CHEM 548: Materials Chemistry

Magnetic induction results in the material

$B_0 = \mu_0 H$   $B = \mu H$   $\mu_r = \mu/\mu_0$   $B = \mu_0 H + \mu_0 M$   $M = (\mu_r - 1)H$

Contribution to the field due to magnetization of the solid

Magnetic susceptibility, χ (dimensionless)

χ measures the material response relative to a vacuum.

$B = \mu_0(H+M) = \mu_0(1+\chi_m)H = \mu H$

$M = \chi_m H$    $\chi_m = \mu_r - 1$

Electrons produce magnetic moments:

Labels: magnetic moments · electron · nucleus · electron · spin

Net magnetic moment:

--Sum of moments from all electrons.

Three types of response...

Chapter 20 - 39

Spins couple and align with no applied field

Table 14.1. Magnetic Constants of Some Materials at Room Temperature.

Table 14.1. Magnetic Constants of Some Materials at Room Temperature.

(3) Thermodynamic Analysis reviews Tc is associated with a 2^nd order transition

free energy (mJ mol⁻¹) · heat capacity (mJ mol⁻¹K⁻¹) · entropy (mJ mol⁻¹K⁻¹) · temperature (K) · C = T (dS/dT) · superconducting · normal · super conducting

Figure 12.5 (a) Free energy, (b) heat capacity, and (c) entropy of Al in the superconducting and normal states of aluminum. Parts (a) and (c) after [5].

PHYSICAL REVIEWVOLUME 108, NUMBER 5DECEMBER 1, 1957

Theory of Superconductivity*

J. BARDEEN, L. N. COOPER,† AND J. R. SCHRIEFFER‡
Department of Physics, University of Illinois, Urbana, Illinois
(Received July 8, 1957)

A theory of superconductivity is presented, based on the fact that the interaction between electrons resulting from virtual exchange of phonons is attractive when the energy difference between the electrons states involved is less than the phonon energy, $\hbar\omega$. It is favorable to form a superconducting phase when this attractive interaction dominates the repulsive screened Coulomb interaction. The normal phase is described by the Bloch individual-particle model. The ground state of a superconductor, formed from a linear combination of normal state configurations in which electrons are virtually excited in pairs of opposite spin and momentum, is lower in energy than the normal state by amount proportional to an average $(\hbar\omega)^2$, consistent with the isotope effect. A mutually orthogonal set of excited states in one-to-one correspondence with those of the normal phase is obtained by specifying occupation of certain Bloch states and by using the rest to form a linear combination of virtual pair configurations. The theory yields a second-order phase transition and a Meissner effect in the form suggested by Pippard. Calculated values of specific heats and penetration depths and their temperature variation are in good agreement with experiment. There is an energy gap for individual-particle excitations which decreases from about 3.5kT_c at T=0°K to zero at T_c. Tables of matrix elements of single-particle operators between the excited-state superconducting wave functions, useful for perturbation expansions and calculations of transition probabilities, are given.

Figure 12.6 (a) Real-space illustration of how electron–phonon coupling in a simple metal system can lead to attractive interaction between two electrons.⁹ (b) Virtual-phonon exchange behind Cooper-pair formation.

Electron Pairing Distance Estimate

Since electrons move much faster than the heavier metal ions, there is a time delay between the motion of the first electron (which distorts the lattice) and the subsequent attraction of the second electron. To estimate the distance d the first electron has traveled by the time the second is attracted, we use:

• v_n: velocity of an electron at the Fermi level
• ω_D: Debye angular frequency, which sets how fast the lattice can respond

The estimated distance is

d = v_F (2π / ω_D)

For aluminum:

• v_F ≈ 2×10⁶ m/s
• ω_D ≈ 5×10¹³ s⁻¹

This gives:
d ≈ 2500 Å

This distance is large compared with atomic spacing, so Coulomb repulsion between the electrons is negligible, especially due to screening by the positively charged metal ions.

Conservation of momentum leads to the relationships:

$k_1 = k_1' + q$ and $k_2 + q = k_2'$

$(k_1 + k_2) = (k_1' + q) + (k_2' - q)$

$k_1 + k_2 = k_1' + k_2' = k_0$

Figure 12.6 (a) Real-space illustration of how electron–phonon coupling in a simple metal system can lead to attractive interaction between two electrons.⁹ (b) Virtual-phonon exchange behind Cooper-pair formation.

Electron Pairing in k-Space

To form a Cooper pair through phonon exchange, each electron must scatter into a different quantum state. The Pauli exclusion principle limits which states are available: electrons deep below the Fermi energy E_F cannot scatter because all nearby states are already occupied. Only electrons close enough to E_F have empty states available.

A phonon of energy $\hbar\omega$ can only change an electron’s energy by at most $\hbar\omega$. Therefore, an electron can scatter only if there is an unoccupied state within that energy range. When you combine this with Pauli exclusion, the result is that only electrons within an energy window of order $\hbar\omega$ around E_F can participate in pairing.

In k-space, this allowed energy window corresponds to a thin shell of states surrounding the Fermi surface, with thickness $\delta k$ determined by $\hbar\omega$. Only electrons whose wave vectors fall within this narrow shell can experience the phonon-mediated attraction, as shown schematically in Figure 12.7.

$k_0$ reduces

Figure 12.7 (a) A k-space sketch of a spherical shell of states within $\delta k$ of Fermi level $k_\mathrm{F}$. Electrons with arbitrary wave vectors $k_1$ and $k_2$ are shown. The change in $k_1$ on phonon exchange is restricted to a small area determined by $\delta k$. (b) The overall condition of $k_1 + k_2 = k_0$ for virtual phonon exchange can be represented geometrically by two such shells offset by $k_0$. For a given value of $k_0$ the overlap region (shaded) gives the states that can be involved in coupling. The maximum coupling will occur when the two shells overlap such that $k_0 = 0$ and $k_1 = -k_2$.

Why $k_0 = 0$ gives the most possible pairs

If $k_0 = 0$, then the rule becomes:

$k_2 = -k_1$

That means every state k in the shell automatically has a partner $-k$ that is also in the shell.

This doubles your options: every point on the Fermi surface has a matching point directly opposite it.

So the number of available pair states is maximized.

If $k_0$ is not zero (say, $k_0$ is some nonzero vector), then the partner of $k_1$ is $k_2 = k_0 − k_1$. Many of those points don’t lie in the allowed shell anymore, so the number of available pairs drops.

$k_0 = k_1 + k_2$, so $k_0 = 0$ means:

$k_2 = -k_1$

In other words, the two electrons in a Cooper pair must have equal and opposite momenta.

Because electrons are fermions, their spins must also be opposite for the overall two-electron wavefunction to be allowed. So a Cooper pair consists of states:

$(k, \uparrow)$ and $(-k, \downarrow)$

This is called s-wave pairing, which is the type found in conventional (BCS) superconductors.

Finally, a Cooper pair is not like two electrons locked together in a fixed chemical bond. Instead, it is a dynamic quantum state in which many electrons near the Fermi surface continuously exchange phonons and momentarily form paired states with opposite momenta and opposite spins.

$$\Delta_0 = \frac{\hbar\omega_D}{\sinh\left[\frac{1}{N(E_F)V}\right]} \approx 2\hbar\omega_D\, e^{-1/(N(E_F)V)}$$

• $N(E_F)$ is the density of states at the Fermi level
• $V$ is the electron–phonon coupling strength
• $\omega_D$ is the Debye frequency

Typical experimental measurements the zero temp energy gap are   Δ₀ ≈ 5×10⁻⁴ eV

BCS Theory further predicts that   $2\Delta_0 = 3.52kT_\mathrm{c}$

T_c = 2Δ₀ 3.52 k_B = 2(5 × 10⁻⁴ eV) 3.52 (8.617 × 10⁻⁵eV/K) ≈ 3.3 K

BCS prediction evaluated at Tc yields:

$$T_\mathrm{c} = 1.14\,\frac{\hbar\omega_\mathrm{D}}{k}\, e^{-1/N(E_\mathrm{F})\cdot V}$$

• The BCS expression for $T_c$ shows that it scales with the Debye frequency $\omega_D$.
  This explains the isotope effect and why materials containing light elements (such as MgB₂) can have high $T_c$ due to their high phonon frequencies.
• A larger electron–phonon interaction $V$ leads to a higher $T_c$.
  This explains why good normal conductors (Cu, Ag, Au), which have weak electron-phonon coupling, are poor superconductors, while poor normal conductors (often oxides with stronger coupling) can be good superconductors.

Axes: Transition temperature (K); Year: 1900, 1920, 1940, 1960, 1980, 2000, 2020.

Labelled: Hg · Pb · Nb · NbC · NbN · V₃Si · Nb₃Sn · Nb-Al-Ge · Nb₃Ge · Liquld nitrogen 77K · LaBaCuO · LaSrCuO · YBaCuO · BiCaSrCuO · TlCaSrCuO · HgCaSrCuO · MgB₂ · LaFePO · LaFeAs · SmFeAs.

Pairing seems to come from electronic interactions, not phonons

In ordinary superconductors, electrons attract each other indirectly by distorting the lattice (phonons). In cuprates, that is not strong enough to explain $T_c$ near 100 K.

Instead, the starting point is that the undoped cuprate is an antiferromagnetic Mott insulator: neighboring Cu spins strongly prefer to be opposite. When you dope holes into this system, you

• destroy long-range antiferromagnetic order
• but keep strong, short-range “up–down” tendencies that fluctuate in space and time

These time-dependent local antiferromagnetic patterns are what people mean by spin fluctuations.

They act as a kind of magnetic glue:

• an electron moving through the lattice disturbs the spin background
• another electron can lower its energy by moving in a way that fits this disturbed pattern
• the net effect is an effective attraction between electrons, even though the bare interaction is repulsive

So instead of an ”electron–phonon” attraction, cuprates likely have an ”electron–spin-fluctuation” attraction that binds electrons into Cooper pairs.

Simple electronic models now reproduce superconductivity

The Hubbard model is the standard “minimal” model to capture strong electronic correlations on a lattice:

• electrons live on a square lattice (like Cu sites)
• they can hop to neighboring sites with amplitude t
• if two electrons land on the same site, they pay a large energy penalty U (strong on-site repulsion)

At half-filling (one electron per site), and for large U, the Hubbard model naturally gives a Mott insulating antiferromagnet, just like undoped cuprates.