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In these lecture notes, I will overview the subject of condensed matter physics, review key ideas from Solids 1 and outline the present course of Solids 2.
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Leo Radzihovsky (Dated: 12 January, 2015)
In these lecture notes, I will overview the subject of condensed matter physics, review key ideas from Solids 1 and outline the present course of Solids 2.
I. INTRODUCTION
A. Condensed matter physics
Condensed matter physics (CMP) is the largest broadly defined area of physics that studies phenomena of strongly interacting, macroscopic (even as large as an Avagadro, 10^23 ) number of degrees of freedom.
After quantum mechanics and its many-degrees of freedom successor, quantum field the- ory were developed in the first quarter of the 20th century, attention turned to application of this scientific breakthrough to the study of solid state materials. The problem is the the quantum mechanical description of an Avagadro number of electrons, carrying spin and charge, moving in the periodic potential ions, that in principle are also a macroscopic num- ber of quantum mechanical degrees of freedom. Since solid state systems are materials with well-packed array of atoms, the typical length scale is on the order of an Angstrom (size of an atom and the associated atomic bonds) and typical energy is on the order of an electron-volt (a Rydberg). Early breakthrough came in 1927 with Arnold Sommerfeld’s appreciation of the key role of Fermi-Dirac statistics, with the subject of conventional solid state physics well-established by late 60s with the pioneeting works of Bravais, Hall, Drude, Einstein, Debye, Kamerlingh Onnes, Max von Laue, Bragg brothers, Meissner, Pauli, Bethe, Wigner, Slater, Bloch, Feynman, Bardeen, Cooper, Schrieffer, Peierls, Seitz, Froelich, Kondo, Hub- bard, Gutzwiller, Pines, Anderson, Kapitsa, Bogoluibov, Landau and his school including Abrikosov, Gorkov, Dzyaloshinski in the Soviet Union. Since then much of the effort has been directed at understanding the effects of deviation from a perfect crystalline lattice, namely the effect impurities and lattice defects, referred to as quenched disorder. These, together with the studies of strong quantum and thermal fluctuations and strong interactions, that drive phase transitions between different states of matter continue to form a central subject of modern solid state physics.
A and B = ∇ × A), electron-electron Coulomb interaction, interaction of electron spin with external magnetic field B (μB = eℏ/(2m), g ≈ −2 are the electron’s Bohr magneton and gyromagnetic ratio) and spin-orbit interaction
HSO = (^2) m^12 c (^21) rdU dr^ (r )(r × p) · s ∝ ` · s, (3)
arising as a relativistic correction that couples spin and orbital degrees of freedom; there are a number of other such corrections (quartic correction to parabolic dispersion and the so-called Darwin term) that we may return to later in the course. In addition to the latter electron spin crucially enters through Pauli principle requiring the electron many-body wavefunction to be totally antisymmetric under the interchange of both orbital and spin electron coordinates. As we will see soon enough, in the presence of Coulomb interaction this quantum statistical constraint on the electronic wavefunction will give rise to the so-called spin exchange interaction JSi · Sj between spins i and j and will lead to the dominant mechanism of magnetism in nature. The ionic Hamiltonian is given by
Hion =
i
P^ ˆ^2 i 2 Mi^ +
i,j
ZiZj e^2 4 π 0 |Ri − Rj | (4)
consisting of the ions’ kinetic and Coulomb interaction energies for charge Zie, and we neglected nuclear spin-orbit and electromagnetic interactions. The final crucial part of the Hamiltonian is the Coulomb interaction between the electrons and ions,
Helectron−ion = −
N,N ∑e i,j
Zie^2 4 π 0 |Ri − rj |.^ (5) Despite a seeming simplicity of the Hamiltonian and the statement of the problem, even with modern-day computers an exact solution of the above Schrodinger’s equation can only be done for at most ten interacting electrons (that’s even when ions are treated as frozen). A classical computer of the size of the universe could at best solve a problem of a pathetic number of 200 electrons[2]. Thus, because of the exponential growth of the Hilbert space with the number of degrees of freedom, a frontal attack on this problem is unimaginable.
C. Approximations to the solid-state problem
Thus to make progress serious imaginative physical insight is needed to inspire appropriate approximate treatments. These include:
In the case of heavy ions ordered into a perfect crystal lattice, one can approximately ignore their quantum character, taking Ri as classical variables forming a lattice:
Rn,s = Rn + rs, (6)
where
Rn = n 1 a 1 + n 2 a 2 + n 3 a 3 (7)
spans a Bravais lattice with lattice vectors ai and a p-atom basis rs, with s = 1,... p. The corresponding reciprocal lattice is spanned by Gh = h 1 b 1 + h 2 b 2 + h 3 b 3 , with reciprocal lattice vectors bi, defined by bi · aj = 2πδij or equivalently eiGh·Rn^ = 1, solved by b 1 = 2 πa 2 × a 3 /v, where v = a 1 · (a 2 × a 3 ) is the unit-cell volume. All the possibilities in two dimensions (5-types) and three dimensions (14 Bravais lattices types and 7 crystal structure classes, characterized by one of the 230 3D space groups) have been completely classified. Examples of current interest of 2D Bravais lattice with a basis are the honeycomb lattice of graphene and the kagome lattice, illustrated in Figs.(3),(4) and in 3D the diamond lattice which is a face-centered cubic Bravais lattice with a 2-atom basis.
FIG. 4: Details of the honeycomb lattice structure illustrating in (a) and (b) its real space triangular unit cell with a 2-atom basis. In (c) its reciprocal lattice and the corresponding Wigner-Seitz cell, i.e., the 1st Brillouin zone is indicated in gray.
FIG. 5: A 3d non-Bravais diamond lattice, which is an FCC Bravais lattice with a 2-atom basis.
Using such crystalline lattice positions inside V (r) ≡ Helectron−ion[Rn,s], defines a periodic ion potential V (r) that the electrons move in. This still leaves electron-electron interaction to contend with that is the main challenge of solid state physics. As we will see, it can be treated in mean-field approximation, perturbation theory (Hartree and Hartree-Fock approximation being the lowest order), or through other inspiring approximations (large-N, order parameter decoupling, numerically). If as a crudest approximation, we ignore the
electron-electron interaction, we are left with a single electron band structure problem ( −ℏ
2 m +^ V^ (r) +^...
ψn(r) = Enψn(r) (8)
for a single-electron wavefunction ψn(r), with the many-body wavefunction give by the antisymmetric Slater determinantal
Ψ(r 1 ,... , rNe ) = √^1 N e!
∏^ Ne n
ψn(rP {n}),
encoding the Pauli principle. The solution of the single particle Schrodinger’s equation, (8) can be laborious, but, because it is afterall a single electron problem, it can in principle be straightforwardly done numerically. Its eigenfunctions satisfy the famous Bloch Theorem,
ψk(r) = eik·ruk(r), (9)
with the Bloch function periodic, uk(r + Rn) = uk(r) and its eigenvalues
E(k) = E(k + Gh)
over the Brillouin zone, leading to the band structure (illustrated in Figs.6,7)), filled accord-
FIG. 6: Band structure of Gallium Nitride, GaN.
ing to Pauli principle, controlling noninteracting properties of the corresponding material.
D. Experimental probes
To study this intricate and rich behavior “hidden” inside the material, a large array of experimental probes is employed. Among the primary ones that we will study (some discussed in the introductory course on solid state physics) include:
E. Solids 2: Advanced solid state physics overview
In this course we will build on the introductory background material outlined above, with a focus on advanced topics that require treatment of interactions and quantum and thermal fluctuations. We will develope and utilize methods of statistical mechanics and quantum-field theory to study a range of phenomena. The outline of the course is as follows.
Course outline:
