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Spins of Conduction electrons in a ferromagnetic and nonmagnetic metals.
Spin and Charge TransportThe conduction electrons in a ferromagnetic metal are spinpolarized. It means that all conduction electrons can be divided into two groups: spinpolarized and spinunpolarized. In the group of spinpolarized electrons, all spins are directed in one direction. In the group of spinunpolarized electrons,Possible 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.in short ferromagnetic metals In a ferromagnetic metal, spins of all localized delectrons are in one direction. The conduction electrons are spinpolarized. All conduction electrons can be divided into two groups: spinpolarized and spinunpolarized electrons. In the group of spinpolarized electrons, spins of all electrons in one direction. In the group of spinunpolarized electrons, the spin directions are equally distributed in all directions. Spins of the localized delectrons are in one direction because of the exchange interaction with a few neighbor electrons. The reason why the conduction electrons are spinpolarized 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 spd exchange interaction between the localized and conduction electrons. nonmagnetic metals In a nonmagnetic metal the conduction electrons are not spinpolarized. 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 spinpolarized (See here). Also, the spinpolarized electrons can be injected or they can diffuse from a ferromagnetic metal to a nonmagnetic metal Nonmagnetic metals include paramagnetic and diamagnetic metals. diamagnetic metal Localized electrons do not have spin. Therefore, there is no spd exchange interaction between the conduction electrons and localized delectrons and the spd 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
Localized delectron: They have a size of one atomic orbital. Their spin direction is determined mainly by the exchange interaction with close neighbor localized delectrons. scatterings between neighbor localized electrons are rare; scatterings between localized and conduction electrons are rare;
Conduction (delocalized) spelectrons: 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.
Pauli paramagnetism in nonmagnetic metals
This classical model explains the paramagnetic susceptibility of a nonferromagnetic 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 spindown and spinup bands. Therefore, the net magnetization due to the conduction electrons is zero. In a magnetic field the spinup and spindown 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 spindown band are flipped into the spinup band. Therefore, the spinup band will be filled by a greater number of electrons than the spindown band. Due to the difference in the number of spinup and spindown electrons, the net magnetization of conduction electrons becomes nonzero. At zero temperature the magnetization can be calculated as where are the total number of conduction electrons with spinup and spindown, 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 nonmagnetic metal is different from the spin accumulation obtained from an injection of spinpolarized electrons from a ferromagnetic metal.
Pauli paramagnetism explained from the model of TIS/TIA assembliesThe Pauli paramagnetism is calculated from model of TIS/TIA assemblies differently than in the classical model
In the absence of a magnetic field all conduction electrons of a nonmagnetic 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 timeinverse symmetrical. The magnetic moment is timeinverse asymmetrical and only time inverse asymmetrical objects may have nonzero magnetic moment. Therefore, the total magnetic moment of an electron gas may be nonzero 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 gfactor, 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
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 nonmagnetic metal is exactly the same as the spin accumulation obtained from an injection of spinpolarized electrons from a ferromagnetic metal.
Ferromagnetic metals
Stoner model of ferromagnetism.Classical modelSee 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 spinpolarized. The density of spindown states is smaller than the density of spinup states. Because of this difference, the conductivity of a ferromagnetic metal becomes spindependent, 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 delectrons is spindependent. Later this assumption was extended for the density of state of the conduction spelectrons.
Note: In model of spinup/spindown bands, the spindependent 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 spindependence 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 spinindependent, 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 spinup/spindown bands.
TIA and TIS assemblies in a ferromagnetic metalProposed modelIn a ferromagnetic metal the dorbitals are partially filled and electrons can be divided into local delectrons with nonzero spin and conduction spelectrons. There is an exchange interaction between delectrons, which aligns spins of all delectrons in one direction. Also, there is an exchange interaction between the local delectrons and the conduction spelectrons, 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 delectrons, 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.1920) is about the Pauli paramagnetism and Chapter 7 (pp.2022) is about ferromagnetic metals.Some explanation can be found in Slides 9 and 10 of this Audio presentation or here

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