Lecture Notes on Solid State Physics - Kevin Zhou

1 Lecture Notes onSolid State PhysicsKevin Notes comprise an undergraduate-level introduction to Solid State Physics . Results fromundergraduate quantum mechanics are used freely, but the language of second quantization is primary sources were: Kittel,Introduction to Solid State Physics . The standard undergraduate-level introduction tosolid State Physics , with consistently good quality. Ashcroft and Mermin, Solid State Physics . The standard graduate-level introduction to solidstate Physics . Relatively dry and difficult to read. Covers essentially the same conceptualmaterial as Kittel, with more detail on specific properties of solids and experimental techniques. Simon,The Oxford Solid State Basics. A new book that covers most of the material in modern than any other Solid State book at this level, with clear, clean explanationsthroughout.

2 Also has the huge advantage of using Dirac notation over wavefunctions, whichshortens and clarifies almost every equation. Has well-written, helpful problems. David Tong s Lecture Notes on Applications of Quantum Mechanics. Covers much of the materialof Simon s book, also in clear modern notation, with a conversational tone. Also contains niceasides on topics like graphene and the Peierls instability. Annett,Superconductivity, Superfluids, and Condensates. A very clear first book on thesephenomena, both describing their basic phenomenology and connecting it to microscopic many body theory, but with a very light touch. The Feynman Lectures on Physics , volume 2. About ten chapters cover properties of bedtime reading, though a number of statements are out of most recent version is here; please report any errors found to Free Electron Introduction.

3 Drude Theory .. Sommerfeld Theory ..82 Crystal Bloch s Theorem .. Bravais Lattices .. The Reciprocal Lattice .. X-ray Diffraction .. 193 Band Bloch Electrons .. Tight Binding .. Nearly Free Electrons .. Phonons .. Band Degeneracy .. 334 Applications of Band Electrical Conduction .. Magnetic Fields .. Semiconductor Devices .. 435 Paramagnetism and Diamagnetism .. Ferromagnetism .. 526 Linear Response Functions .. Kramers Kronig .. The Kubo Formula .. 6331. Free Electron Models1 Free Electron these Notes , we investigate properties of solids. We would like to ask: What is the global ground State of the atoms? Is it a periodic crystal, and if so, what is thecrystal structure? Does it agree with the results of X-ray diffraction?

4 Given the crystal structure, what are the properties of the Solid ? For example, can we calculatethe thermal and electrical conductivity, color, hardness, magnetic susceptibility, resistivity, Resistivity is an especially interesting quantity, since it can range over 30 orders of magnitudebetween metals and insulators. Can we explain this dramatic difference in behavior?In principle, we have a theory of everything for solids, given by the HamiltonianH= j~22m 2j ~22M 2 j, Z e2|ri R |+ j<ke2|ri rk|+ < Z Z e2|R R |where capital letters/Greek indices denote lattice ions and lowercase letters/Latin indices denoteelectrons. However, in a Solid , withO(1023) nuclei and electrons, solving this Hamiltonian exactly isinfeasible. In fact, the situation is more like QCD than perturbative QFT. Couplings are generallystrong, and perturbation theory can fail.

5 Instead, we must use better approximations. First, we can use adiabatic approximations. Since the electrons are much less massive than theions, the ions move very slowly, and we may treat their effect on the electrons yields the Born-Oppenheimer approximation. Second, we may use independent particle approximations. By neglecting electron-electron inter-actions, we may approximate the behavior of the fullN-body interacting system by consideringthe behavior of a single electron. Third, we may use field theory methods. Suppose we can write the Hamiltonian in the formH =E0+ k k k k+f(.., k,.., k ,..).Then the low-lying degrees of freedom behave like independent harmonic oscillators. Iffissmall, we can treat it perturbatively. This approach is useful because many properties of solidsonly depend on the low-lying excitations. There will generally be two kinds of excitations: quasiparticles, or dressed particles, thatresemble a single free particle, and collective excitations which are due to many , we ll consider very basic free electron models, which completely neglect interactions ofthe electrons with the lattice ions; such an approach can only be a good approximation for , we reintroduce the lattice ions, leading to a discussion of band structure.

6 Much later, we llreintroduce interactions between electrons, leading to many body/field theory methods. We llsee that the structure of the Fermi sea tends to make interactions unimportant in some contexts,retroactively justifying the neglect of Free Electron TheoryThe Drude theory of metals, introduced in 1900, models a metal as a classical gas of electrons,assumed to be the valence electrons of the atoms used to form the metal. We assume the electrons don t interact with each other at all, the independent electron ap-proximation . However, we will allow collisions with the lattice ions. We take the free electronapproximation , assuming that in between collisions, the electrons are completely free, with theexception that the ions act as a wall preventing the electrons from leaving the metal. We assume that collisions instantaneously randomize the velocity of an electron, so that itsmean final velocity is zero, and that they occur in a timedtwith probabilitydt/ , where isthe relaxation time.

7 If we wanted to treat the collisions more carefully, we could choose the speed distribution aftera collision so that the electrons thermalize appropriately over time. It is difficult to calculate the collision rate 1. There are many contributions, includingscattering off impurities, phonons, and other electrons. One might estimate = 1/n vwhere is the cross section, but the cross section is infinite for the Coulomb fields of electron-electronscattering; the collisions simply aren t sharp as assumed by Drude theory. Instead, we take as a phenomenological , we consider the effects of static fields. Suppose the electrons experience an external forceFand have average momentum p . Thend p dt= p +Fwhere the first term accounts for collisions. For simplicity we drop the brackets below. The average current density is given byj= a static electric field, we thus havep= eE ,j= E, DC=ne2 m.

8 Next, we consider the Hall effect. Suppose we apply an electric fieldExand a transversemagnetic fieldBz. Then the Lorentz force should deflect electrons in theydirection, causing abuildup of charge on the side of the metal and creating a transverse fieldEy. More formally, letE= jwhere is the resistivity tensor. Again working in the steady State ,and restricting to a two-dimensional sample in thexyplane,E=(1nej B+mne2 j)51. Free Electron Modelsfrom which we read off , =(m/ne2 B/ne B/ne m/ne2 )=1 DC(1 B B 1)where B=eB/mis the cyclotron frequency. Note that had to be antisymmetric by rotationalinvariance. We conventionally report the Hall coefficientRH= xyB= is especially useful because , which depends on messy details, cancels out. Another usefulfact is that in practice, we measure transverse resistances, butRxy=VyIx= LEyLJx= EyJx= xyso this is equivalent to a measurement ofRH.

9 Note this is particular to two-dimensional samples. Finally, it is sometimes useful to know the conductivity, = DC1 + 2B 2(1 BT B 1). Strikingly, the Hall coefficient depends on the sign of the charge carriers, so it can be used toshow that charge carriers have negative charge. But even more strikingly, this fails! The Hallcoefficient is measured to have the opposite of the expected sign in some common materials,such as Be and Mg. It is also measured to be anisotropic. These results will be explained bycrystal and band structure below. In practice, the Hall effect can be used in reverse to detect magnetic fields, using a Hall make the measured voltages large, Hall sensors use materials with a low density of electrons,such as semiconductors. Drude theory works very well for , we consider AC fields. We consider an electric field with frequency , sodpdt= p eE0e i momentum is also sinusoidal, and we find the frequency-dependent conductivity ( ) = DC1 i ,j( ) = ( )E( ).

10 Note that this is only sensible if `where`is the mean free path, since we ve neglected thespatial variation of the Free Electron Models It is useful to consider the limit . Naively we would havelim ( ) = ne2i mbut the real part requires a bit more care. We haveRe ( ) =ne2m 1 + 2 2which is a Lorentzian with width 1/ and total area ne2/m. Hence we havelim ( ) = nse2me ( ) ne2i indicates thatjandEare 90 out of phase except for DC fields, where they are in phase. Now suppose we pass an electromagnetic wave through the material. It can be shown that thedielectric constant is related to the conductivity by ( ) = 1 +4 i ( ).At low frequencies, the imaginary part of yields an exponential damping, corresponding toabsorption of light. For high frequencies, we have ( ) 1 2p 2, 2p=4 ne2mwhere pis called the plasma frequency, and is notably independent of.