Brief Review of Solid State Physics

8/14/2019 Brief Review of Solid State Physics

http://slidepdf.com/reader/full/brief-review-of-solid-state-physics 1/27

A (Far Too) Brief Review of Solid State Physics

August 9, 2001

These first lectures will be an attempt to summarize some of the keyideas in solid state physics. These ideas form the intellectual foundationupon which most nanoscale physics has b een based. By “summarize” Imean just that. It’s extraordinarily difficult to do a complete and moderntreatment of these ideas even in a full two-semester sequence; therefore, we’llbe sacrificing depth and careful derivations as a trade-off for breadth anddeveloping a physical intuition.

We will supplement this introduction with additional discussions of par-ticular subjects as they arise in our look through the literature. At the endof these notes is a list of some references for those who want a more in-depthtreatment of these subjects.

It’s pretty amazing that we can understand anything at all about theproperties of condensed matter. Consider a cubic centimeter of copper, for

example. It contains roughly 1023 ion cores and over 1024 electrons, allof which interact (a priori not necessarily weakly) through the long-rangeCoulomb interaction. However, because of the power of statistical physics,we can actually understand a tremendous amount about that copper’s elec-trical, thermal, and optical properties.

In fact, because of some lucky breaks we can even get remarkably far bymaking some idealizations that at first glance might seem almost unphysical.In the rest of this section, we’ll start with a simple model system: nonin-teracting electrons at zero temperature in an infinitely high potential well.Gradually we will relax our ideal conditions and approach a more realisticdescription of solids. Along the way, we’ll hit on some important concepts

and get a better idea of why condensed matter physics is tractable at all.

1



1 Ideal Fermi Gas

By starting with noninteracting electrons, we’re able to pick a model Hamil-tonian for the single particle problem, solve it, and then pretend that themany-particle solution is simply related to that solution. Even withoutCoulomb interactions, we need to remember that electrons are fermions.Rigorously, the many-particle state would then be a totally antisymmetrizedproduct of single-particle states. For our purposes, however, we can get theessential physics out by just thinking about filling up each single-particlespatial state with one spin-up(↑) and one spin-down (↓) electron.

Intro quantum mechanics tells us that an eigenfunction with momentump for a free particle is a plane wave with a wavevector k = p/h. Consider aninfinitely tall 1d potential well with a flat bottom of length L, the standardintro quantum mechanics problem (the generalization to 2d and 3d is sim-ple, and we’ll get the results below). The eigenfunctions of the well have tobe built out of planewaves, and the boundary conditions are that the wave-function ψ(x) has to vanish at the edge of the well and outside (the infinitepotential step means we’re allowed to relax the usual condition that ψ(x)has to be continuous). The wavefunctions that satisfy these conditions are:

ψn(x) =

2

Lsin

π

Lnx (1)

where n > 0 is an integer. So, the allowed values of k are quantized due to theboundary conditions, and the states are spaced in k by π/L. In 3 dimensions,we have states described by k = (kx, ky , kz), where kx = (π/L)nx, etc. Noweach spatial state in k each takes up a “volume” of (π/L)3. As usual, theenergy of an electron in such a state is

E =h2

2m(k2x + k2y + k2z). (2)

An important feature here is that the larger the box, the closer the spacing in energy of single particle states, and vice-versa.

Suppose we dump N of our ideal electrons into the box. This system iscalled an ideal Fermi gas. Now we ask, filling the single particle states fromthe bottom up (i.e. in the ground state), what is the energy of the highestoccupied single-particle state? We can count states in k-space, and for valuesof k that are large compared to the spacing, we can do this counting using anintegral rather than a sum. Calling the highest occupied k value the Fermi wavevector , kF, and knowing that each spatial state can hold two electrons,

2



we can write N in terms of kF as:

N = 2 ×

L

π

3 kF0

1

84πk2dk.

=1

3π2k3FL3. (3)

If we define n3d ≡ N/V and the Fermi energy as

E F ≡hk2F2m

, (4)

we can manipulate Eq. 3 to find the density of states at the Fermi level :

n3d(E ) = 13π2

2mE

h2

3/2 , → (5)

ν 3d(E = E F) ≡dn3ddE

|E =E F

=1

2π2

2m

h2

3/2E 1/2F . (6)

The density of states ν 3d(E ) as defined above is the number of single-particle states available per unit volume per unit energy. This is a very important quantity because, as we will see later (in Sect. 2.4), the ratesof many processes are proportional to ν (E F). Intuitively, ν (E ) represents

the number of available states into which an electron can scatter from someinitial state. Looking at Eqs. (6,6) we see that for 3d increasing the densityof electrons increases both E F and ν 3d.

Let’s review. We’ve dumped N particles into a box with infinitely highwalls and let them fill up the lowest possible states, with the Pauli restrictionthat each spatial state can only hold two electrons of opposite spin. We thenfigured out the energy of the highest occupied single-particle state, E F, andthrough ν (E F and the sample size we can say how far away in energy thenearest unoccupied spatial state is from E F.

Thinking semiclassically for a moment, we can ask, what is the speed of the electron in that highest occupied state? The momentum of that state iscalled the Fermi momentum , pF = hkF, and so we can find a speed by:

vF =hkFm

in 3d =h(3π2n3d)

1/3

m. (7)

3



dim. nd(E F) ν d(E F) vF

3 13π2

2mE Fh2

3/212π2

2mh2

3/2E 1/2F

h(3π2n3d)1/3

m

2 1πmE Fh2

mπh2

h(2πn2d)1/2

m

1 2π

2mE Fh2

1/21π

2mh2

1/21

E 1/2F

πhn1d2m

Table 1: Properties of ideal Fermi gases in various dimensionalities.

So, the higher the electron density, the faster the electrons are moving . Aswe’ll see later, this semiclassical picture of electron motion can often be auseful way of thinking about conduction.

We can redo this analysis for 2d and 1d, where n2d ≡ N/A and n1d ≡N/L, respectively. The results are summarized in Table 1. Two remarks:notice that ν 2d(E F) is independent of n2d; further, notice that ν 1d actuallydecreases with increasing n1d. This latter property is just a restatement of something you already knew: the states of an infinite 1d well get farther

and farther apart in energy the higher you go.The results in Table 1 are surprisingly more general than you might

expect at first. One can redo the entire analysis starting with Eq. (1) anduse periodic boundary conditions (ψ(x = L) = ψ(x = 0) = 0;ψ(x = L) =ψ(x = 0)). When this is done carefully, the results in Table 1 are reproducedexactly .

From now on, we will generally talk about the periodic boundary condi-tion case, which allows both positive and negative values of kx, ky, and kz,and has the spacing between states in k be 2π/L. In the 3d case, this meansthe ground state of the ideal Fermi gas can be represented as a sphere of radius kF in k-space, with each state satisfying k < kF being occupied by a

spin-up and a spin-down electron. States corresponding to k > kF are un-occupied. The boundary between occupied and unoccupied states is calledthe Fermi surface.

Notice that the total momentum of the ideal Fermi gas is essentially zero;the Fermi sphere (disk/line in 2d/1d) is centered on zero momentum. If an

4



k

k

x

y

F e r m i s p h e r e ( z e r o f i e l d )

F e r m i s p h e r e ( a f t e r f i e l d )

Figure 1: Looking down the z-axis of the 3d Fermi sphere, before and afterthe application of an electric field in the x-direction. Because there wereallowed k states available, the Fermi sphere was able to shift its center to anonzero value of kx.

electric field is applied in, say, the x-direction, because there are availablestates, the Fermi sphere will shift, as in Fig. 1. The fact that the sphere re-mains a sphere and the picture represents the case of an equilibrium currentin the x-direction is discussed in various solid state books.

One other point. Extending the above discussion lets us introduce thefamiliar idea of a distribution function , f (T, E ), the probability that a par-

ticular state with energy E is occupied by an electron in equilibrium at aparticular temperature T . For our electrons-in-a-box system, at absolutezero the ground state is as we’ve been discussing, with filled single-particlestates up to the Fermi energy, and empty states above that. Labeling spin-up and spin-down occupied states as distinct, mathematically,

f (0, E ) = Θ(E F − E ), (8)

where Θ is the Heaviside step function:

Θ(x) = 0, x < 0

= 1, x > 0. (9)

Note that f is normalized so that ∞

0f (T, E )ν (E )dE = N. (10)

5



Figure 2: The Fermi distribution at various temperatures.

At finite temperature the situation is more complicated. Some stateswith E < E F are empty and some states above E F are occupied becausethermal energy is available. In general, one really has to do the statisti-cal mechanics problem of maximizing the entropy by distributing N indis-tinguishable electrons among the available states at fixed T . This is dis-cussed in detail in many stat mech books, and corresponds to minimizingthe Helmholtz free energy. The answer for fermions is the Fermi distribution function :

f (T, E ) =1

exp[(E − µ(T ))/kBT ] + 1, (11)

where µ is the chemical potential . The chemical potential takes on the valuewhich satisfies the constraint of Eq. (10). At T = 0, you can see thatµ(T = 0) = E F. See Fig. 2.

A physical interpretation of µ is the average change in the free energy of a system caused by adding one more particle. For a thermodynamic spin onµ, start by thinking about why temperature is a useful idea. Consider twosystems, 1 and 2. These systems are in thermodynamic equilibrium if, whenthey’re allowed to exchange energy, the entropy of the combined system isalready maximized. That is,

δS tot =

∂S 1∂E 1

δE 1 +

∂S 2∂E 2

δE 2

=

∂S 1∂E 1

−

∂S 2∂E 2

δE 1

= 0, (12)

6



implying ∂S 1∂E 1

=

∂S 2∂E 2

. (13)

We define the two sides of this equation to be 1/kBT 1 and 1/kBT 2, andsee that in thermodynamic equilibrium T 1 = T 2. With further analysis onecan show that when two systems of the same temperature are brought intocontact, on average there is no net flow of energy between the systems.

We can run through the same sort of analysis, only instead of allowingthe two systems to exchange energy such that the total energy is conserved,we allow them to exchange energy and particles so that total energy andparticle number are conserved. Solving the analogy of Eq. (12) we find thatequilibrium between the two systems implies both T 1 = T 2 and µ1 = µ2.

Again, with further analysis one can see that when two systems at the sameT and µ are brought into contact, on average there is no net flow of energyor particles between the systems.

So, in this section we’ve learned a number of things:

• One useful model for electrons in solids is an ideal Fermi gas. Startingfrom simple particle-in-a-box considerations we can calculate proper-ties of the ground state of this system. We find a Fermi sea , with fullsingle particle states up to some highest occupied level whose energyis E F.

• We also calculate the spacing of states near this Fermi energy and the

(semiclassical) speed of electrons in this highest state.

• We introduce the idea of a distribution function for calculating finite-temperature properties of the electron gas.

• Finally, we see the chemical potential, which determines whether par-ticles flow between two systems when they’re brought into contact.

2 How ideal are real Fermi gases?

Obviously we wouldn’t spend time examining the ideal Fermi gas if it wasn’ta useful tool. It turns out that the concepts from the previous section

generally persist even when complications like actual atomic structure andelectron-electron interactions are introduced.

7



2.1 Band theory

Clearly our infinite square well model of the potential seen by the electrons isan oversimplification. When the underlying lattice structure of (crystalline)solids is actually included, the electronic structure is a bit more complicated,and is typically well-described by band theory .

Start by thinking about two hydrogen atoms very far from one another.Each atom has an occupied 1s orbital, and a number of unoccupied higherorbitals ( p,d,etc.). If the atoms are moved sufficiently close (a spacing com-parable to their radii), a more useful set of energy eigenstates can be formedby looking at hybrid orbitals. These are the σ and σ∗ bonding and antibond-ing orbitals. When a perturbation theory calculation is done accounting forthe Coulomb interaction between the electrons, the result is that instead of

two single-electron states (1s) of identical energy, we get two states (σ, σ∗)that differ in energy.

Now think about combinations of many-electron atoms. It’s reasonableto think about an ion core containing the nucleus and electrons that arefirmly stuck in localized states around that nucleus, and valence electrons,which are more loosely bound to the ion core and can in principal overlapwith their neighbors. For small numbers of atoms, one can consider the dif-ferent types of bonding that can occur. Which actually takes place dependson the details of the atoms in question:

• Van der Waals bonding. This doesn’t involve any significant change tothe electronic wavefunctions; atom A and atom B remain intact and

interact via fluctuating electric dipole forces. Only included on thislist for completeness.

• Ionic bonding. For Coulomb energy reasons atom A donates a valenceelectron that is accepted by atom B. Atom A is positively charged andits remaining electrons are tightly bound to the ion core; atom B isnegatively charged, and all the electrons are tightly bound. Atoms Aand B “stick” to each other by electrostatics, but the wavefunctionoverlap between their electrons is minimal.

• Covalent bonding. As in the hydrogen case described above, it becomesmore useful to describe the valence electrons in terms of molecular or-

bitals, where to some degree the valence electrons are delocalized overmore than one ion core. In large molecules there tends to be clusteringof energy levels with intervening gaps in energy containing no allowedstates. There is a highest occupied molecular orbital (HOMO) and alowest unoccupied molecular orbital (LUMO).

8



• Metallic bonding. Like covalent bonding only more extreme; the delo-

calized molecular orbitals extend over many atomic spacings.

Now let’s really get serious and consider very large numbers of atomsarranged in a periodic array, as in a crystal. This arrangement has lotsof interesting consequences. Typically one thinks of the valence electronsas seeing a periodic potential due to the ion cores, and to worry aboutbulk properties for now, we’ll ignore the edges of our crystal by imposingperiodic boundary conditions. When solving the Schrodinger equation forthis situation, the eigenfunctions are plane waves (like our old free Fermigas case) multiplied by a function that’s periodic with the same period asthe lattice:

ψk(r) = uk(r)exp(ik · r),uk(r) = uk(r + rn). (14)

These wavefunctions are called Bloch waves, and like the free Fermi gaswavefunctions are labeled with a wavevector k. See Fig. 3

The really grungy work is two-fold: finding out what the the functionuk(r) looks like for a particular arrangement of particular ion cores, and fig-uring out what the corresponding allowed energy eigenvalues are. In practicethis is done by a combination of approximations and numerical techniques.It turns out that while getting the details of the energy spectrum right isextremely challenging, there are certain general features that persist.

First, not all values of the energy are allowed. There are bands of energy

for which Bloch wave solutions exist, and between them are band gaps, forwhich no Bloch wave solutions with real k are found. Plotting energy vs. kin the 1d general case typically looks like Fig. 4.

The details of the allowed energy bands and forbidden band gaps areset by the interaction of the electrons with the lattice potential. In fact,looking closely at Fig. 4 we see that the gaps really “open up” for Blochwaves whose wavevectors are close to harmonics of the lattice potential. TheCoulomb interactions between the electrons only matter here in the indirectsense that the electrons screen the ion cores and self-consistently contributeto the periodic potential.

Now we consider dropping in electrons and ask what the highest occupied

single-particle states are, as we did in the free Fermi gas case. The situationhere isn’t too different, though the properties of the ground state will end updepending dramatically on the band structure. Notice, too, that the Fermisea is no longer necessarily spherical, since the lattice potential felt by theelectrons is not necessarily isotropic.

9



Figure 3: The components of a Bloch wave, and the resulting wavefunction.From Ibach and Luth.

10



Figure 4: Allowed energy vs. wavevector in general 1d periodic potential,from Ibach and Luth.

Figure 5 shows two possibilities. In the first, the number of electronsis just enough to exactly fill the valence band . Because there is an energygap to the nearest allowed empty states, this system is a band insulator : if an electric field is applied, the Fermi surface can’t shift around as in Fig.1 because there aren’t available states. Therefore the system can’t developa net current in response to the applied field. A good example of a bandinsulator is diamond, which has a gap of around 10 eV. The second caserepresents a metal, and because of the available states near the Fermi level,it can support a current in the same way as the ideal Fermi gas in Fig. 1.

Other possibilities exist, as shown in Fig. 6. One can imagine a bandinsulator where the gap is quite small, so small that at room temperature adetectable number of carriers can be promoted from the valence band intothe conduction band. Such a system is called an intrinsic semiconductor . Agood example is Si, which has a gap of around 1.1 eV, and a carrier densityat room temperature of around 1.5 × 1010 cm−3.

Further, it is also possible to dope semiconductors by introducing im-purities into the lattice. A donor such as phosphorus in silicon can addan electron to the conduction band. At zero temperature, this electron isbound to the P donor, but the binding is weak enough to be broken at higher

11



E

v a l e n c e b a n d

c o n d u c t i o n b a n d

i n s u l a t o r m e t a l

Figure 5: Filling of allowed states in two different systems. On the left,the electrons just fill all the states in the valence band, so that the nextunoccupied state is separated by a band gap; this is an insulator. On theright, the electrons spill over into the conduction band, leaving the systemmetallic.

E

v a l e n c e b a n d

c o n d u c t i o n b a n d

i n s u l a t o r

s e m i c o n d u c t o r

( i n t r i n s i c , f i n i t e T )


( n - d o p e d , f i n i t e T )


( p - d o p e d , f i n i t e T )

Figure 6: More possibilities. On the left is a band insulator, as before. Next

is an intrinsic semiconductor, followed by two doped semiconductors.

12



temperatures, leading to usual electronic conduction. This is the third case

shown in Fig. 6. Similarly, an acceptor such as boron in silicon can grab anelectron out of the valence band, leaving behind a positively charged hole.This hole acts like a carrier with charge +e; at zero temperature it is weaklybound to the B acceptor, but at higher temperatures it can be freed, leadingto hole conduction. One more exotic possibility, not shown, is semimetallicbehavior, as in bismuth. Because of the funny shape of its Fermi surface,parts of Bi’s valence band can have holes at the same time that parts of Bi’sconduction band can have electrons (!).

While the existence of Bloch waves means it is possible to label eacheigenstate with a wavevector k, that doesn’t necessarily mean that the en-ergy of that state depends quadratically on k as in Eq. (2). The approxi-

mation that E (k

) ∼ k

2

is called the effective mass approximation . As youmight expect, the effective mass is defined by:

E (k) = h2

k2x + k2y2m∗

t

+k2z

2m∗

l

, (15)

where we’re explicitly showing that the effective mass m∗ isn’t necessarilyisotropic.

One final point: there is one more energy scale in the electronic structureof real materials that we explicitly ignored in our ideal Fermi gas model. Thework function , Φ, is defined as the energy difference between the vacuum (afree electron outside the sample) and the Fermi energy, E vac− E F. Our toy

infinite square well model artificially sets Φ = ∞. Unsurprisingly, Φ alsodepends strongly on the details of the material’s structure, and can varyfrom as low as 2.4 eV in Li to over 10 eV in band insulators.

The important ideas to take away from this subsection are:

• Bonding in small systems is crucially affected by electronic bindingenergies and Coulomb interactions.

• Large systems typically have energy level distributions well-describedby bands and gaps.

• Eigenstates in systems with periodicity are Bloch waves that can belabeled by a wavevector k.

• Whether a system is conducting, insulating, or semiconducting de-pends critically on the details of its band structure, including the num-ber of available carriers. Systems can exhibit either electronic or holeconduction depending on structure and the presence of impurities.

13



Figure 7: Surface states on copper, imaged by Crommie and Eigler at IBMwith a scanning tunneling microscope.

• The energy needed to actually remove an electron from a material tothe vacuum also reflects the structure of that material.

2.2 Structural issues

The previous section deliberatly neglected a number of what I’ll call “struc-tural issues”. These include nonidealities of structure such as boundaries,impurities, and other kinds of structural disorder. Further, we’ve treatedthe ion cores as providing a static backgroundd potential, when in fact theycan have important dynamics associated with them.

Let’s deal with structural “defects” first. We’ll treat some specific effectsof these defects later on. First, let’s ask what are the general consequencesof not having an infinite perfect crystal lattice.

Bloch waves infinite in extent are no longer exact eigenstates of thesystem. That’s not necessarily a big deal. Intuitively, an isolateddefect in the middle of a large crystal isn’t going to profoundly alter

the nature of the entire electronic structure. In fact, the idea thatthere are localized electronic states around the ion cores and can bedelocalized (extended ) states which span many atomic spacings is stilltrue even without any lattice at all; this happens in amorphous andliquid metals.

14



Special states can exist at free surfaces. Unsurprisingly, these are called

surface states. The most famous experimental demonstration of thisis shown in Fig. 7. Suppose the surface is the x − y plane. Because of the binding energy of electrons in the material, states exist which arepinned to the surface, having small z extent and wavelike (or localized,depending in surface disorder) character in x and y. Surface states canhave dramatic implications when (a) samples are very small, so thatthe number of surface states is comparable to the number of “bulk”states; and (b) the total number of carriers is very small, as in somesemiconductors, so that an appreciable fraction of the carriers can endup in surface states rather than bulk states.

Interfaces between different materials can also produce dramatic ef-

fects. The boundary between a material and vacuum is just the lim-iting case of the interface between two materials with different workfunctions. When joining two dissimilar materials together there aretwo conditions one has to keep in mind:

(a) The “vacuum” for materials A and B is the same, though ΦA = ΦBin general. That means that prior to contact the Fermi levels of A andB are usually different.

(b) Two systems that can exchange particles are only in equilibriumonce their chemical potentials are equal. That implies that when con-tact is made between A and B, charge will flow between the two to

equalize their Fermi levels.The effect of (a) and (b) is that near interfaces “space charge” layerscan develop which bend the bands to equalize the Fermi levels acrossthe junction. The details of these space charge layers (e.g. how thickare they, and what is the charge density profile) depend on the avail-ability of carriers (is there doping?) and the dielectric functions of the two materials. In general one has to solve Poisson’s equation self-consistently while considering the details of the band structure of thetwo materials.

A term related to all this is a Schottky barrier . It is possible to haveband parameters of two materials be such that the space charge layer

which forms upon their contact can act like a substantial potentialbarrier to electronic transport from one material to the other. Forexample, pure Au on GaAs forms such a barrier. These barriers havevery nonlinear I − V characteristics, in contrast to “Ohmic” contactsbetween materials (an example would be In on GaAs). A great deal of

15



semiconductor lore exists about what combinations of materials form

Schottky barriers and what combinations form Ohmic contacts.

We also need to worry about the dynamics of the ion cores, rather thannecessarily treating them as a static background of positive charge. Thequantized vibrations of the lattice are known as phonons. We won’t go intoa detailed treatment of phonons, but rather will highlight some importantterminology and properties.

A unit cell is the smallest grouping of atoms in a crystal that exhibits allthe symmetries of the crystal and, when replicated periodically, reproducesthe positions of all the atoms.

The vibrational modes of the lattice fall into two main categories, dis-tinguished by their dispersion curves, ω(q), where q is the wavenumber of

the wave. When q = 0, we’re talking about a motion such that the displace-ments of the atoms in each unit cell is identical to those in any other unitcell. Acoustic branches have ω(0) = 0. There are three acoustic branches,two transverse and one longitudinal. Optical branches have ω(0) = 0, andthere typically three optical branches, too. The term “optical” is historic inorigin, though optical modes of a particular q are typically of higher energythan acoustic modes with the same wavevector.

For our purposes, it’s usually going to be sufficient to think of phononsas representing a bath of excitations that can interact with the electrons. In-tuitively, the coupling between electrons and phonons comes about becausethe distortions of the lattice due to the phonons show up as slight variations

in the lattice potential through which the electrons move. Electron-phononscattering can be an important process in nanoscale systems, as we shallsee.

The Debye model of phonons does a nice job at describing the low energybehavior of acoustic modes, which tend to dominate below room tempera-ture. The idea is to assume a simple dispersion relation, ω = vL,Tq, where vis either the longitudinal or transverse sound speed. This relation is assumedto hold up to some high frequency (short wavelength) cutoff, ωD, the Debyefrequency, set by the requirement that the total number of acoustic modes= 3rN uc, where r is the number of atoms per unit cell, and N uc is the num-ber of unit cells in the solid. Without going into details here, the main result

of Debye theory for us is that at low temperatures (T < T D ≡ hωD/kB), theheat capacity of 3d phonons varies as T 3.One more dynamical issue: for extremely small structures, like clusters of

tens of atoms, figuring out the equilibrium positions of the atoms requiresa self-consistent calculation that also includes electronic structure. That

16



is, the scales of electronic Coulomb contributions and ionic displacement

energies become comparable. Electronic transitions can alter the equilibriumconformations of the atoms in such systems.

The important points of this subsection are:

• Special states can exist at surfaces, and in small or low carrier densitysystems these states can be very important.

• Interfaces between different materials can be very complicated, involv-ing issues of charge transfer, band bending, and the possible formationof potential barriers to transport.

• The ion cores can have dynamical degrees of freedom which couple to

the electrons, and the energy content of those modes can be stronglytemperature dependent.

• In very small systems, it may be necessary to self-consistently accountfor both the electronic and structural degrees of freedom because of strong couplings between the two.

2.3 Interactions?

We have only b een treating Coulomb interactions between the electronsindirectly so far. Why have we been able to get away with this? As we shallsee, one key is the fact that our electronic Bloch waves act so much like a

cold (T < T F ≡ E F/kB) Fermi gas. Another relevant piece of the physicsis the screening of point charges that can take place when the electrons arefree to move (e.g. particularly in metals).

Let’s look at that second piece first, working in 3d for now. Supposethere’s some isolated perturbation δU (r) to the background electrical po-tential seen by the electrons. For small perturbations, we can think of thisas causing a change in the local electron density that we can find using thedensity of states:

δn(r) = ν 3d(E F)|e|δU (r). (16)

Now we can use Poisson’s equation factor in the change in the potentialcaused by the response of the electron gas:

∇2(δU (r)) = e0

δn(r) = ν 3d(E F)|e|δU (r). (17)

17



In 3d, the solution to this is of the form δU (r) ∼ (1/r)(exp(−λr)), where

λ ≡ 1/rTF, defining the Thomas-Fermi screening length :

rTF =

e2

0ν 3d(E F)

−1/2

. (18)

If we plug in our results from Eqs. (6,6) for the free Fermi gas in 3d, we find

rTF 0.5

n

a0

−1/6

, (19)

where a0 is the Bohr radius. Plugging in n ∼ 1023 cm−3, typical for a metallike Cu or Ag, we find rTF ∼ 1 A. So, the typical lengthscale for screeningin a bulk metal can be of atomic dimensions, which explains why the ion

cores are so effectively screened.Looking at Eq. (18) it should be clear that screening is strongly affected

by the density of states at the Fermi level. This means screening in bandinsulators is extremely poor, since ν ∼ 0 in the gap. Further, because of the changes in ν with dimensionality, screening in 2d and 1d systems isnontrivial.

Now let’s think about electron-electron scattering again. We have toconserve energy, of course, and we have to conserve momentum, thoughbecause of the presence of the lattice in certain circumstances we can “dump”momentum there. The real “crystal momentum” conservation that must beobeyed is:

k1 + k2 = k3 + k4 + G, (20)where G is a reciprocal lattice vector . For zero temperature, there just aren’tany open states the electrons can scatter into that satisfy the conservationconditions! At finite temperature, where the Fermi distribution smears outthe Fermi surface slightly, some scattering can now occur, but (a) screeningreduces the scattering cross-section from the “bare” value; and (b) the Pauliprinciple reduces it further by a factor of (kBT/E F)

2. This is why ignor-ing electron-electron scattering in equilibrium behavior of solids isn’t a badapproximation. We will come back to this subject soon, though, becausesometimes this “weak” scattering process is the only game in town, and canhave profound implications for quantum effects.

The full treatment of electron-electron interactions and their consequencesis called Landau Fermi Liquid Theory . We won’t get into this in any signif-icant detail, except to state some of the main ideas. In LFLT, we considerstarting from a noninteracting Fermi gas, and adiabatically turn on electron-electron interactions. Landau and Luttinger argued that the ground state of

18



the noninteracting system smoothly evolves into the ground state of the in-

teracting system, and that the excitations above the ground state also evolvesmoothly. The excitations of the noninteracting Fermi gas were (electron-above-E F, hole-below-E F), so-called particle-hole excitations. In the inter-acting system, the excitations are quasiparticles and quasiholes. Rather thana lone electron above the Fermi surface, a LFLT quasiparticle is such an elec-tron plus the correlated rearrangement of all the other electrons that thentakes place due to interactions.

This correlated rearrangement of the other electrons is called dressing .An electron injected from the vacuum into a Fermi liquid is then “dressed” toform a quasiparticle. The effect of these correlations is to renormalize certainquantities like the effective mass and the compressibility. The correlations

can be quantified by a small number of “Fermi liquid parameters” that canbe evaluated by experiment. Other than these relatively mild corrections,quasiparticles usually act very much like electrons.

Note that this theory can break down under various circumstances. Inparticular, LFLT fails if a new candidate ground state can exist that haslower energy than the Fermi liquid ground state, and is separated fromthe LFLT ground state by a broken symmetry. The symmetry breakingmeans that the smooth evolution mentioned above isn’t possible. The classicexample of this is the transition to superconductivity.

In 1d systems with strong interactions, Fermi liquid theory can alsobreak down. The proposed ground state in such circumstances is called aLuttinger liquid . We will hopefully get a chance to touch on this later.

The main points here are:

• Screening depends strongly on ν (E F), and so on the dimensionalityand structure of materials.

• Electron-electron scattering is pretty small in many circumstances be-cause of screening and the Pauli principle.

• LFLT accounts for electron-electron interactions by dressing the bareexcitations.

2.4 Transitions and rates

Often we’re interested in calculating the rate of some scattering process thattakes a system from an initial state |i takes it to a final state |f . Also, oftenwe’re not really interested in the details of the final state; rather, we wantto consider all available final states that satisfy energy conservation. The

19



result from first order time-dependent perturbation theory is often called

Fermi’s Golden Rule. If the potential associated with the scattering is V s,the rate is approximately:

S =2π

h|i|V s||2δ(∆E ), (21)

where the δ-function (only strictly a δ-function as the time after the collision→ ∞) takes care of energy conservation. If we want to account for all finalstates that satisfy the energy condition, the total rate is given by the integralover energy of the right hand side of Eq. (21) times the density of states forfinal states.

The important point here is, as we mentioned above, the density of states

plays a crucial role in establishing transition and relaxation rates. Onegood illustration of this in nanoscale physics is the effect of dimensionalityon electron-phonon scattering in, for example, metallic single-walled carbonnanotubes. Because of the peculiarities of their band structure, these objectsare predicted to have one-dimensional densities of states. There are onlytwo allowed momenta, forward and back, and this reduction of the Fermisurface to these two points greatly suppresses the electron-phonon scatteringrate compared to that in a bulk metal. As a result, despite large phononpopulations at room temperature, ballistic transport of carriers is possiblein these systems over micron distances. In contrast, the scattering time dueto phonons in bulk metals is often three orders of magnitude shorter.

We won’t go into detail calculating rates here; any that we need laterwe’ll do at the time. Often this simple perturbation theory picture gives im-portant physical insights. More sophisticated treatments include diagram-matic expansions, equivalent to higher-order perturbation calculations.

One other thing to bear in mind about transition rates. If multipleindependent processes exist, and their individual scattering events are un-correlated and well-separated in time, it is safe to add rates:

1

τ tot=

1

τ 1+

1

τ 2+ .... (22)

If a single effective rate can be written down, this implies an exponentialrelaxation in time, as we know from elementary differential equations. Thisis not always the case, and we’ll see one major example later of a nonexpo-nential relaxation.

20



3 Transport

One extremely powerful technique for probing the underlying physics innanoscale systems is electrical transport, the manner in which charge flows(or doesn’t) into, through, and out of the systems. Besides the simple slap-on-ohmmeter-leads dc resistance measurement, one can consider transport(and noise, time-dependent fluctuations of transport) with multiple leadconfigurations as a function of temperature, magnetic field, electric field,measuring frequency, etc.

We’ll begin with some general considerations about transport, and progresson to different classical and quantum treatments of this problem.

3.1 Classical transport

Let’s start with a collection of n classical noninteracting electrons per unitvolume, whizzing around in random directions with some typical speed v0.Suppose each electron typically undergoes a collision every time interval τ that completely scrambles the direction of its momentum, and that an elec-tric field E acts on the electrons. Without the electric field, the averagevelocity of all the electrons is zero because of the randomness of their di-rections. In the steady state with the field, the average electron velocityis:

vdft =−eEτ

m. (23)

The current density is simply j = n(−e)vdft, and we can find the conductiv-ity σ and the mobility µ (not to be confused with the chemical potential...):

σ =ne2τ

m

µ ≡eτ

m. (24)

This is a very simplified picture, but it introduces the idea of a charac-teristic time scale and the notion of a mobility. Notice that longer timesbetween momentum-randomizing collisions and lower masses mean highermobilities and conductivities.

Now think about including statistical mechanics here. One can definea classical distribution function by saying that probability of finding a par-ticle at time t with within dr of r with a momentum within dp of p isf (t, r,p)drdp. We know from the Liouville theorem that as a function of time the distribution function has to obey a kind of continuity equation

21



that expresses conservation of particles and momentum. This is called the

Boltzmann equation and looks like this:∂f

∂t+ v · ∇rf + F · ∇pf =

∂f

∂t

coll

. (25)

Here we’ve used the idea that forces F = p and that v = r. This is allexplicitly classical, since it assumes that we can specify both r and p. Theright hand side is the contribution to changes in f due to collisions likethe ones mentioned above. In the really simple case we started with, oneeffectively replaces that term with (f − f 0)/τ , where f 0 is the equilibriumdistribution function at some temperature. For more complicated collisionprocesses, naturally one could replace τ with some function of r,p, t.

A good treatment of this is in the appendices to Kittel’s solid state book,as well as many other places. If you assume that τ is a constant, you cando the equilibrium problem with a classical distribution function and findthat this treatment gives you Fick’s law for diffusion. That is, to first ordera density gradient produces a particle current:

jn = −D∇n = −1

3v2τ n, (26)

in 3d, where we’ve defined the particle diffusion constant D.

3.2 Semiclassical transport

We can do better than the above by incorporating what we know aboutband theory and statistical mechanics. Note that to preserve our use of the Boltzmann equation for quantum systems, in general we need to thinkhard about the fact that r,p don’t commute. For now, we’ll assume thatquantum phase information is lost between collisions. We’ll see later thatpreservation of that phase info leads to distinct consequences in transport.We also need to assume that the disorder that produces the scattering isn’ttoo severe; one reasonable criterion is that

kFvFτ >> 1. (27)

This says that the electron travels many wavelengths between scattering

events, so our Bloch wave picture of the electronic states makes sense.Remembering to substitute hk for p, one can solve the Boltzmann equa-tion using the Fermi distribution function, and again find the diffusion con-stant:

D =1

3v2Fτ. (28)

22



Here τ is the relaxation time at the Fermi energy.

In general, we can relate the measured conductivity to microscopic pa-rameters using the Einstein relation:

σ = e2νD, (29)

where ν is again our density of states at the Fermi level. In principal, one canmeasure ν by either a tunneling experiment or through the electronic heatcapacity. If ν is known, D can be inferred from the measured conductivity.

3.3 Quantum transport

A fully quantum mechanical treatment of transport is quite involved, though

some nanoscale systems can be well understood through models involvingsimple scattering of incident electronic waves off various barriers. This sim-ple scattering picture is called the Landauer-B¨ uttiker formulation, and wewill return to it later.

Another approach to quantum transport is to use Greens functions, ofteninvolving perturbative expansions, the shorthand for which are Feynmandiagrams. The small parameter often used in such expansions is (kF)−1, ashinted at in Eq. (27).

We’ll leave our discussion of quantum corrections to conductivity untilwe’re examining the relevant papers.

3.4 Measurement considerations and symmetriesJust a few words about transport measurements. These can be nicely dividedinto two-terminal and multi-terminal configurations. In the world of theory,one measures conductance by specifying the potential difference across thesample, and measuring the current that flows through the sample with aperfect ammeter. In practice, it is often much easier to measure resistance.That is, a current is specified through the sample, and an idea volt meter isused to measure the resulting potential difference.

In two-terminal measurements, the voltage is measured using the sameleads that supply the current, as in Fig. 8a. This is usually easier to do thanmore complicated configurations, especially if the device under examination

is extremely small. A practical problem can result, though, because usuallythe sample is relatively far away from the current source and volt meter(i.e. at the bottom of a cryogenic setup). The voltage being measuredcould include funny junction voltages at joints between the leads and thesample, as well as thermal EMFs due to temperature gradients along the

23



leads, etc. Some of these complications can be avoided by actually doing

the measurement at some low but nonzero frequency. Contact and leadresistances, however, remain, and can be significant.

Four-terminal resistance measurements can eliminate this problem, thoughcare has to be taken in interpreting the data in certain systems. The ideais shown in Fig. 8b. The device under test has four (or more) leads, twoof which are used to source and sink current, and two of which are used tomeasure a potential difference. Ideally, the voltage probes are at the same(chemical) potential as their contact points on the sample, and those con-tact points are very small compared to the sample size. The first conditionimplies that no net current flows between the voltage probes and the sam-ple, avoiding the problem of measuring contact and lead resistances. The

second condition is necessary to avoid having the voltage probes “shortingout” significant parts of the sample and seriously altering the equipotentiallines from their ideal current-leads-only configuration. A third condition,that the voltage probes be “weakly coupled” to the sample, ensures thatcarriers are not likely to go back and forth between the voltage probes andthe sample. This is particularly important when the carriers in the sampleare very different than those in the probes (such as in charge density wavesystems, or Luttinger liquids). In such cases the nature of the excitationsin the sample near the voltage probes could be strongly changed if carrierscould be exchanged freely.

While more complicated to arrange, especially in nanoscale samples,multiprobe measurements can provide more information than two-probe

schemes. An example: in 2d samples it is possible to use the additionalprobes to measure the Hall voltage induced in the presence of an externalmagnetic field, as well as to measure the longitudinal (along the current)resistance. The classical Hall voltage (balancing the Lorentz force by a Hallfield transverse to the longitudinal current) is proportional to B, and canbe used to find the sign of the charge carriers as well as their density.

In general, with multiple probes it is possible to measure with all sortsof combinations of voltage and current leads. When the equations of mo-tion are examined, certain symmetry relations (Onsager relations) must bepreserved. For two terminal measurements, we find that the measured con-ductance G must obey G(B) = G(−B). In the four probe case, it is possible

to show (the argument treats all four probes equivalently) that:

Rkl,mn(B) = Rmn,kl(−B), (30)

where Rkl,mn means current in leads k,l, voltage measured between leadsm,n.

24



V

R

0

> > R

s

I

~

_

V / R

0

I + , V +

I - , V -

s a m p l e

V

V

R

0

> > R

s

I

~

_

V / R

0

I +

I -

s a m p l e

V

V + V -

1

2 3

4

Figure 8: Schematics of 2 and 4 probe resistance measurements.

25



4 Concluding comments

That’s enough of an overview for now of basic solid state physics results.Hopefully we’re all on the same page, and this has gotten you thinking aboutsome of the issues that will be relevant in the papers we’ll be reading.

The next two meetings we’ll turn to the problems of characterizing andfabricating nanoscale systems. This should familiarize you with the stateof the art in this field, again to better prepare you for the papers to come.After those talks, we’ll start to examine quantum effects in transport, andthen it’s time to start doing seminars.

5 References

General solid state references:

H. Ibach and H. Luth. Solid State Physics, an Introduction to Theory and Experiment . Springer-Verlag.

This is a good general solid state physics text, with little experimentalsections describing how some of this stuff is actually measured. Itsbiggest flaw is the number of typographical mistakes in the exercises.

N. Ashcroft and N.D. Mermin. Solid State Physics.

The classic graduate text. Excellent, and as readable as any physics

book ever is. Too bad that it ends in the mid 1970’s....C. Kittel. Introduction to Solid State Physics.

Also a classic, and also fairly good. Like A&M, the b est parts werewritten 25 years ago, and some of the newer bits feel very tacked-on.

W. Harrison. Solid State Physics. Dover.

Very dense, written by a master of band structure calculations. Hasthe added virtue of being quite inexpensive!

P.M. Chaikin and M. Lubensky. Condensed Matter Physics.

More recent, and contains a very nice review of statistical mechanics.Selection of topics geared much more toward “soft” condensed matter.

Nanoelectronics and mesoscopic physics:

26



Y. Imry. Introduction to Mesoscopic Physics. Oxford University Press.

Very good introduction to many issues relevant to nanoscale physics.Occasionally so elegant as to be cryptic.

D.K. Ferry and S.M. Goodnick. Transport in Nanostructures. Cam-bridge University Press.

Also very good, and quite comprehensive.

27

Documents

Brief Review of Solid State Physics