Magnetism in Solids

Abstract

We begin with a brief review of some elementary electromagnetism. A current distribution $\mathbf j(\mathbf r)$ produces a magnetic dipole moment at the origin that is given by $\mathbf m = \frac{1}{2c} \int \mathbf r \times \mathbf j(\mathbf r) d^3 r.$

Appendices

Problems

9.1

Consider a volume V bounded by a surface S filled with a magnetization $\mathbf M(\mathbf r^\prime )$ that depends on the position $\mathbf r^\prime$. The vector potential $\mathbf A$ produced by a magnetization $\mathbf M(\mathbf r)$ is given by

$$ \mathbf A (\mathbf r) = \int d^3 r^\prime \frac{\mathbf M(\mathbf r^\prime ) \times \left( \mathbf r-\mathbf r^\prime \right) }{\left| \mathbf r-\mathbf r^\prime \right| ^3}. $$

• (a) Show that $\mathbf {\nabla }^\prime \frac{1}{|\mathbf r-\mathbf r^\prime |}=\frac{\mathbf r-\mathbf r^\prime }{|\mathbf r-\mathbf r^\prime |^3}$.

• (b) Use this result together with the divergence theorem to show that $\mathbf A (\mathbf r)$ can be written as

  $$ \mathbf A (\mathbf r) = \int _\mathrm{V} d^3 r^\prime \frac{\mathbf \nabla _{r^\prime } \times \mathbf M(\mathbf r^\prime )}{\left| \mathbf r-\mathbf r^\prime \right| } +\oint _\mathrm{S} dS^\prime \frac{\mathbf M(\mathbf r^\prime ) \times \hat{\mathbf n}^\prime }{\left| \mathbf r-\mathbf r^\prime \right| }, $$

  where $\hat{\mathbf n}$ is a unit vector outward normal to the surface S. The volume integration is carried out over the volume V of the magnetized material. The surface integral is carried out over the surface bounding the magnetized object.

9.2

Demonstrate for yourself that Table 9.1 is correct by placing $\uparrow$ or $\downarrow$ arrows according to Hund’s rules as shown below for Cr of atomic configuration $(3d)^5(4s)^1$.

Table 9.2 The ground state atomic configuration of Cr

Clearly $S=\frac{1}{2}\times 6=3$, $L=0$, $J=L+S=3$, and

$$ g = \frac{3}{2} + \frac{1}{2} \frac{3(3+1)-0(0+1)}{3(3+1)}=2. $$

Therefore, the spectroscopic notation of Cr is $^7S_3$.

Use Hund’s rules (even though they might not be appropriate for every case) to make a similar table for $\mathrm Y^{39}$, $\mathrm Nb^{41}$, $\mathrm Tc^{43}$, $\mathrm La^{57}$, $\mathrm Dy^{66}$, $\mathrm W^{74}$, and $\mathrm Am^{95}$.

9.3

A system of N electrons is confined to move on the $x-y$ plane confined within a rectangular strip with sides of $L_x$ and $L_y$. A magnetic field $\mathbf B=B\hat{z}$ is perpendicular to the plane.

• (a) Show that the eigenstates of an electron are given by

  $$ \varepsilon _{n\sigma } (k)=\hbar \omega _\mathrm{c}(n+\frac{1}{2}-g^*\sigma _z/2) $$

  and

  $$ \psi _{n\sigma }(k,x, y)=\mathrm e^{iky}\phi _n(x+\frac{\hbar k}{m \omega _\mathrm{c}})\eta _\sigma , $$

  where $g^*$ is the effective g-factor of an electron and $\sigma _z=\pm 1$. Here $k=\frac{2\pi }{L}\times j$, where $j=-\frac{N}{2}$, -$\frac{N}{2}+1$, $\ldots$, $\frac{N}{2}-1$, and $\eta _\sigma$ is a spin eigenfunction.

• (b) Determine the density of states $g_{\sigma }(\varepsilon )$ for electrons of spin $\sigma$. Remember that each cyclotron level can accommodate $N_\mathrm{L} = \frac{BL^2}{hc/e}$ electrons.

• (c) Determine $G_\sigma (\varepsilon )$, the total number of states per unit area.

• (d) Describe qualitatively how the chemical potential at $T=0$ changes as the magnetic field is increased from zero to a value larger than $(\frac{hc}{e})\frac{N}{L^2}$.

9.4

Consider the system of electrons sitting in the potential well $V(x)=\frac{1}{2}m\omega _0^2x^2$. Then apply a magnetic field $\mathbf {B}$ in such a way that $\mathbf {A}=(0, xB, 0)$.

• (a) Write down the Hamiltonian of the system.

• (b) Get the energy eigenvalues $\varepsilon _n$ and eigenstates $\psi _n(x)$.

• (c) Examine the cases (i) $\omega _0 \rightarrow 0$ and (ii) $\omega _0 \simeq \omega _c$, where $\omega _c$ denotes the cyclotron frequency of an electron.

9.5

Demonstrate that $S_\mathrm{m} (B, T) < S_\mathrm{m} (0,T)$ by showing that $dS(B,T) = \partial _B S|_{T,V} + \partial _T S|_{B, V} dT$ and that $\partial _B S(B,T)|_{T, V} < 0$ for all values of $\frac{g_\mathrm{L}\mu _\mathrm{B}B}{k_\mathrm{B}T}$ if $J \ne 0$. Here $\partial _T S|_{B, V}$ is just $\frac{c_\mathrm{v}}{T}$.

Summary

The total angular momentum and magnetic moment of an atom are given by

$$ \mathbf J = \mathbf L +\mathbf S. ; \text { } \mathbf m = -\mu _\mathrm{B} \left( \mathbf L+2\mathbf S\right) = -\hat{g} \mu _\mathrm{B} \mathbf J. $$

Here the eigenvalue of the operator $\hat{g}$ is the Landé g -factor written as

$$ g_\mathrm{L} = \frac{3}{2} + \frac{1}{2} \frac{s(s+1)-l(l+1)}{j(j+1)}. $$

The ground state of an atom or ion with an incomplete shell is determined by Hund’s rules:

• (i) The ground state has the maximum S consistent with the Pauli exclusion principle.

• (ii) It has the maximum L consistent with the maximum spin multiplicity $2S+1$ of Rule (i).

• (iii) The J-value is given by $|L-S|$ when the incomplete shell is not more than half filled and by $L+S$ when more than half filled.

In the presence of a magnetic field $\mathbf B$ the Hamiltonian describing the electrons in an atom is written as

$$ H = H_0 + \sum _i \frac{1}{2m}\left( \mathbf p_i + \frac{e}{c}\mathbf A(\mathbf r_i)\right) ^2 + 2\mu _\mathrm{B} \mathbf B\cdot \sum _i\mathbf s_i, $$

where $H_0$ is the non-kinetic part of the atomic Hamiltonian and the sum is over all electrons in an atom. For a homogeneous magnetic field $\mathbf B$ in the z-direction, we have $\mathbf A = -\frac{1}{2}B_0\left( y\hat{i} - x \hat{j} \right) .$ In this gauge, the Hamiltonian becomes

$$ H = \mathcal {H}- m_z B_0 +\frac{e^2B_0^2}{8m_\mathrm{e}c^2}\sum _i\left( x_i^2+y_i^2\right) , $$

where $\mathcal {H} = H_0 + \sum _i \frac{p_i^2}{2m_\mathrm{e}}$ and $m_z=\mu _\mathrm{B}(L_z+2S_z)$. In the presence of $\mathbf B_0$, the z-component of magnetic moment of the atom becomes

$$ \mu _z = m_z -\frac{e^2B_0}{6m_\mathrm{e}c^2}\sum _i \overline{r_i^2}. $$

The second term on the right hand side is the origin of diamagnetism. If $J=0$ (so that $\overline{J_z} =0$), the (Langevin) diamagnetic susceptibility is given by

$$ \chi _\mathrm{DIA} = \frac{M}{B_0} =-N\frac{e^2}{6m_\mathrm{e}c^2}\sum _i \overline{r_i^2}. $$

The energy of an atom in a magnetic field $\mathbf B$ is $E = g_\mathrm{L} \mu _\mathrm{B}Bm_{J},$ where $m_\mathrm{J} = -J, -J+1, \ldots , J-1, J$. The magnetization of a system containing N atoms per unit volume is written as $M=Ng_\mathrm{L}\mu _\mathrm{B}J B_J(\beta g_\mathrm{L}\mu _\mathrm{B}B J),$ where the function $B_J(x)$ is called the Brillouin function. If the magnetic field B is small compared to 500 T at room temperature, M becomes

$$ M \simeq \frac{Ng_\mathrm{L}^2 \mu _\mathrm{B}^2 J(J+1)}{3k_\mathrm{B}T} B, $$

and we obtain the Curie’s law for the paramagnetic susceptibility:

$$ \chi _\mathrm{PARA} = \frac{M}{B} = \frac{N\langle \mathbf m^2\rangle }{3k_\mathrm{B}T} $$

at high temperature, $\left( g_\mathrm{L}\mu _\mathrm{B}B J \ll k_\mathrm{B}T\right)$.

In the presence of the magnetic field $\mathbf B$, the number of electrons of spin up (or down) per unit volume is

$$ n_{\pm } = \frac{1}{2} \int _0^\infty dE f_0(E) g\left( E\mp \mu _\mathrm{B}B\right) . $$

For $\zeta \gg \mu _\mathrm{B}B$ and $k_\mathrm{B}T \ll \zeta$, the magnetization $M (=\mu _\mathrm{B}(n_- -n_+))$ reduces to

$$ M \simeq \mu _\mathrm{B}^2 B\left[ g(\zeta ) +\frac{\pi ^2}{6}(k_\mathrm{B}T)^2 g^{\prime \prime }(\zeta )\right] , $$

with $\zeta =\zeta _0-\frac{\pi ^2}{6}(k_\mathrm{B}T)^2 \, \frac{g^{\prime }(\zeta _0)}{g(\zeta _0)}.$ Since $g(\zeta )=\frac{3}{2}\frac{n_0}{\zeta _0}\left( \frac{\varepsilon }{\zeta _0}\right) ^{1/2}$, we obtain the (quantum mechanical) expression

$$ \chi _\mathrm{QM} = \frac{3n_0\mu _\mathrm{B}^2}{2\zeta _0} \left[ 1-\frac{\pi ^2}{12}\left( \frac{k_\mathrm{B}T}{\zeta _0}\right) ^2 + \cdots \right] $$

for the Pauli spin (paramagnetic) susceptibility of a metal.

In quantum mechanics, a dc magnetic field can alter the distribution of the electronic energy levels and the orbital states of an electron are described by the eigenfunctions and eigenvalues given by

$$ \left| n k_y k_z\right. \rangle = L^{-1} \mathrm e^{ik_yy+ik_zz} \phi _n\left( x+\frac{\hbar k_y}{m\omega _\mathrm{c}}\right) ; \text { } E_n(k_y, k_z) = \frac{\hbar ^2 k_z^2}{2m} + \hbar \omega _\mathrm{c} (n+\frac{1}{2}). $$

The quantum mechanical (Landau) diamagnetic susceptibility of a metal becomes

$$ \chi _\mathrm{L} = -\frac{n_0}{2\zeta _0}\left( \frac{e\hbar }{2m^*c}\right) ^2 = -\frac{n_0\mu _\mathrm{B}^2}{2\zeta _0}\left( \frac{m}{m^*}\right) ^2. $$

Appearance of $m^*$ (not m) indicates that the diamagnetism is associated with the orbital motion of the electrons.

In a metal, as we increase $\mathbf B$, the Landau level at $k_z=0$ passes through the Fermi energy $\zeta$ and the internal energy abruptly decreases. Many physically observable properties of the system are periodic functions of the magnetic field. The periodic oscillation of the diamagnetic susceptibility of a metal at low temperatures is known as the de Haas–van Alphen effect. Oscillations in electrical conductivity are called the Shubnikov–de Haas oscillations.