spin relaxation is proportional to the number of spin states in the TIA assembly:

where is the spin life time.

The electrons of the TIS assembly are converted into the TIA assembly due to the spin pumping. The conversion rate is linearly proportional to the number of the spin states in the TIS assembly:

where is effective time of conversion, which is reverse proportional to

  1. magnitude of the applied magnetic field or/and
  2. the magnitude of the effective magnetic field due to exchange interaction of delocalized electrons of the electron gas with localized d- or f electrons
  3. the magnitude of the effective magnetic field due to the spin-orbit interaction

From Eqs.(12.1) and (12.2), the conversion rate between the TIS and TIA assemblies is obtained as

 

 



 

 

The electrical and spin currents. The number of electrons in assemblies

Knowing the distribution functions for spin, full and empty states allows to calculate the number of states in the assemblies

The number of quantum states in a volume of the phase space is equal to . Integrating over all states in a special volume, the number of electron is calculated as

where index "i" indicates half-filled states of the TIS assembly, full-filled states, "empty" states and half-filled states of the TIA assembly. For example, the number of half-filled states in the TIA assembly (spin accumulation) can be calculated as

where F_spin is the distribution function for the half-filled states in the TIA assembly.

 

Since each state transforms charge "-e", the current of the i- states can be calculated as

 

In an equilibrium the number of the i- states in an assembly can be calculated as

where D(E) is the density of states. Since in the equilibrium there is no current

where e is the charge of an electron

When an external perturbation applied to the electron gas, the currents may flow in a metal and and the electron can be converted either in or out from the an assembly

The currents can be calculated as

note: only the term of running-wave electrons and the scattering term of the Boltzmann transport equation give contribution to the currents Eq.(11.8)

 

It is often important to calculate a current along some direction, for example, along a gradient of the chemical potential or a gradient of the spin polarization. In this case the cylindrical coordinate needs to be used. If theta is the angle between the electron movement direction and the direction of the current flow. Then, the number of electrons can be calculated as

where D(E) is the density of the states.

The current can be calculated as

The conversion rate can be calculated as

note: only the conversion term o of the Boltzmann transport equation gives contribution to current Eq.(11.8)

Note the currents Eq.(11.8) is not zero only in the case when the distribution function depends on electron speed as

The running-wave-electron term and the scattering term of the Boltzmann transport equation are of this type.

The conversion is not zero only in the case when the distribution function depends on electron speed as

The conversion terms of the Boltzmann transport equation is of this type.


 

Continuity equations

The spin/charge transport equations are derived from the Boltzmann transport equations using the continuity equations.

The charge and spin are conserved values. They can not appear from nowhere and can not disappear to nowhere. The half-filled states can be converted between assemblies. The spin and/or charge can diffuse from a point to a point. This continuity equations describes these facts as

where the spin conversion and the spin and charge currents can be calculated from Eqs. (11.4), (11.5), (11.8), (11.9). For details, see here.

note 1: The continuity equations can be obtained by integrating the Boltzmann equations over all quantum states.

note 2: The spin/charge transport equations is a form of the continuity equations.

 



 

 

The same content can be found in this paper and this paper

 

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