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Spin vs. Orbital moment in a solid. Quenching of orbital moment
Spin and Charge TransportAbstract:The inluence of the orbital moment on magnetic properties of conduction and localized electrons in a solid is discussed.Orbital moment of an electron is corresponded to the spacial distribution of electron orbital. In contrast to a gas, in a solid the electron distribution is spicially fixid due to bondings between neibour atoms. Since all bondings are fixid in the space according to atom position, the orbital electron distribution and therefore orbital moment and corresponded magnetic moment are fixed in space
Contentclick on the chapter for the shortcut(1)..........In short:
 The orbital moment of electrons of the conduction band is zero and the electrons of light and heavy hole bands is 1.  Because of the periodicity of the crystal lattice, the orbital of electrons (Bloch function) might not be able to rotate freely. In this case the orbital moment is piratically or fully quenched (frozen). In the case of the unquenched orbital moment, the orbital moment is parallel to the spin for electrons of the heavyhole band and the orbital moment is opposite to the spin direction for electrons of the lighthole band. The different orientation of the orbital moment in the respect to the spin direction for heavy and light holes is the reason for the reduction of the magnetooptical response and the reduction of the efficiency for excitation of spin polarized electrons by circular polarized light.
How to know whether the orbital moment is quenched or not?A. A good sign that the orbital moment is fully quenched is when the gfactor of electron is equal exactly 2 as in the case of a free electron in vacuum. The gfactor could be measured by the Electron Paramagnetic Resonance (EPR). Some researchers believe that difference of the gfactor from 2 means that the orbital moment is partiality unquenched. It is not correct. The gfactor may be significantly enlarged or be reduced due to the spinorbit interaction (See here). The spinorbit interaction is not related to the quenching of the orbital moment. Orbital moment in an ordinary atomic gas and in a solid
In the gas the orbit of an atom can be in any direction as well as corresponded orbital moment. Even though the orbital moment of each individual may be nonzero and the timeinverse symmetry is broken for each atoms, for the atomic gas as whole the timeinverse symmetry is not broken, because of the random orientation of the orbitals moments for each atom in the atomic gas.
The precession of the orbital moment literally means the precession of electron orbit as well.
"frozen" or "quenched" orbital moment in the solid.
The effect of "quenching" of the orbital moment is the known effect in magneto chemistry (See here).
Q. In the case when there is a spin accumulation in the electron gas, is there any accumulation of the orbital moment? Short answer: Yes. There is an accumulation of the orbital moment. Q. The accumulations of the spin and the orbital moments are of the same kind. Is it correct? Short answer: No. The accumulations of the spins and orbital moment are of different kinds, but often they coexist in the electron gas at the same time.
Orbital moment & spacial symmetry & timeinverse symmetryIn an atomic gas the timeinverse symmetry may be broken for each individual atom. Since the random orientation of atoms in the gas, even though this is the case, the timeinverse symmetry for the atomic gas is not broken. The case of the solid is different. The electron orbitals are fixed in the space and the timeinverse symmetry should not be broken even locally. In contrast, the spin direction of electrons in a solid is not fixed in the space. Therefore, locally the timeinverse symmetry can be broken for an electron state. It is a "spin" state, which has nonzero spin (See here) . Since the spins of "spin" states are directed randomly in all directions, the timeinverse symmetry is not broken for the electron gas as whole. Even though the electron gas contains the "spin" states and for each individual "spin" state the timeinverse symmetry is broken. In a nonmagnetic metal without a spin accumulation the timeinverse symmetry can not be broken even locally. Therefore, the Hamiltonian for the electron gas should not change for inverse time.
Luttinger Hamiltonian & orbital moment & symmetry
Band structure of semiconductorFollowing explanations are based on the Luttinger Hamiltonian. The Luttinger Hamiltonian assumes that delocalized electrons in a solid have similar orbital moment as the free atoms of an atomic gas. More realistic Hamiltonian is the LuttingerKohn Hamiltonian, which is based on the symmetry of wave functions rather than the features of the orbital moment. Despite of some "rough" assumptions, the Luttinger Hamiltonian is still useful for understanding of features of delocalized electrons. This explanation is based on an assumption that the orbital moment is not "frozen". S,L and J denote the spin, the orbital moment and the total moment. j denotes the projection of the total moment on chosen axis. Conduction band: slike symmetry; each electron state can be occupied either by two or one electrons or be not occupied. The orbital moment is zero. The electron orbital is spherical. Since the orbital is spherical, the conduction electrons should not experience the spinorbit interaction. It is far from truth. The Bloch function of conduction electrons is not spherical, especially in the case of a compound semiconductor. The spinorbit interaction can be huge for the conduction electrons. For example, in the case of GaAs the gfactor can be even negative for the conduction electrons. Valence band: plike symmetry; each electron state can be occupied either by two or one electrons or be not occupied. There are 3 different bands: heavy holes, light holes and splitoff (SO) band splitoff (SO) band The total orbital momentum is zero. The electron orbital is spherical. Each electron state can be occupied either by two electrons of opposite spins or by one electron or be not occupied. The projection of the total moment on chosen axis, the orbital moment and spins of two electrons, which could occupy one electron state can be Since the orbital is spherical, the conduction electrons should not experience the spinorbit interaction. According to the LuttingerKohn model the orbitals of electrons of SO band is not spherical.
Heavyhole band The total orbital momentum is one. Each electron state can be occupied either by two electrons of opposite spins and opposite orbital moments or by one electron or be not occupied. Spin is directed along the orbital moment. The projection of the total moment on chosen axis, the orbital moment and spins of two electrons, which could occupy one electron state can be Lighthole band The total orbital momentum is one. Each electron state can be occupied either by two electrons of opposite spins and opposite orbital moments or by one electron or be not occupied. The projection of the total moment on chosen axis, the orbital moment and spins of two electrons, which could occupy one electron state can be Spin is directed opposite to the orbital moment. The projection of the total moment on chosen axis, the orbital moment and spins of two electrons, which could occupy one electron state can be Luttinger Hamiltonian for light and heavy holes from M.I. Dyakonov "Spin Physics in Semiconductors" Springer Series in SolidState Sciences ISSN 01711873The following proves that the orbital moment of heave holes is parallel to their spins and the orbital moment of light holes is opposite to their spins. Assumption of fully unquenched orbital moment is used.The timeinverse should not change the Hamiltonian. The quantummechanical momentum operator p changes sign with time reversal. Therefore, in the Hamiltonian it can be only in a product with other operator, which changes its sign with time reversal. If we assume that there is no any quenching of the orbital moment, due to LScoupling (Russell–Saunders coupling) the total moment J should be used as such operator. In the Hamiltonian the timeinversesymmetrical product should be included. In the case of a critical point (band min or max) at p=0, the Hamiltonian should be symmetrical for the reversal of the direction of p, then the product should be included into the Hamiltonian. The Hamiltonian, which describes both the heavy and light holes bands, should have 4 eigenvalues corresponded to the projections of the total moment 3/2; 1/2;1/2;3/2. Therefore, the Hamiltonian for the heavy and light holes bands can be expressed as where A and B are constants and
Along the direction of electron movement (along the direction of p), the projections the total moment can be only 3/2; 1/2;1/2;3/2. Therefore and Solving Eq.(2.1), the energy of heavy holes (j=+/3/2)can be calculated as the energy of light holes (j=+/1/2)can be calculated as since B<0 , the effective electron mass for heavy holes is larger than for the light holes. Orbital moment of electrons in a solid. Quenched or unquenched ???
Note: There is no "black or white" answer for this question. There are some effects, which are completely not influenced by the orbital moment of electrons (for example, EPR), but there are some effects, which are still influenced by the fullyquenched orbital moment (for example, the Zeeman effect for delocalized conduction electrons)Note: Often it is difficult to separate between influences of spinorbit interaction and partially unquenched orbital moment on electron properties.Delocalized conduction electrons. The orbital moment is (1) nearlyfully quenched in a monocrystal metal (2) nearlyfully unquenched in a amorphous metal (3) partiality quenched in a polycrystal metal The effective length of delocalized electrons may be over 1000 lattice periods. Therefore, the periodicity of crystal strongly fixes the orbits of the delocalized electrons. In the crystal there is a strong force against any rotation or deformation of orbit of delocalized electron. Since the orbit can not be changed, it literally means that the orbital moment is "frozen" and it can not be changed. Localized delectrons. The orbital moment (1) nearlyfully quenched in a monocrystal metal and a polycrystal metal (2) partially unquenched in a amorphous metal
Even though the delectrons are localized, their size is substantial. There is a substantial exchange interaction between the delectrons and delocalized conduction electrons and between neighbor delectrons. The delectrons have t and e symmetrical states according to the ligand field theory. The energy separation between d and t state is called the ligandfield splitting parameter delta. Only in the case of a smaller delta, the delectrons can be in the high spin state. It proves that there is a strong interaction between delectrons and the lattice. The orbits of the delectron are fixed by the lattice symmetry and it can not freely rotated. Definitely, the orbital moment of the delectrons are quenched. Localized felectrons. The orbital moment is only partially quenched The felectrons only weakly interact with neighbors and with delocalized conduction electrons. Orbital moment & special symmetryEach orbital moment corresponds to the specific symmetry of electron orbit (the symmetry of the Bloch function). Even though the orbital moment is quenched, still the electron orbit has the symmetry corresponded to the orbital moment. The symmetry often defines many properties of electrons. For example, the symmetry of atomic orbitals (Bloch functions) of electrons of the splitoff band (orbital moment L=0) is nearly spherical. In contrast, the symmetry of atomic orbitals of electrons of the lighthole and heavyhole bands (orbital moment L=1) is only rotational around one axis. For this reason, the energy of electrons of splitoff band is different (smaller) than the energy of heavy and light holes even at p=0.
Ligand fieldsigma bond and pi bond is here
In my understanding, the dorbital moment is nearlyfully quenched in a single or polycrystalline phase, because the orbital is strongly coupled the bonding which is anisotropic due to crystal structure. Otherwise, the crystal would break itself due to orbital dynamics. In this respect, I assume that it could be possible that the orbital is partially unquenched in an amorphous phase, as you wrote in your webpage. I intuitively imagine that an amorphous solid is “soft” in the sense that the orbital may rotate. But I think it would be nice to have another argument to explain why an amorphous crystal is “orbitalactive”. Is there any experimental evidence that proved this statement? I’m not an expert on experiments, but I guess it might be possible to measure the orbital moment through the selection rule in the optical excitation. Or, do you know any theory/simulation paper that discusses the orbital quenching effect in amorphous and single/poly crystals?
A. As you mentioned, the orbital moment is related to the electron wave function and therefore the shape and symmetry of electron orbitals. As a result, in case when the orbital moment changes its direction or precesses, the orbital distribution and therefore the bonding is modified. Rotation of the quenched orbital moment will not cause a crystal explosion. However, the modulation of the strength of the bonding may cause some magnetostriction effect. Therefore, the degree of quenching of orbital moment might be possible to estimate from the features of the magnetostriction effect. However, it is not easy to do because there other mechanisms, which contribute substantially to the magnetostriction, and it is hard to distinguish the weaker contribution from orbital quenching. For a bulk material, the magnetostatic interaction between magnetic domains is the major contributor to the magnetostriction. For a nanomagnet (singledomain state) the major contributor to the magnetostriction is the spin orbit (SO) interaction. See here . Even though there is a word "orbital" in SO name, the relation of this effect to orbital in this effect is not direct I don't know, even it is difficult, but it might be possible to extract the degree of orbital quenching from a magnetostriction measurement Another problem to measure the degree of of orbital quenching is the electron spin. The spin is fully unquenched and it make distinguish the magnetic features due to the quenched/ unquenched orbital moment. Again the spinorbit interaction makes the magnetic energy of electron substantially dependent on the spin direction (See PMA or magneto anisotropy) There are effects, which are substantially influenced or even originated due to orbital moment of electrons. The following effect are due to the orbital moment of conduction electrons: (effect 1) light and heavy holes in a semiconductor (effect 2) Anomalous Hall effect (AMR) see here Spin orbit torque (SOT effect) is related to the Spin Hall effect and therefore to the spin orbit interaction. As I mentioned the SO interaction is not very directly related to orbital moment (even though it has the same tendency). In many cases (both the amorphous and the single crystal materials) the orbital moment is either fully unquenched or zero. Even it is not 100% true. It is a good assumption.

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