The 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.

 

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The same content can be found in this paper (V.Zayets JMMM 2018 )

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Content

click on the chapter for the shortcut

(1).

 

(12). Questions & Answers

 

 

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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 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.

 

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. Spin-diffusion conductivity σ_spin: (Spin diffusion).

The σ_spin is always larger than the σ_charge.

In a semiconductor the spin-diffusion conductivity is nearly equal to the charge (ordinary) conductivity σ_spin≈σ_charge;

in a metal the spin-diffusion conductivity σ_spin is 10-60% 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 non-zero 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 p-type 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

 

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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 d-electrons do not participate. Also, the standing-wave conduction electrons do not participate in the band current (See here).

All conduction (delocalized) electrons in a solid can be either standing-wave electrons or running-wave electrons. Only the running-wave electrons contributes to this current, the standing-wave 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 low-resistivity 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 2-4 orders in magnitude less effective than the band current.

In a vicinity of interface and contact between two metals, the running-electron-wave conductivity significantly decreases or even can become zero, because of the decrease of number of running-wave 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 low-resistivity contact between two metals.

Even though at a contact between two metals, the conductivity of the band current always become smaller, for a low-resistivity contact it is still larger than the conductivity of the scattering current.

Due to the existence of the standing-wave 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.

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Drift and diffusion band currents

Drift 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 spin-up/spin-down bands is here.

This page describes only the when there are no standing-wave electrons and the band electrons are running-wave electrons (Ideal case)

The standing-wave 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 currents

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Drift 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 almost-equal 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 non-zero 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 non-zero 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 non-degenerate n-type 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 negatively-charged particles in vacuum, when the spin and the negative charge are carried in the same direction. In a non-degenerate p-type 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 positively-charged particles in vacuum, when the spin and the positive charge are carried in the same direction.

 

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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 spin-polarized 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 spin-polarized electrons diffuse from a region of a higher spin-accumulation to a region a lower spin accumulation, additionally there is a current of spin-unpolarized electrons, which flows exactly in the opposite direction. When the opposite currents of the spin-polarized and spin-unpolarized electrons are not equal, the charge is accumulated along the spin-diffusion (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 defect-free conductor there is always an exact balance between opposite currents of spin-polarized and spin-unpolarized electrons.

 

 

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Properties of the bulk

Metals.

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 spin-accumulation. 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.

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Semiconductors

The 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 n-type semiconductor and only electrons of half-filled states contribute in the case of a p-type 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.

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Solution of the Boltzmann equation for the band current

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

 

From the Boltzmann Transport Equations, the ordinary charge conductivity σ_charge, spin-diffusion conductivity σ_spin, detection conductivity σ_detection and injection conductivity σ_injection can be found

 

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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 spin-polarized electrons as "TIA", group of spin-unpolarized electrons as "TIS" and electrons of full-filled 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 spin-unpolarized electrons and is the band current of electrons occupying the full-filled states.

 

are are equilibrium distribution functions of groups of spin-polarized electrons, spin-unpolarized electrons and electrons occupying the full-filled states, correspondingly. They are calculated here.

 

All three currents , and transport the charge, but the spin is only transported by the current of the spin-polarized 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

 

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 standing-wave electrons . All delocalized electrons are the running-wave electrons and all of them contribute to the running-wave 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.

 

 

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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):

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Conductivities, which describes spin and charge currents

Drift current (Fig. 43) currents along gradient of spin accumulation ∇μ Fig. 23 Click on image to enlarge it. The ordinary conductivity σcharge describes transport of the charge in an electrical current. The injection conductivity σinjection describes transport of the spin in an electrical current. The data were calculated from Eqs.(1.5) and Eqs.(1.7) Diffusion current (Fig. 44) currents along gradient of spin accumulation ∇sp Fig. 23 Click on image to enlarge it. The detection conductivity σcharge describes transport of the charge in a diffusion spin current. It is zero for all energies. The spin-diffusion conductivity σspin describes transport of the spin in a diffusion spin current. The data were calculated from Eqs.(1.5) and Eqs.(1.7)

 

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.

Spin-diffusion conductivity σ_spin

The spin-diffusion 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 spin-polarized 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 non-zero 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 non-zero 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 non-degenerate n-type 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 negatively-charged particles in vacuum, when the spin and the negative charge are carried in the same direction. In a non-degenerate p-type 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 positively-charged 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 spin-polarized electrons diffuse from a region with a higher spin-accumulation to a region with a lower spin accumulation, additionally there is a current of spin-unpolarized electrons, which flows exactly in the opposite direction. When the opposite currents of the spin-polarized and spin-unpolarized electrons are not equal, the charge is accumulated along the spin-diffusion. 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 defect-free conductor there is always an exact balance between opposite currents of spin-polarized and spin-unpolarized electrons

 

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Spin-polarization 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 spin-polarization 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 spin-polarization 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.

 

 

 

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Conductivity of spin diffusion σ_spin

The spin-diffusion 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 spin-polarized electrons at the Fermi energy (See here)

 

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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 here

Notes:

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 (1-3)kT above the Fermi energy. The injection conductivity σ_injection is equal but of opposite sign to the charge conductivity for energy less than (1-3)kT below the Fermi energy. This means that in the bulk of non-degenerated semiconductors the transport of the spin by a drift current is the most effective. In p- and n-semiconductors 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 decreases

5) In both the semiconductors and the metals the detection conductivity σ_detection is zero.

note: the classical model of the spin-up/spin-down bands incorrectly predicts that the detection conductivity is always equal to the injection conductivity (See here)

6) The spin-diffusion conductivity σ_spin is larger than the charge conductivity.

 

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Electrons and Holes

for more details about the holes and electrons click here

Both the “electrons” and the “holes” are absolutely identical particles. They both are half-filled 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 half-filled 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 negative-charged 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 positively-charged 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 non-degenerate n-type 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 full-filled states is small and the major transport mechanisms is the transport of the half-filled states. A half-filled state is negatively charged and its spin is ½. Therefore, the half-filled 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 n-type semiconductor is called the electron conductivity.

In a non-degenerate p-type 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 full-filled and half-filled states substantially contribute to the transport. Since σ_μ,full is positive and σ_μ,TIE,σ_μ,TIS are negative, the full-filled and half-filled 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 full-filled state is zero and only half-filled 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 p-type 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.

 

 

 

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Charge, spin-diffusion, 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 non-degenerate semiconductor the conductivity ratios are

 

 

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The spin-diffusion length

 

Substituting Eq. (15.3) into Eq. (4.8) gives the spin-diffusion length in a non-magnetic metal as

 

 

 

 

 

 

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The conductivities in a conductor with defects and multi-layers

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The transport properties, which are described above on this page, are only valid for the case of transport in the bulk of a high-conductivity conductor without defects and multi-layers. This is the case when the periodicity of the crystal is not broken and all conduction electrons are running-wave electrons. When the number of defects decreases, the number of the standing-wave electrons increases as well. It significantly changes the properties of the spin transport. When the number of defects increases further, the scattering current become dominated. The spin transport properties of the scattering current are substantially different from that of the band current. In the following we discuss how the existence of the standing-wave electrons influences all conductivities .

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Running-wave electrons and standing-wave 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: running-wave and standing-wave 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 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.

 

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

 

 

How do we know how many standing-wave electrons and how many running-wave electrons??

The mean-free path is an important parameter, which determines the number of the standing-wave 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 standing-wave 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 full-filled 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 spin-polarization sp, when the λ_mean of spin-polarized 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 full-filled states, which have no unoccupied places where an electron can be scattered into. There are only a few of half-filled states, which have one unoccupied place. The probability of an electron to be scattered from a full-filled states is very low, because most of time a full-filled state is surrounded by other full-filled states, into which an electron can not be scattered. Rarely a half-filled state is nearby and a scattering event may happen. Therefore, the life time of full-filled state is long and the λ_mean is long as well. In contrast, a half-state is always surrounded by full-filled states, from which an electron can be scattered into the half-filled state with a high probability. Therefore, both the life time and λ_mean are short for a half-filled 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 spin-polarized electrons. As was mentioned above, electrons can not be scattered between states occupied by spin-polarized electrons. It makes the life time and λ_mean of the spin-polarized electrons longer.

Influence of the standing-wave electrons on conductivities:
The existence of the standing-wave electrons affects all four conductivities σ_charge, σ_spin, σ_injection and σ_detection. Since the standing-wave electrons do not participate in the band current, both the ordinary charge conductivity σ_charge and the spin-diffusion conductivity σ_spin decrease. For example, the ordinary conductivity σ_charge decreases in a conductor with defects and in the vicinity of an interface. Due to the existence of the standing-wave electrons at an interface between two metals, there is always a contact resistance between the metals even in the case when there is no energy barrier between them. When a thickness of a metal decreases, its conductivity σ_charge becomes smaller because of an increase of the number of the standing-wave electrons. However, when the thickness becomes thinner than the mean-free path, the conductivity may become larger again because the transport changes to the 2D type. Because of the electron confinement, the electrons can not be scattered across the interface and they move only in one direction. As result, the number of standing-wave electrons decreases and the conductivities σ_charge and σ_spin increase. For more details, see here and here

detection conductivity σ_detection

As can be seen from Fig.3, the number of the standing-wave electrons significantly dependents on the electron energy. As consequence, the spin properties of the conductivity may change substantially. For example, the detection conductivity σ_detection may become non-zero. As can be seen from Fig. 3, the mean-free path λ_mean of full-filled states is rather long for energies below E_F-2kT. Therefore, many of electrons of these states are standing-wave electrons and they do not participate in the band current. The reason, why the detection conductivity σ_detection is zero in Fig.44, is the exact balance of the opposite flows of spin-polarized and spin-polarized electrons along a gradient of spin accumulation. In a conductor with defects or in the vicinity of an interface, more spin-unpolarized electrons and less spin-polarized electrons become the standing-wave electrons. It breaks the balance and the σ_detection becomes non-zero. This is the reason why the spin detection effect is only observed at a contact between metals, but not in the bulk of a metal.

injection conductivity, σ_injection
The injection conductivity σ_injection may increase due to the existence of the standing-wave electrons. As was explained above, in an electrical current the electrons of energy above and below the Fermi energy EF transport the spins in opposite directions. Since a metal has almost the same amounts of electrons with energies above and below the Fermi energy E_F , the opposite spin currents nearly compensate each other and the σ_injection in a metal is small. Such balance may be broken due to standing-wave electrons. The number of standing-wave electrons is not the same for energies above and below the Fermi energy E_F . It makes σ_injection larger. Such an enlargement of σ_injection is more profound in the vicinity of an interface. It has an important consequence. Due to the enlargement of the σ_injection at an interface, the spin injection is more effective through a contact between metals rather than within a single metal. For this reason, the threshold current for magnetization reversal by the spin-transfer torque in a magnetic junction is smaller than the threshold current for a current-induced domain movement. In the first case the spin-transfer torque is due to spin injection between metals, when the spin injection is more effective. In the second case the spin-transfer torque is due to the spin injection within one metal, when the spin injection is less effective.

scattering current
In a metal with a high density of defects and a high resistance or in the vicinity of a high- resistance contact, the scattering current may become a dominated transport mechanism. The spin transport properties of the scattering current are substantially different from those of the band current. For example, the ordinary charge conductivity σ_charge may significantly depend on the spin polarization of the electron gas, which is a rare effect for the drift current. Both the injection conductivity σ_injection  and the detection conductivity σ_detection are larger for a scattering current. All spin-dependent effects are more profound for a scattering current.

 

 

 

 

 

 

 

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