Miscellaneous Aspects Related to the Electronic Structure

Abstract

In the light of the main aspects involving the electronic properties, a classification of crystalline solids can be devised.

Appendix 13.1 Magnetism from Itinerant Electrons

The magnetic properties associated with localized magnetic moments, therefore of crystals with magnetic ions, have been addressed at Chap. 4. At Sect. 12.7 and Problem 12.10 the paramagnetic susceptibility of the Fermi gas has been described.

The issue of the magnetic properties associated with an ensemble of delocalized electrons, with no interaction (Fermi gas) or in the presence of electron-electron interactions, is much more ample. In this Appendix we first recall the diamagnetism due to free electrons (Landau diamagnetism). Then some aspects of the magnetic properties of interacting delocalized electrons (ferromagnetic or antiferromagnetic metals) are addressed, in a simplified form.

The conduction electrons in metals are responsible of a negative susceptibility, associated with orbital motions under the action of external magnetic field. To account for this effect one has to refer to the generalized momentum operator (see Eq.  (1.26)) $-i\hbar \mathbf {\nabla }+ (e/c)\mathbf {A}$, with $\mathbf {A}= (0, Hx,0)$ (second Landau gauge),² for a magnetic field $\mathbf {H}$ along the z axes.

Then the Schrodinger equation takes the form

$$\begin{aligned} -\frac{\hbar ^2}{2m}\biggl [\biggl (\frac{\partial }{\partial x}\biggr )^2 + \left( \frac{\partial }{\partial y} + \frac{ieHx}{\hbar c}\right) ^2+ \left( \frac{\partial }{\partial z}\right) ^2\biggr ] \psi = E\psi \end{aligned}$$

(A.13.1.1)

Since describe constants of motion with eigenvalues $\hbar k_{y,z}$ one can rewrite this equation in the form

$$\begin{aligned} \biggl [ - \frac{\hbar ^2}{2m} \frac{\partial ^2}{\partial x^2} + \frac{1}{2} \frac{e^2H^2}{mc^2} \left( x- \frac{\hbar k_yc}{eH}\right) ^2 + \frac{\hbar ^2 k_z^2}{2m} \biggr ] \psi = E\psi \end{aligned}$$

(A.13.1.2)

where the first two terms represent the Hamiltonian for a displaced linear oscillator, with characteristic frequency

$$\begin{aligned} \omega _c= \frac{eH}{mc}= \frac{2\mu _BH}{\hbar }= 2 \omega _L \end{aligned}$$

(A.13.1.3)

($\omega _L$ Larmor frequency, see Problem 3.4). $\omega _c$ is the cyclotron frequency, while $x_o= \hbar ck_y/ eH$ is the center of the oscillations.

Therefore, from Eq. (A.13.1.2) the eigenvalues turn out

$$\begin{aligned} E_{n k_z}= \frac{\hbar ^2 k_z^2}{2m} +\left( n+ \frac{1}{2}\right) \hbar \omega _c \end{aligned}$$

(A.13.1.4)

where the quantum number n labels the Landau levels.

The one-electron eigenfunctions in the presence of the magnetic field are plane waves along one direction (dependent on the choice of the gauge for $\mathbf {A}$) multiplied by the wavefunctions for the harmonic oscillator.

The semiclassical view of the result given at Eq. (A.13.1.4) is that under the Lorenz force $\mathbf {F}_L= -(e/c) v_g\times \mathbf {H}$ (with $\mathbf {v}_g$ the group velocity) the evolution of the crystal momentum$\hbar d\mathbf {k}/dt= \mathbf {F}_L$ induces a cyclotron rotational motion in the xy plane while the electron propagates along the z direction (see Problem 12.2).

It is noticed that each Landau level is degenerate, the degeneracy depending on the number of possible values for $x_o$. For a volume $V=L_x . L_y . L_z$, then $0\le x_0\le L_x$, while one has $0\le k_y\le L_xeH/\hbar c\equiv k_y^{max}$. Therefore, $k_y$ being quantized in steps $\varDelta k_y= 2\pi /L_y$, the degeneracy of each Landau level, given by the number of oscillators with origin within the sample, is

$$\begin{aligned} N_L(H)= \frac{k_y^{max}}{\varDelta k_y}= L_xL_y H \frac{e}{hc} = \frac{\varPhi (H)}{\varPhi _o}, \end{aligned}$$

(A.13.1.5)

where $\varPhi (H)$ is the flux of the magnetic field across the crystal and $\varPhi _o= hc/e\simeq 4\times 10^{-7}$ Gauss cm$^2$ is the flux quantum .³

It is observed that the degeneracy, the same for all the n levels, increases linearly with H. Hence, by increasing H one can vary the population of each level and eventually when H is very high (and for moderate electron densities) all electrons will occupy just the first $n=0$ level. Accordingly, on increasing H different Landau levels will cross the Fermi energy.

By resorting to the results outlined above one can calculate the energy of the electrons E(H) in presence of the field and then the magnetization. One can conveniently distinguish two regimes, for $k_BT$ large or small compared to $\hbar \omega _c$. For $k_BT\ll \hbar \omega _c$ an oscillatory behaviour of E(H) is observed. The oscillations occur when the Landau level pass through the Fermi surface and cause changes in the energy of the conduction electrons, namely for

$$\begin{aligned} (n+ \frac{1}{2})\hbar \omega _c= E_F, \end{aligned}$$

(A.13.1.6)

Characteristic oscillations in the magnetization, known as De Haas-Van Alphen oscillations can be detected.

For $k_BT\gg \hbar \omega _c$ the discreteness of the Landau levels is no longer effective and the energy increases with $H^2$:

$$E(H)\propto \hbar \omega _c [\hbar \omega _c D(E_F)], $$

corresponding to an increase by $\hbar \omega _c$ of the energy for all the $\hbar \omega _c D(E_F)$ electrons in a Landau level ($D(E_F)$ density of states at the Fermi level, see Sect. 12.7.1). Therefore, the susceptibility turns out

$$\begin{aligned} \chi _L= -\frac{1}{12} \left( \frac{e\hbar }{mc}\right) ^2 \frac{D(E_F)}{Nv_c}= - \frac{1}{12\pi ^2} \frac{e^2}{mc^2} k_F, \end{aligned}$$

(A.13.1.7)

$k_F$ being the Fermi wave vector. From the Pauli susceptibility$\chi _P$ (see Problem 12.10) one can write

$$\begin{aligned} \chi _L= -\frac{1}{3} \chi _P. \end{aligned}$$

(A.13.1.8)

Modifications in $\chi _L$ (as well as in $\chi _P$) have to be expected when the effective mass $m^*$ of the electrons is different from $m_e$. For instance, when $m^*\ll m_e$ (as for example in bismuth, where $m^*\sim 0.01 m_e$) the metal can become diamagnetic. In fact, the total susceptibility for non-interacting delocalized electrons has to be written

$$ \chi _{total}= \mu _B^2 D(E_F) \left[ 1- \frac{1}{3}\left( \frac{m_e}{m^*}\right) ^2\right] \equiv \chi _P \left[ 1- \frac{1}{3}\left( \frac{m_e}{m^*}\right) ^2\right] $$

For further insights on the behaviour of the Fermi gas in the presence of constant magnetic field, Chap. 15 in the book byGrosso and Pastori Parravicini should be read.

In transition metals, with partially occupied d bands, the electrons involved in the magnetic properties are itinerant, with relevant many-body correlation effects. The Fermi-gas picture for the conduction electrons is no longer adequate and significant modifications to the Pauli susceptibility have to be expected, including the possibility of the transition to an ordered state. In these cases one often speaks of ferro (or antiferro)magnetic metals. For example, an experimental evidence of a particular itinerant ferromagnetism is iron metal: the magnetic moment per atom is found around $2.2 \mu _B$. This value cannot be justified in terms of localized moments on Fe$^{2+}$ ion, in the $^5D_4$ state (see Sect. 3.2.3).

The simplest model to account for the correlation effects on the magnetic properties of itinerant electrons is the one due to Stoner and Hubbard. In this model the electron-electron Coulomb interaction is replaced by a constant repulsive energy U between electrons on the same site, with opposite spins according to Pauli principle. Then the total Hamiltonian is written

$$\begin{aligned} \mathcal{H}= \sum _{\mathbf {k}}E(\mathbf {k})(n_{\mathbf {k}, \uparrow } + n_{\mathbf {k}, \downarrow })+ U \sum _m p_{m, \uparrow }p_{m, \downarrow } \end{aligned}$$

(A.13.1.9)

where the first term is the usual free electron kinetic Hamiltonian, while the second term describes the repulsive on-site interaction, with the sum running over all lattice sites.

The total magnetization can be derived in a way analogous to the one used for the Pauli susceptibility (Problem 12.10), by estimating the numbers of electrons with spin up and spin down, following the application of the magnetic field. For N electrons per cubic cm, in the conduction band of width larger than U, $N_{\uparrow }$ and $N_{\downarrow }$ are the numbers of electrons of spin up and spin down respectively. Then the energy for spin-up electrons turns out

$$\begin{aligned} E(\mathbf {k})_{\uparrow }= E(\mathbf {k}) + U n_{\downarrow }+ \mu _BH \end{aligned}$$

(A.13.1.10)

while for electrons with spin-down

$$\begin{aligned} E(\mathbf {k})_{\downarrow }= E(\mathbf {k}) + U n_{\uparrow }- \mu _BH. \end{aligned}$$

(A.13.1.11)

where $n_{\uparrow ,\downarrow }= N_{\uparrow ,\downarrow }/N.$

The decrease of the energy of the spin-down band with respect to the spin-up band yields an increase in the population of spin-down electrons and a non-zero magnetization. Since (see again Problem 12.10) for $N_{\uparrow ,\downarrow }$ one writes

$$\begin{aligned}&N_{\downarrow }= \frac{1}{2}\int ^{\infty }_{U n_{\uparrow }- \mu _BH } f(E) D(E-U n_{\uparrow }+ \mu _BH)dE \simeq \nonumber \\&\quad \simeq \frac{1}{2}\int ^{\infty }_0 f(E) D(E)dE + \frac{1}{2}(\mu _BH -U n_{\uparrow })D(E_F) \end{aligned}$$

(A.13.1.12)

while

$$\begin{aligned} N_{\uparrow }\simeq \frac{1}{2}\int ^{\infty }_0 f(E) D(E)dE - \frac{1}{2}(\mu _BH +U n_{\downarrow })D(E_F). \end{aligned}$$

(A.13.1.13)

The magnetization (per unit volume) becomes

$$\begin{aligned} M= \mu _B\frac{(N_{\downarrow }-N_{\uparrow })}{V}\simeq \frac{\mu _B U D(E_F)}{2N}(N_{\downarrow }-N_{\uparrow }) + \mu _B^2D(E_F)H \,\,\, \end{aligned}$$

(A.13.1.14)

(V the reference volume). Therefore the magnetic susceptibility becomes

$$\begin{aligned} \chi =\frac{M}{H}= \frac{\mu _B^2D(E_F)}{1- \frac{U D(E_F)}{2N} }= \frac{\chi _P}{1 - ({U\chi _P}/{2\mu _B^2N}) }\,\, , \end{aligned}$$

(A.13.1.15)

with $\chi _P$ Pauli susceptibility (for bare electrons) and $D(E_F)$ the density of states per unit volume.

It is noted that when $U D(E_F)/2N\rightarrow 1$ (Stoner criterium) the susceptibility diverges and ferromagnetic order is attained.

Even if the Stoner condition is not fulfilled, Eq. (A.13.1.15) shows that the susceptibility is significantly modified with respect to the one for bare free-electrons. Equation (A.13.1.15) can be considered a particular case of Eq. (4.33), where the enhancement factor corresponds to the mean field acting on a particular electron due to the interaction with all the others. Stoner criterium rather well justifies the ferromagnetism in metals like Fe, Co and Ni, as well as the enhanced susceptibility (about 5 $\chi _P$) measured in Pt and Pd metals.

Finally a few words are in order about the magnetic behaviour of itinerant electrons when the concentration n is reduced (diluted electron fluid in the presence of electron-electron interaction).

As shown in Problem 13.4 the Coulomb repulsive energy of the electrons goes as $<E_C>\propto e^2 n^{1/D}$ (D the dimensionality), while for the kinetic energy (for $T\rightarrow 0$) one has $<E>\propto n^{2/D}$. Thus the electron dilution causes a decrease of the average kinetic energy $<E>$ which is more rapid than the one for the average repulsion energy. Eventually, below $n_{3D}= 1.77\times 10^{-1}/a_o^3$ and below $n_{2D}= 0.4/a_o^2$, when $<E_C>$ becomes dominant, a spontaneous “crystallization” could occur, in principle (Wigner crystallization).

Monte Carlo simulations predict a three-dimensional crystallization into the bcc lattice at densities below $2\times 10^{18}$ cm$^{-3}$, while at densities below $2\times 10^{20}$ cm$^{-3}$ the Coulomb interaction should be strong enough to align all the spins, according to the Stoner criterium. Charge or spin ordering are hard to be experimentally tested, mainly because of the difficulty of the physical realization of the electron fluid at low density sufficiently free from impurities and/or defects.

Problems

Problem 13.3

Silver is a monovalent metal, with density $10.5\,\mathrm {g}/\mathrm {cm}^3$ and fcc structure. From the values of the resistivity at $T=20\,\mathrm {K}$ and $T=295\,\mathrm {K}$ given by $\rho _{20} = 3.8 \cdot 10^{-9}\,\varOmega \,\mathrm {cm}$ and $\rho _{295} = 1.6 \cdot 10^{-6}\,\varOmega \,\mathrm {cm}$, estimate the mean free paths $\lambda$ of the electrons .

Solution: The Fermi wavevector turns out $k_F = 1.2 \cdot 10^{8}$ cm$^{-1}$ and the Fermi energy is $E_F= 64390$ K. The electron density is $n= 5.86\times 10^{22}$ cm$^{-3}$.

From $\rho = {m}/{n e^2 \tau }$, $\lambda = <v> \tau$ and $<v> \sim \sqrt{E_F/m}$ (see Sect. 13.4), one derives

$$\lambda = 3.6 \cdot 10^{-6}\,\mathrm {cm} \,\,\,\, \text {at} \,\,\,295\,\mathrm {K}\,\,\,\, \mathrm {and}\,\,\,\, \lambda = 1.53 \cdot 10^{-3}\,\mathrm {cm} \,\,\,\, \text {at} \,\,\,20\,\mathrm {K}\,.$$

Problem 13.4

For three-dimensional and for two-dimensional metals, in the framework of the free-electron model and for $T\rightarrow 0$, evaluate the electron concentration n at which the average kinetic energy coincides with the average Coulomb repulsion (which can be assumed $U=e^2/d,$ with d the average distance between the electrons).

Solution: In 3D $d=1/(4\pi n/3)^{1/3}$, while in 2D    $d=1/n^{1/2}$. Thus

$$ U^{3D}= e^2 \left( \frac{4\pi }{3}\right) ^{1/3} n^{1/3}\,\,\,\, \mathrm {and}\,\,\,\, U^{2D}= e^2 n^{1/2} $$

The average kinetic energy per electron (for $T\rightarrow 0$) is $<E>=\int _0^{E_F} D(E) E dE$, with $D(E)^{3D}= (3/2)E^{1/2}/E_F^{3/2}$ and $D(E)^{2D}= 1/E_F$.

Then $<E>^{3D}=(3/5)E_F= (3\hbar ^2/10 m)(3\pi ^2 n)^{2/3}$

and $<E>^{2D}=(1/2)E_F= \hbar ^2\pi n/2m$.

The average kinetic energy coincides with the Coulomb repulsion for $n^{3D}=1.77\times 10^{-1}/a_0^3$ and $n^{2D}=0.4/a_0^2$, with $a_0$ Bohr radius.

Problem 13.5

A magnetic field is applied on an atom with a single p electron in the crystal field at the octahedral symmetry (Sect. 13.3), with six charges Ze along the $\pm x, \,\pm y, \, \pm z$ axes. Show that without the distortion of the octahedron (namely $a=b$, with a the distance from the atom of the charges in the xy plane and b the one along the z axis) only a shift of the p levels would occur. Then consider the case $b \ne a$ and discuss the effect of the magnetic field (applied along the z axis) deriving the eigenvalues (neglect the spin magnetic moment).

Solution: By summing the potential due to the six charges, analogously to the case described at Sect. 13.3, for $r \ll a$ the crystal field perturbation turns out (see footnote 1 in this chapter)

$$V_{CF}= -Ze^2 \left\{ \left( \frac{1}{a^3} - \frac{1}{b^3}\right) r^2 + 3 \left( \frac{1}{b^3} - \frac{1}{a^3}\right) z^2\right\} + ..... =A(3z^2 -r^2) + \text {const}$$

where $A\ne 0$ only for $b\ne a$.

From the unperturbed eigenfunctions the matrix elements of $V_{CF}$ are

$$\begin{aligned}<\phi _{p_x}\vert V_{CF}\vert \phi _{p_x}>&= A\int r^2 \vert \mathcal {R}(r)\vert ^2 r^2 dr \int \sin ^2 \theta \cos ^2 \phi (3\cos ^2 \theta -1)\sin \theta \, d\theta \, d\phi \\&= -A<r^2>\frac{8\pi }{15} =<\phi _{p_y}\vert V_{CF}\vert \phi _{p_y}> \end{aligned}$$

while

$$<\phi _{p_z}\vert V_{CF}\vert \phi _{p_z}> = A <r^2>\frac{16\pi }{15}\,.$$

In the absence of magnetic field the energy levels are

This effect can be interpreted in terms of quenching of angular momentum (see Problem 4.12). It can be observed that for orthorombic crystal symmetry, where the lowest degree ploynomial solution of the Laplace equation yields $V_{CF}= Ax^2 + By^2 - (A+B)z^2$, with A and B constants (with $A\ne B$), total quenching of the components of the angular momentum would occur.

For the electron in octahedral symmetry and in the presence of the field, the total perturbative Hamiltonian becomes $V_{CF} + \mu _B \,l_z \,H$.

The diagonal matrix elements of $l_z$ in the basis of the unperturbed eigenfunctions are zero. In fact,

$$<\phi _{2p_x}\vert l_z \vert \phi _{2p_x}> = + i\hbar \int _0 ^{\infty } f(r) dr \int _0 ^{\pi } \sin ^3 \theta \,d\theta \int _0 ^{2\pi } \sin \phi \cos \phi \, d\phi =0$$

(Problems 4.11 and  4.12) and, analogously,

$$<\phi _{2p_y}\vert l_z \vert \phi _{2p_y}>\,=\,<\phi _{2p_z}\vert l_z \vert \phi _{2p_z}>=0.$$

The non-diagonal matrix elements are $<\phi _{2p_y}\vert l_z \vert \phi _{2p_x}> =$

$i\hbar =-<\phi _{2p_x}\vert l_z \vert \phi _{2p_y}>$.

The secular equation becomes

$$\left| \begin{array}{ccc} E_0 -E &{} -i\mu _B H &{} 0\\ i\mu _B H &{} E_0 -E &{} 0\\ 0 &{} 0 &{} E_1 -E \end{array} \right| =0$$

yielding $E'= E_1$ and $E''= E_0 \pm \mu _B H,$ as sketched below