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Spins of Conduction electrons in a ferromagnetic and non-magnetic metals.

Spin and Charge Transport

The conduction electrons in a ferromagnetic metal are spin-polarized. It means that all conduction electrons can be divided into two groups: spin-polarized and spin-unpolarized. In the group of spin-polarized electrons, all spins are directed in one direction. In the group of spin-unpolarized electrons,


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.

in short

ferromagnetic metals

In a ferromagnetic metal, spins of all localized d-electrons are in one direction. The conduction electrons are spin-polarized. All conduction electrons can be divided into two groups: spin-polarized and spin-unpolarized electrons. In the group of spin-polarized electrons, spins of all electrons in one direction. In the group of spin-unpolarized electrons, the spin directions are equally distributed in all directions.

Spins of the localized d-electrons are in one direction because of the exchange interaction with a few neighbor electrons.

The reason why the conduction electrons are spin-polarized is:

1) Scattering between localized and conduction electrons. Therefore, spins of some conduction electrons become in the same direction as spins of localized electrons.

2) The sp-d exchange interaction between the localized and conduction electrons.

non-magnetic metals

In a non-magnetic metal the conduction electrons are not spin-polarized. The spin directions of the conduction electrons are equally distributed in all directions and the total spin is zero.

In a magnetic field the conduction electrons become spin-polarized (See here).

Also, the spin-polarized electrons can be injected or they can diffuse from a ferromagnetic metal to a non-magnetic metal

Non-magnetic metals include paramagnetic and diamagnetic metals.

diamagnetic metal

Localized electrons do not have spin. Therefore, there is no sp-d exchange interaction between the conduction electrons and localized d-electrons and the sp-d scatterings is spin- independent. As a result,, the spin life for conduction electrons in a diamagnetic metal is longest.

paramagnetic metal

Localized electrons do have spin. The spin directions of the localized electrons are equally distributed in all directions and the total spin is zero.

In a magnetic field the localized electrons tend to align their spins along magnetic field. However, thermo- fluctuation realign the spins. The resulting spin distribution is described by the the Brillouin function.



Spins of localized d- electrons and conduction sp- electrons in metals

metals localized electrons magnetization of localized electrons conduction electrons spin-polarization of conduction electrons spin life time spin torque

Non-magnetic metals

none none none

long, moderate (diamagnetic)

short, moderate (paramagnetic)

none

Ferromagnetic metals

in one direction

large

large long/moderate moderate

antiferromagnetic metals

in opposite directions, full compensation none none short strong
ferrimagnetic metals in opposite directions, uncompensated

small

small short strong

 

 

Spins of localized d- electrons and conduction sp- electrons in metals

 

Localized d-electron:

-They have a size of one atomic orbital.

-Their spin direction is determined mainly by the exchange interaction with close neighbor localized d-electrons.

-scatterings between neighbor localized electrons are rare;

-scatterings between localized and conduction electrons are rare;

 

Conduction (delocalized) sp-electrons:

-They have a size of a thousand and more atomic orbitals;

-scatterings between conduction electrons are very frequent;

-spin of each conduction electron frequently rotates after each frequent scattering (See here);

- total spin of all conduction electrons is nearly constant. It my decrease slowly within the spin relaxation time.

 

 

 

 

 

 

 

 


 

Spin precession

Period of spin precession of a free electron around a magnetic field

 
 
   
click on image to enlarge it

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Pauli paramagnetism in non-magnetic metals


 

Fig.1 Pauli paramagnetism. Classical explanation by the model of spin-up/spin-down bands. (left) Conduction electrons when there is no magnetic field. There is an equal number of spin-up and spin-down electrons. (right) Magnetic field is applied, The spin-down electrons have a larger energy. Since in equilibrium the Fermi energy is equal for spin-up and spin down bands, the spin-up band will be filled by a greater number of electrons.

This classical model explains the paramagnetic susceptibility of a non-ferromagnetic metal due to the conduction electrons (For more details, see Kittel, Charles - Introduction to Solid State Physics page 433).

In the absence of a magnetic field there is an equal number of electrons in the spin-down and spin-up bands. Therefore, the net magnetization due to the conduction electrons is zero.

In a magnetic field the spin-up and spin-down electrons have different energy. The energy difference is

In order to make the Fermi energy equal for both bands, after a magnetic field is applied, some electrons from the spin-down band are flipped into the spin-up band. Therefore, the spin-up band will be filled by a greater number of electrons than the spin-down band. Due to the difference in the number of spin-up and spin-down electrons, the net magnetization of conduction electrons becomes non-zero.

At zero temperature the magnetization can be calculated as

where are the total number of conduction electrons with spin-up and spin-down, respectively. D is the density of states near the Fermi energy.

 

Note: according to the classical model, the spin accumulation induced by a magnetic field in a non-magnetic metal is different from the spin accumulation obtained from an injection of spin-polarized electrons from a ferromagnetic metal.

 

Pauli paramagnetism explained from the model of TIS/TIA assemblies

The Pauli paramagnetism is calculated from model of TIS/TIA assemblies differently than in the classical model

Fig.2 The Pauli paramagnetism explained from the model of TIA/TIS assemblies. (left) Conduction electrons when there is no magnetic field. All electrons in TIS assembly (right) When magnetic field is applied some electrons are converted into TIA assembly.

In the absence of a magnetic field all conduction electrons of a non-magnetic metal are in the TIS assembly. Even though there are “spin” states in the TIS assembly, the total magnetic moment of the TIS assembly is zero, because this assembly is time-inverse symmetrical. The magnetic moment is time-inverse asymmetrical and only time- inverse asymmetrical objects may have non-zero magnetic moment. Therefore, the total magnetic moment of an electron gas may be non-zero only in the case when some electrons are in the TIA assembly.

In a magnetic field there is a conversion of “spin” states from the TIS assembly into the TIA assembly at the rate (Eq.20 here)

where is the precession damping time and k=4/3 in the case of a weak magnetic field and k is smaller than 4/3 in the case of a larger magnetic field (See condition (13) here).

There is a back conversion of “spin” states from the TIA assembly into the TIS assembly due to spin relaxation mechanisms. The rate of this conversion is proportional to the number of electrons in the TIA assembly

where is the spin life time.

In equilibrium the rate of conversion of electrons from the TIS assembly into the TIA assembly should be equal to the rate of reverse conversion

From Eq. (12) the equilibrium number of electrons in the TIA assembly can be calculated as

The magnetic moment per “spin” state of the TIA assembly is

where g is the g-factor, is the Bohr magneton.

The induced magnetic moment of an electron gas is equal to the total magnetic moment of the TIA assembly and it can be calculated as

Fig.3. Animated pictures. Comparison of mechanisms responsible for the Pauli paramagnetism as represented by the model of the spin-up/spin-down bands (left) and the model of TIA/TIS assemblies (right)

Therefore, the induced magnetic moment of an electron gas is calculated as

where lambda is a phenomenological damping parameter of spin precession. The number of “spin” states in the TIS assembly should be calculated using the spin statistics as described here.

In the case of a weak magnetic field (See condition (13) here) the induced magnetization is linearly proportional to the magnetic field. This means that in this case the permeability does not depend on the intensity of the magnetic field. For example, in the case of a metal with constant density of states (See Eq. (18b) here), the induced magnetic moment of the electron gas is calculated as

where D is the density of states.

For a stronger magnetic field the permeability decreases when the magnetic field increases. This is because the coefficient k decrease (See Fig.7 here) and the number of “spin” states in the TIS assembly decreases. The TIS assembly has a smaller number of "spin" states, because some “spin” states have been converted into the TIA assembly.

In the case of an even larger magnetic field, nearly all “spin” states may be converted into the TIA assembly (the metal is near 100 % spin polarized) and the induced magnetization saturates at a value

where is the total number of spin states.

Note: in contrast to the classical model, according to the proposed model, the spin accumulation induced by a magnetic field in a non-magnetic metal is exactly the same as the spin accumulation obtained from an injection of spin-polarized electrons from a ferromagnetic metal.

 

Ferromagnetic metals


 

Stoner model of ferromagnetism.

Classical model

See wikipedia explanation here

According to the Stoner model, in a ferromagnetic metal the spin direction of all conduction electrons is either along or opposite to the metal magnetization direction. An exchange interaction has split the energy of states with different spins, and states near the Fermi level are spin-polarized.

The density of spin-down states is smaller than the density of spin-up states. Because of this difference, the conductivity of a ferromagnetic metal becomes spin-dependent, which leads to several interesting effects for transport in the ferromagnetic metal such as a charge accumulation, the shortening of spin diffusion length and the blocking of spin diffusion (See details see here).

Note: At first, in the Stoner model it was assumed that only the density of states of the local d-electrons is spin-dependent. Later this assumption was extended for the density of state of the conduction sp-electrons.

Fig.4. Conduction electrons in a ferromagnetic metal. (left) Model of spin-up/spin-down bands. The blue band is the majority band and the yellow band is the minority band. (right) Model of TIA/TIS assemblies. The blue assembly is the TIA assembly and the yellow assembly is the TIS assembly .

Note: In model of spin-up/spin-down bands, the spin-dependent density of states near the Fermi level is a key feature of the ferromagnetic metals. In contrast, the model of TIA/TIS assemblies does not require the spin-dependence of the density of states in a ferromagnetic metal. According to the model of TIA/TIS assemblies, a ferromagnetic metal, in which the density of states is spin-independent, is possible.

The difference of the density of states for minority and majority spins in ferromagnetic metals has been verified experimentally.

The Stoner model is based on the classical model of spin-up/spin-down bands.

 

 


 

TIA and TIS assemblies in a ferromagnetic metal

Proposed model

In a ferromagnetic metal the d-orbitals are partially filled and electrons can be divided into local d-electrons with non-zero spin and conduction sp-electrons. There is an exchange interaction between d-electrons, which aligns spins of all d-electrons in one direction.

Also, there is an exchange interaction between the local d-electrons and the conduction sp-electrons, which leads to the conversion of “spin” states from the TIS assembly into the TIA assembly. In addition, there is a back conversion from the TIA assembly into the TIS assembly, because of a finite spin life time. In equilibrium the conversion of electrons from the TIS to the TIA assembly is balanced by back conversion.

In equilibrium the conduction electrons in a ferromagnetic metal are in one TIA assembly and in one TIS assembly.

The conversion rate from the TIS assembly to the TIA assembly, which originates from the exchange interaction between sp- and d-electrons, is proportional to the number of “spin” states in the TIS assembly (See Eq.(21) here)

where is the precession damping time for exchange interaction and k=4/3 in the case of a weak exchange field and k is smaller than 4/3 in of case of a larger exchange field (See condition (13) here).

The electrons are converted from the TIA assembly into the TIS assembly due to spin relaxation mechanisms. The rate of this conversion is proportional to the number of electrons in the TIA assembly

where is the spin life time.

In equilibrium the rate of conversion of electrons from the TIS assembly into the TIA assembly should be equal to the rate of the reverse conversion

From Eq. (12) the relative number of “spin” states in the TIA assembly to the number of “spin” states in the TIS assembly can be calculate as

The spin polarization of conduction electrons in a ferromagnetic metal can be calculated as

Note: The model of TIA/TIS assemblies does not necessarily require a difference of the density of states for electrons with spin parallel and anti parallel to the metal magnetization.

 

Distribution of "spin" states in TIA and TIS assemblies can be calculated by the spin statistic.

 

 

 

 

 


The same content can be found in V. Zayets JMMM 356 (2014)52–67 (click here to download pdf) or (http://arxiv.org/abs/1304.2150 or this site) . Chapter 6 (pp.19-20) is about the Pauli paramagnetism and Chapter 7 (pp.20-22) is about ferromagnetic metals.
Some explanation can be found in Slides 9 and 10 of this Audio presentation or here

 

 

 

 

 

 

 

 

 

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