Condensed Matter Physics

Crystal Structure

Basic crystal structure  Crystal: an infinite repeated pattern of identical groups of atoms, consisting of a basis and a lattice  Basis: a group of atoms which forms the crystal structure  Lattice: the three-dimensional set of points on which the basis is attached to form the crystal  Primitive cell: the cell of the smallest volume which can be translated to map out all the points on the lattice. The primitive cell is not unique for a given crystal. There is always exactly one lattice point per primitive cell

 Conventional cell: a more convenient unit which also translates to map out the entire lattice, often a cube  Wigner-Seitz primitive cell: this is a special form of the primitive cell constructed by connecting a given lattice point to all its neighbours, then drawing new lines as the mid-points of these lines

 Lattice point group: a collection of symmetry operations which map the lattice back onto itself, such as translation, reflection, and rotation

Miller indices Miller indices are an indexing system for defining planes and directions in a crystal. The method for defining them is as follows:

 Find the intercepts of the plane with the axes in terms of lattice constants  Take the reciprocal of each intercept, then reduce the ratio to the smallest possible integer ratio

 A particular plane is denoted  A set of planes equivalent by symmetry is denoted  The direction perpendicular to is denoted by

The reciprocal lattice If we have a repeating lattice we can represent the electron density by a periodic function, which we can write as a Fourier series:

Where the factors of ensure that has the required periodic relation . Points in the reciprocal lattice are then given by , for positive integer coefficients.

In three dimensions the lattice is defined by:

The Fourier transform of the original lattice series gives the series of the inverse lattice:

In three dimensions this becomes (for cell volume ):

The reciprocal lattice vector is given by:

Where are the primitive vectors of the crystal lattice and are the primitive vectors of the reciprocal lattice.

Diffraction conditions The total amplitude of a scattered wave from one point in the crystal in direction is proportional to:

Where is the electron density at a given point, and the exponential term is the phase factor difference between the incident wave with wavevector and the reflected wave with wavevector . Integrating over the entire crystal we get the scattering amplitude :

Substituting in the Fourier decomposition of we have:

The exponential will be negligible if is much different from , so is only really non-zero when (the Bragg scattering condition), meaning that we have:

The diffraction condition can now be found starting with:

Using the elastic scattering condition that we can write this as:

The distance between two parallel lattice planes normal to is . Thus if if is the angle between (the incident wavevector) and (the crystal lattice), and noting that is perpendicular to so the angle between and is , then we can write this as:

Using the fact that we have:

Likewise and . Substitute and we arrive at the Laue equations:

For diffraction to occur, must simultaneously satisfy all three of these equations. Each equation is satisfied for a conic region, so this corresponds to lying at the centre of a cone about each of the directions , meaning that must lie at the intersection of three cones at once.

Brillouin zones The Brillouin zone is just the Wigner-Seitz primitive cell in the reciprocal lattice. It is formed in the same way as in the real lattice: draw a vector from the origin to each neighbouring reciprocal lattice point, then at the midpoint of this line draw a second line perpendicular to each of these lines, and shade the central area bounded by perpendicular lines. This central area is the Brillouin zone. Only incident waves whose wavevector drawn from the origin terminates on a bounding surface of the Brillouin zone will be diffracted by the crystal.

To see that any such vector will satisfy the diffraction condition, observe that:

Note that a simple cubic is its own reciprocal lattice, while bcc and fcc lattices are the reciprocals of each other.

Structure factor When the scattering condition is met, the scattering amplitude for an array of N cells can be written:

We call the structure factor, which is the electron density multiplied by the phase, integrated over a single cell with directed from the center of the cell. We can recast this as an integral over each of the atoms in the cell if we write for the electron concentration at each of the atoms in the basis. Then we have:

We thus define the atomic form factor as:

This is a measure of the scattering power of the th atom in the unit cell. Using this definition, the structure factor of the basis then has the form:

Substituting in the lattice coordinates of atom :

We then have:

Theory of Phonons

Classical theory of phonons

Total kinetic energy (in one dimension) of a chain of atoms each displaced a distance from equilibrium is given by:

The potential energy is similarly given by:

Where is the restoring force constant.The Lagrangian for atom then is:

We use this Lagrangian with the Euler-Lagrange equation:

To arrive at the equation of motion:

Consider the solution (where is the wavenumber):

Using the identity , we find the dispersion relation:

The wavenumber is effectively the ‘distance’ in the reciprocal lattice, so we can plot these dispersion relations as a function of the frequency of phonons at various positions in the reciprocal lattice:

In the case where there are two atoms with masses and and spring constants and , the dispersion relation becomes:

Using solutions:

We arrive at the equations:

In matrix form:

Which has the solution:

The negative solution gives the acoustic phonon mode, while the positive solution gives the optical phonon mode.

In the full three-dimensional case, there will be degrees of freedom for a unit cell with different atoms, leading to three acoustic and optical branches.

Quantum theory of phonons While normal modes are wave-like phenomena in classical mechanics, they have particle-like properties in the wave-particle duality description of quantum mechanics. For the full quantum mechanical description the atomic displacements and atomic momenta become operators:

For a particle in a parabolic potential we use the Hamiltonian:

We now define quantum raising and lowering operators:

We use the standard canonical commutation relations:

The Hamiltonian then becomes:

The number operator gives the excitation level of normal mode .

The eigenstates of this operator are denoted:

Photon emission or absorption follows the rule:

Total energy is then:

Thermal statistics of phonons Each phonon has energy , yielding a total lattice energy of vibrational mode is:

The probability distribution of the number of phonons in level is:

This is the Bose-Einstein distribution, and allows us to calculate the average phonon number in equilibrium:

We thus have for mode :

Thus we see that the average number of phonons depends both on the temperature and on the phonon frequency . The average energy of the entire lattice at temperature is then given by:

It would be more convenient to write this as an integral:

If we use the simple dispersion relation for for constant velocity then this becomes:

The density of states is the number of phonon modes per unit frequency per unit length. It is a way of expressing the number of different values of that can be generated by various combinations of

etc, divided by . It is usually not an easy function to evaluate, but becomes simpler for small values of . This is the approximation used above.

We need it to convert the sum to an integral. Note that is the Debye frequency, the cutoff frequency representing the highest frequency that can propagate within the lattice (depends on atomic separation a). We can compute it by noting the integral over the density of states is just the volume density of states, and so:

We also note that each of the phonon branches (LA, TA, and TO) has its own set of frequencies and its own Debye frequency, so we need to multiply the answer by three. We can now use this integral result to calculate the thermal energy of the lattice per unit volume at temperature :

Neglecting the factor of :

Let , then we have:

By the definintion of , we are led to the definition of a natural temperature scale:

We can then write the specific heat capacity as:

At low temperatures we can use the approximation that the upper limit goes to infinity:

Hence the specific heat capacity becomes:

Phonon anharmonicity The potential energy Taylor series expansion around the equilibrium position takes the form:

For the harmonic oscillator case . Including higher order terms leads to anharmonicity. This means that photons will decay over time and can interact to create a third phonon.

The Origin of Band Gaps

Introduction to free electron theory The free electron model assumes that the valence electrons of atoms in the crystal lattice become conduction electrons that can move freely throughout the volume of the metal without interacting with either the ion cores or each other (Pauli exclusion principle).

The energy levels of free electrons in one dimension are given by the TISE:

To solve this we can consider both fixed boundary conditions where and periodic boundary conditions where . The solution for periodic boundary conditions is:

In one dimension, the Fermi energy is the energy of the uppermost filled electron state in the ground state configuration of an N electron system. These energies are given by:

In three dimensions with periodic boundary conditions, the solutions take the form of plane waves:

The periodic boundary conditions enforce the allowed values of k:

Substituting our solution back into the three-dimensional TISE yields:

As electrons are added to the system, points progressively further out from the centre of the three- dimensional momentum space diagram are filled:

The energy of the surface of this sphere when all electrons have been added (i.e. the valence electron energy) is called the Fermi energy:

Thus the volume of the region occupied by electrons is . Since there is one lattice point per

unit cell, each lattice point occupies the volume , which translates to lattice points per unit volume. The number of states is twice the number of occupied lattice points (to account for spin), thus yielding:

This allows us to solve for the Fermi wavelength:

We can then write the Fermi energy in terms of the electron density :

Thus the number of states is:

The number of states per unit volume per unit of energy is then found by:

Thus, the number of orbitals per unit energy at the Fermi energy is similar to the total number of conduction electrons divided by the Fermi energy.

Free electron theory beyond ground state The foregoing only applies to the ground state, which occurs at . Average occupation of an energy level at higher temperatures is given by the Fermi-Dirac distribution:

We define as the chemical potential. For , this simplifies to the Maxwell- Boltzmann distribution:

We now define as the probability that the quantum state with wavevector is occupied by an electron. The total number of electrons, since each state can hold two, is then:

Writing this as an integral it becomes:

We can substitute out using the Fermi-Dirac distribution and the relation:

Simplifying using the relation we get:

Drude model of metals Assumptions

 Conduction electrons only scatter from ionic cores of atoms, electrons do not interact  Collisions are instantaneous and result in a change in electron velocity  After a scattering event the momentum of electron is completely random in direction

If the average time between collisions is then the collision rate is . Current density in the metal due to electrions with average drift velocity is then given by:

Consider electrons in an electric field . In time interval there is probability of a collision .

The average electron momentum at time is then given by:

Where the last term is because the average momentum is zero if a collision occurs. Expanding out we get:

Neglecting the higher order term we can write this as a derivative:

In the steady state we have:

We can then find the average electron drift velocity:

We define as the electron mobility. We have the electron current density as:

So the conductivity . While this result is correct, this classical model grossly underestimates the typical electron speeds, since it assumes they follow the Maxwell-Boltzmann distribution.

The Drude model also predicts the Hall effect (production of a voltage difference across a conductor), though the magnitude and even the sign of the effect is not always predicted correctly. The effect is predicted as a simple result of the Lorentz force on e:

Taking the y-component:

Hence we find:

Sommerfeld model of metals This model incorporates a quantum mechanical treatment of the free electron gas. As before we have:

But now the velocities of the electrons is given by the Fermi distribution:

Hence we can write:

Using the quantum relation (which now yields the correct electron velocities) this can be written:

When there is an applied electric field , the equation of motion for an electron becomes:

This means that the electric field shifts the wavevector of each electron by the same amount.

Now substituting this into our expression for the electron current we have:

Shifting the integration variable:

The first term is equal to zero because of the symmetry of terms. Hence we are left with:

This is the same as the Drude result, as it should be. Thus we have shown that a full quantum treatment yields the same result for as the classical Drude model, while also giving a more accurate treatment of electron velocity. 21

Periodic potentials We can represent a one dimensional lattice as a series of delta functions, located at the point of each of the atoms. Thus we have the representation:

The fourier transform of this gives us the reciprocal lattice:

Since we also know that the reciprocal lattice itself can be described by a sum of delta functions, we determine that:

If we define the lattice translation vector we can write the Fourier transform as:

In two dimensions we need to take two delta functions:

We then have for the reciprocal lattice:

Three dimensions is similar:

The reciprocal lattice is then:

We can combine this delta function approach with a periodic potential function , such that we can write as a convolution of its value in one cell with a lattice of delta functions:

Using the Fourier transforms derived just above, and also the convolution theorem (convolution becomes multiplication in Fourier space) we can write this as:

Writing a generic Fourier transform we have:

Now substituting in our result for this becomes:

This gives us a potential which we can then substitute into the TISE:

By the lattice symmetry solutions must satisfy:

It turns out (see notes page 9) that the form of this function is:

This leads us to Bloch’s Theorem which states that every solution of the TISE with a periodic potential has an associated wavevector such that:

This result tells us that the phase of a wavefunction solution cannot be arbitrary, but is dependent upon the translation and the wavevector .

Nearly free electron theory Without a periodic potential caused by the lattice points, just considering a free electron in a box, we have the solutions as plane waves:

Now let’s add in the periodic potential that we found above:

Combine this with the kinetic Hamiltonian:

Now we apply perturbation theory:

We get the new eigenfunctions:

Let’s compute the numerator term of the fraction:

The integral is equal to zero unless by the Bragg condition, hence we have:

Substituting this back into the perturbation theory expansion we have:

Thus we have solutions that are a superposition of plane waves with wavevectors differing from by the reciprocal lattice vectors. When the denominator approaches zero and so perturbation theory breaks down:

Using the result from before this is:

This means that perturbation theory breaks down precisely at the Bragg scattering condition. This means that we need to find an alternative means for calculating the wavefunctions near . Let

us assume a variational solution for the perturbed state near of the form:

We can determine these constants and the energies by substituting into the TISE:

We take the bra with and with - see notes for full details. It turns out that we get:

Note that key result here is that there are two solutions. The difference between these solutions corresponds to the band gap. The same argument applies to values of near – .

Using these energies and the normalisation property, we can solve for the wavefunctions. It turns out (see notes for details) that we get:

Energy degeneracies occur when:

Which is Bragg’s law.

Band structure plots Band structure diagrams show the energy levels on the vertical axis against the reciprocal lattice wavevector on the horizontal axis. Because the wavevector has three components and we only have one horizontal axis dimension, we typically plot only a single dimension of variation in one segment of the graph – e.g. the segment will plot one plane in the reciprocal lattice, while the will plot another plane, etc. These diagrams are plotted using the dispersion relation for the lattice in question. For a free lattice this is simply:

In the case of a square 2D reciprocal lattice, the Brillouin zone is given by:

Bandgaps open up at the Brillouin zone boundaries:

We commonly use the reduced zone scheme by mapping all the higher values back to the first Brillouin zone.

In the reduced zone scheme, the band structure is represented in diagrams such as the following:

The numbers in parentheses represent the degeneracy of each energy band – that is the number of reciprocal lattice vectors that give rise to the same energy band. 28

Band filling In one dimension, we have primitive cells with lattice constant . The allowed values of within the first Brillouin zone only are:

Thus there are allowed values of within each primitive cell (i.e. the same as the number of primitive cells in the lattice as a whole). We can also see this by calculating directly:

The result is the same for a 3D lattice:

Since band gaps occur precisely at the boundaries between Brillouin zones, we consider how many electrons the first Brillouin zone has. Note that there are orbitals in each Brillouin cell because of spin up and spin down electrons.

If we have a single atom basis with 1 valence electron available, there will be orbitals filled and hence the first Brillouin cell (i.e. the lowest energy band) will only be half full. Electrons are thus available for conduction, and the resulting material is a conductor.

By contrast, if we have a two atom basis with 1 valence electron per atom then orbitals will be filled, and the first Brillouin cell (lowest energy band) will be completely full, leaving no electrons available for conduction. Hence the resulting material is an insulator.

Effective mass Bloch states are superpositions of plane waves:

They have group velocity:

For a free electron in 1D we have:

If an external force is applied to an electron in an energy band, the rate at which work done is:

From the expression above we can write this as:

Using the results above we can write in an alternative form:

We know from Newton that:

This suggests that we define the effective mass as:

For a free electron , and hence:

Away from band edges the effective mass will approach the inertial mass, however near Brillouin zone boundaries, where the bands ‘bend’, the effective mass can differ significantly from the inertial mass as a result of energy interactions with the lattice ionic cores.

Effective masses are usually quoted as fractions of the free electron mass, which can be greater or less than one, positive or negative. A negative effective mass means that in response to an applied force, the electron accelerates in the opposite direction to what it would classically. This applies to electrons at the top of a band, which behave like holes.

Generally we are interested in energies close to the extrema (near the middle of the band gap), where the dispersion relation can be approximated as a parabola in k:

Where and are values at the band extremum.

Holes If an electron is excited out of the valence band, there will no longer be an equal number of positive and negative wavenumbers , so now:

We call the net obtained by the valence band , and it is always equal and opposite to the wavenumber of the excited electron:

The lower down in the band that the electron came from, the higher will be the excitation energy of the system afterward:

As such, the group velocity of the hole is the same as the group velocity of the missing electron:

Since the wavenumber is equal and opposite that of the excited electron we have also:

Since and it follows that:

Semiconductor Physics

Semiconductor basics The most common semiconductors have an FCC lattice with either a one or two atom basis. Pure froms of these semiconductors are insulators at 0K, as their valence bands are full at this temperature.

Often the valence band maximum occurs for (at on the horizontal axis). If the conduction band minimum also occurs at , we call the material a direct band semiconductor. Silicon has an indirect bandgap, meaning that excitation of an electron requires a change in k. Such materials are anisotropic, and have an effective mass tensor with different components in different directions.

Calculating electric fields

Given electron and hole drift velocities and for electron or hole mobilities , we can write the total current as:

Note that electron drift velocity is defined as:

Note that in these sorts of calculatations we must be careful to use the correct form of the effective mass:

Intrinsic semiconductor statistics At 0K, the valence band of an instrinsic semiconductor is full, so there are no electrons in the conduction band and it behaves as an insulator. The Fermi level is defined as the energy level at which the probability of being occupied is equal to one half, and is temperature dependent:

For an instrinsic semiconductor sits exactly in the middle of the band gap, even though there are no states there to occupy. This is possible because the average energy of an excitation over the band gap will be halfway through the band gap.

For silicon, the band gap energy at 300K is 1.12eV, compared to . This is very convenient as when we can approximate the Fermi distribution with the Boltzmann distribution:

If we use our result from the previous section that the energy in the conduction band is:

Also from a previous result we have the density of states in 3D given by:

Substituting in our energy we have:

We can use this, along with our Boltzmann distribution, to calculate the concentration of electrons in the conduction band:

Since a hole is simply the absence of an electron, the distribution function for holes has the form:

It follows that the distribution of holes in the valence band is likewise given by:

If we multiply these two numbers together we find that the result is actually independent of :

This means that the product np at a given temperature is a constant. We can get some intuition for why this is the case by realising that holes and electrons can annihilate each other to release a photon, a process which at a given set temperature will occur at a constant rate. 35

In the case of intrinsic semiconductors the only way to get an electron is to leave a hole behind, and so we must have:

We thus arrive at a series of related equalities:

Where we have defined the effective density of states in the conduction and valence bands as:

Extrinsic semiconductors An extrinsic semiconductor is one that is produced by adding impurities, or dopants, to the base substance (often silicon). There are two main types of dopants: n and p type. N-type dopants have five valence electrons; one of each is easily excited into the conduction bands at any measurable (positive) temperature. These donors thus introduce an energy band level close to the conduction band edge, allowing conduction of charge. P-type dopants have three valence electrons, with one bonding orbital to its silicon neighbours missing, forming a hole. These acceptors thus introduce an energy level close to the valence band edge. These energy levels are placed roughly:

Even very small concentrations of dopants can have a dramatic effect on the number of conduction band electrons.

In a uniformly doped semiconductor the electric field in equilibrium is related to the charge density and semiconductor dielectric constant by Poisson’s equation:

Inside the semiconductor the charge density is equal to:

Where is the concentration of donor ions and is the concentration of acceptor ions. These donor and acceptor ions are located in the energy band gap, and are responsible for providing the electrons or holes that give rise to the new energy band. The energy is just halfway between and .

Note that a doped semiconductor (unlike an intrinsic semiconductor) does NOT have to have equal numbers of majority and minority charge carriers. The charge carriers from dopant atoms are ‘extra electrons’ or ‘electron deficiencies’ which result from ionisation and so do not leave behind a charge carrier of the opposite sign (ions are too big to carry charge themselves).

If the field is zero everywhere then , and hence:

This is called the charge neutrality relationship. Note that as long as for n-type or for p-type, then we can use the Boltzmann distribution approximation.

In the case of an n-type semiconductor we will have and so:

Likewise in the case of a p-type semiconductor we will have and so:

If both n and p type impurities are present then we substitute into the charge neutrality equation to find:

From this we obtain the solutions:

This yields:

In all cases therefore the energy gap between the intrinsic Fermi level and the actual (i.e. with dopant) Fermi level is a linear function of temperature:

There are three main temperature zone behaviours of a semiconductor:

Semiconductor junctions When two dissimilar semiconductors are brought together to form a junction, the equilibrium condition is determined by a balance between charge carriers diffusing from regions of higher carrier concentration to regions of lower carrier concentration, and drift of charge carriers as a result of moving along electric field lines.

Concentration-dependent diffusion is governed by Hick’s law. This states that the number of atoms crossing a unit area in unit time is equal to the divergence of atom concentration times the diffusion coefficient:

For a concentration gradient of electrons in the x-direction this becomes:

Similarly for holes:

When an electric field is present, the total current density at any point is a sum of drift and diffusion components:

Electrons tend to diffusive from n-type into p-type materials, while holes tend to diffuse from p-type to n-type materials. In either case this results in a charge separation, giving rise to an intrinsic electric field. Interestingly this build-in electric field tends to push electrons and holes back in the direction they came from, thereby tending to cancel out the effect of diffusion. At equilibrium they exactly balance resulting in a zero net current density:

We then can apply the equation:

In neutral regions there is no net charge density , and hence we have:

In the p-type region there are hardly any electrons so this becomes:

Using the potential equation derived previously we can then write:

This is directly related to the electrostatic potential in the neutral p-type region with respect to the mid- gap energy :

By exactly the same argument we have the potential for the n-side:

The difference between p-side and n-side potentials is the built-in potential of the junction:

It takes typical values of around 1 V.

In the depletion region electrons and holes annihilate each other such that . Using the equation from above we then have:

On the p-side of the semiconductor this becomes:

This produces an electric field:

Similarly on the n-side:

Which produces the electric field (where is the field at ):

The total width of the depletion region is then:

Overall charge neutrality requires that the total charge in each region is equal:

By which we can write:

Since net charge only exists in the depletion region, we can find the built in potential simply by integrating over the electric field in the depletion region:

Substitute in the charge neutrality relationship and we get for the built-in potential:

Equilibrium IV characteristics of junctions In equilibrium, drift and diffusion currents cancel out, leaving a net current of zero. When a forward bias is added, the barrier to diffusion is reduced (as the battery replaces the diffusing charge with the same type of charge, facilitating continued flow), meaning that diffusion of both holes and electrons is increased. The diffusion current now dominates, resulting in an overall net current in the forward (diffusion) direction, with the charges supplied by the battery. As only a fixed voltage difference is required in order to offset the doping-induced concentration differences in n and p charges, and part of this voltage difference is supplied by the battery, only a smaller depletion region is required to generate the remaining voltage difference. A reverse bias does the exact opposite, increasing the barrier to diffusion and thus reducing diffusion currents of holes and electrons (as the battery now replaces the diffusing charge with the opposite charges, inhibiting continued flow). The drift current now slightly dominates (though still limited by availability of minority carriers), thereby resulting in a small net reverse current.

Remember that charge flows UP the potential since electrons are negatively charged.

Let and be the equilibrium electron densities on the n and p sides respectively. Let and be the equilibrium hole densities on the p and n sides respectively. We can substitute these definitions into the equation previously derived for to yield:

Now using also the definition:

We can write this as:

From which we find the pair of expressions:

Thus the charge densities on either side of the depletion region are related by the intrinsic voltage . This expression still applies when an external bias field is applied, just with the modification:

Where is positive for a forward bias and negative for a reverse bias. However the injected minority charge density is much too low to have much effect on either majority charge density, so even with a bias potential we have and . We thus can combine the above to sets of equations:

These equations tell us that the change in minority charge carrier density is exponentially dependent upon the applied voltage .

Diode IV relation for pn junctions To model time-dependent processes occurring at pn junctions we need to use the continuity equations:

Here are the electron and hole generation rates from the voltage source, are the recombination rates, and are the given by:

If we consider the steady state where , outside the depletion region so , and treat the charge generation rate as negligible, then these equations simplify to:

These equations have the solutions:

With the diffusion lengths and .

Substituting these results into the current equations on the boundary of the depletion region we have:

The total current on the boundary is then:

As shown in the figure below, this equation describes a situation in which driving a voltage forward through the junction results in a significant current increasing in V, while driving a voltage back through the junction only results in a small, constant current in the opposite direction. This asymmetry arises because a reverse bias increases the potential difference between p and n sides of the junction, thus reducing current (since the charges are moving against the potential difference under the influence of diffusion). Conversely, a forward biases reduces the potential difference, ‘working with’ the junction by adding positive charge just where it is needed, and hence facilitating current flow. Both processes are exponential, hence the drastically different forward and backward behaviour.

Metal-semiconductor contact

Let be the metal work function (ionisation energy of metal), and be the semiconductor’s electron affinity (the energy difference between the conduction band edge and the vacuum). At the junction between metal and semiconductor, their Fermi levels must align (in equilibrium as otherwise charge will flow). This leads to ‘band bending’ at the interface, and a built-in potential difference of:

This built-in potential is the result of the discontinuity in accumulated charge at the boundary, as electrons or holes equalise the Fermi energies by flowing from the material with higher energy to the one with lower energy. The triangle potential at the boundary arises because electrostatic potential and chemical potentials needing to balance out in equilibrium, but chemical potential goes to zero a large distance away from the boundary. Thus to balance the constant electrostatic potential there must be a ‘bump’ at the boundary where chemical potential difference is nonzero.

The ‘ski ramp’ is caused by electrons ‘bunching up’ as they try to move from the higher potential to lower potential sides of the junction (holes move in the opposite direction). This results in a surface charge and hence a discontinuity in the potential.

Because the metal is a perfect conductor of charge carriers, it serves as a source/sink of charges from the attached semiconductor. As such, the behaviour of the semiconductor in a metal-n-type semiconductor junction (for example) is just the same as if it were part of a p-n semiconductor junction.

In the case of an ideal metal oxide semiconductor (MOS), we chose the metal specifically so that its Fermi level aligns with that of the doped semiconductor:

More typically, however, there is a small offset between the work functions of the metal and the semiconductor. This results in bending of the conduction and valence energy levels as they are (in the case pictured below) pulled upwards by the higher potential of the metal. Note that the Fermi level of the semiconductor is not bent in the same way because it must always be the same level everywhere in equilibrium.

We can then apply external potential biases of varying magnitudes and signs, with three main resulting types of behaviour.

For a p-type semiconductor, a positive potential represents an accumulation of positive charge in the conductor (and hence is a negative potential for the electron as plotted above), which thereby attracts electrons from the nearby portions of the semiconductor (inversion). When instead a negative potential is present, holes are instead gathered (accumulation). These are reversed in the case of an n-type semiconductor (see diagrams above).

Metal oxide semiconductor

For the p-type MOSFET, when (accumulation), the source to drain path consists of two back to back diodes, as current tends to flow from p to n type semiconductors but not the reverse (see above). One of these diodes is always reverse biased regardless of the drain voltage polarity. Thus n-p-n transistors are off when is low and on when is high. The exact reverse is true for p-n-p transistors (see diagram below).

When is increased so that , an inversion layer forms in the channel beneath the gate, in the manner illustrated above (inversion diagram). This electrically connects the source and the drain, allowing a current to flow. The depletion region nearest to an n-region is positively charged, while that nearest to a p-region is negatively charged. The inversion layer inverts this, so in this case will be negatively charged, thus connecting the two n-regions together.

For a while as is increased, there is a linear relationship between and . However eventually the saturation point is reached and current does not increase with additional increases in voltage. This occurs because with fixed , a higher increases the width of the depletion layer around the drain region (see diagram above), thereby reducing the concentration of charge carriers in the channel and limiting the ability of current to flow. Note that the charge carriers already set in motion at the left-hand terminal will continue to flow to the right, thus current won’t decrease, but it won’t be able to increase any further.

Pinch-off point is location at which minority carrier concentration in channel reduced to zero, due to reduced voltage drop across oxide arising from increased drain voltage. With increasing the pinch-off point moves further away from the drain end. We can increase to compensate and achieve larger and larger saturation voltage.

Heterostructures A heterostructure is a semiconductor composed of more than one material. They are very useful for building devices such as LEDs, lasers, and quantum wells. A common application is quantum confinement, which can be achieved by producing a potential well in a material by sandwiching layers with differently sized band gaps. Growth of heterostructures is a difficult process, as it requires adding one layer of atoms at a time to achieve atomic-level precision in the transition from one material to the next.

Note that this is a potential well because the conduction electrons in the central region are at a lower energy than in the surrounding regions. Also the holes in the valence band have lower energy (their energy is opposite to that of electrons) in the hole than in surrounding regions.

The other forms of heterostructures are shown in the diagrams below.

Type I: Straddling Alignment, Type II: Staggered Alignment, Type III: Broken-gap Alignment

Anderson's rule is a technique used for the construction of energy band diagrams of the heterojunction between two semiconductor materials. It states that we should first align the vacuum levels of our two materials, and then offset the conduction band edges by a value , where is the ‘work function’ of the material in question.

The steps for drawing band diagrams are specified below. Note that in diagram ii represents an external potential bias applied over the junction – if this is zero, the Fermi levels will be aligned on both sides.

Doping in regions where electrons/holes are desired leaves charged donors/acceptors. These ionised impurities act as scattering centres which can reduce device speed. To overcome this, experimentalists employ remote modulation doping, in which carriers placed in one region migrate to another undoped region which is the active region of the device.

Band-bending produces a triangular potential well at the interface, where the energy levels in z- direction can be quantised while being unconstrained in x-y plane. This is a 2D electron gas.

Low Dimensional Structures

Quantum wells A finite square well has TISE given by:

This has stationary state solutions inside the well, and exponential decay outside.

The penetration depth is the distance over which the amplitude of the wavefunction decreases by

1/e. For a square well of height it is given by:

To find the energy levels for a finite square well we can use an iterative procedure:

1) First calculate an approximate using an infinite square well of the same width 2) Then calculate the value of for this 3) Now use this to calculate a new value for using the formula:

A triangular potential well has TISE given by:

We can solve this making the judicious change of variables:

Which changes the equation to the form:

With boundary conditions and , leads to solutions of the form .

This has energy levels given by:

Where is the th zero of .

A three dimensional system with confinement in the -direction but free electron motion on the x-y plane has a TISE:

This can be rewritten as a one dimensional SE in z:

With . This has full solutions of the form:

Transfer matrices – potential barrier A transfer matrix is a useful formalism for finding the transmission coefficients for a series of potential barriers or wells by simply multiplying together matrices for individual barriers.

The transfer matrix works ‘backwards’, giving the transmission coefficients moving from region 1 to region 2 (to the right).

Note that we find the form of the transfer matrix by using continuity of the wavefunctions and their derivatives, which yields the conditions:

Consider now the case of a potential barrier:

Propagation over a region can be represented by the matrix . Note that in simplifying this we use the fact that .

When we have a potential change that is translated away from the origin, we need to use the following adjustment:

Now we can put all these components together to solve for the case of two equally spaced edges:

Having found the matrix elements in terms of the wavenumber and the barrier separation , we can now compute the transmission amplitude and reflection amplitude as follows:

The actual reflection and transmission coefficients, which satisfy , are given by:

Thus we find at last:

Having solved for and we may wish to substitute out the values of in terms of , as obtained from the appropriate solutions of the TISE. The solution will depend upon the relative values of and .

If the plateaus on either side of the barrier are different in height, it is necessary to normalise the initial propagating states in order that they carry equal currents:

Transfer matrices – potential step We can re-express the results for the potential step in terms of the (in general complex) values and :

In this case we have amplitudes:

And also the coefficients:

And the transfer matrix then becomes:

Note that if we reverse the direction the transfer matrix becomes (working out omitted):

This means that reflection and transmission amplitudes differ only be a phase, and so the reflection and transmission coefficients (the observables) are the same for going across a barrier in either direction.

Current and conductance Current will flow across a barrier if we apply a potential difference, thereby producing a difference in Fermi levels equal to:

In this setup the current flowing from left to right is equal to:

Where is the electron density at wavenumber , is electron charge, is average charge velocity, and the initial fact of 2 is to account for the two possible electron spins. Since is equal to the Fermi distribution times the transmission coefficient, we can write this as:

We can now use the relation to rewrite the integral in terms of energies:

We can also rewrite velocity using the relation and hence . Thus we have:

Noting that the transition amplitude is always equal on both sides of a given barrier, the current due to electrons impinging from the right is likewise:

Noting that electrons on the right with energies lower than will not be able to contribute to the current, since there are no sufficiently low energy states available for them to move to, the total current is given by:

There is in general no exact solution to this expression, however we can consider the solution in a number of approximations.

1) For a large enough bias, all the states on the RHS are below and hence do not contribute to the current. In this case we can drop them from the expression, leaving:

2) At , becomes a step function at the Fermi level, so the current becomes:

3) For small bias we can expand the different in Fermi functions to lowest order in a Taylor series:

4) If we combine 2 and 3 to consider the situation for a small bias at low temperature we can use

the approximation – and hence:

Resonant tunnelling Instead of a potential well with infinitely wide barriers and a definite bound state, we consider a well surrounded by finite width barriers, constituting a quasi-bound state.

Uncertainties of the energy of the bound state with lifetime are given by the uncertainty relation:

Using the same method as before, we have the transfer matrix:

The transmission amplitude is then:

And the coefficient becomes:

If and are small we can use the Taylor series approximation:

Whence we find:

We are interested in where the maximum occurs, which obviously happens with (note that this corresponds to a particular k and hence to a particular energy gap):

If the barriers are identical then and so . This perfect transmission is the result of resonance across the two barriers. If we now consider small perturbations away from , we note that . Thus we have:

We note that this is a Lorentzian peak which has a FWHM at . The energy width is given by:

Using and the usual we write this as:

From this we can compute the Breit-Wigner formula:

Here we see that resonance is sharply peaked about the regions where the system exhibits resonance. On the right is a logarithmic version of the left plot. The dotted line shows the situation without resonance.

The current flowing through such a device depends on the size of the potential difference.

Note that there must be positive density of states on the left side barrier in order for tunnelling to occur. This is why we get no resonance effect in a and d. Maximum resonance effect occurs in c where the bound state is at the bottom of the left hand side bands, where density of states is highest (in 1D).

Superlattices and minibands Superlattices involve an infinite number of barriers separated by a distance :

If we define:

Then transfer matrix for such a lattice is given by:

Since this is a periodic structure Bloch’s Theorem applies, which recall states for a periodic structure:

We note that now the wavenumber refers to phase changes from one cell to the next, while describes phase changes within each cell. Hence we can write:

We have thus shown that is an eigenvalue of with eigenvector . We find the matrix as:

Thus our eigenvalue problem becomes:

Solving for the eigenvalue yields:

Note that we have the property:

Combining this with the fact that we can write:

The sum of all these eigenvalues cancels all the terms on the square root:

The figure below shows an example of the solution of this equation for -function barriers with particular values of and substituted in.

We note that the amplitude oscillates and decays but always remains greater than 1 outside the shaded allowed bands, since only in these regions are eigenvalues consistent with . This means that only inside the shaded regions where can solutions propagate, with regions in between corresponding to band gaps. This solution leads to the pattern of energy bands as shown below in the extended (left) and reduced (right) schemes. Note the band gaps at the zone boundaries.

Quantum Point Contact A quantum point contact is a device which channels a 2D electron flow through a 1D constricted point in the center. In the leads (away from the constriction) the electrons are confined in the vertical direction but not in the plane of the 2DEG. In the constriction the electrons are confined to quasi-1D transport. So constriction acts as a barrier, transmitting electrons that have the right characteristics but rejecting 70 others. The constriction is controlled via the gate bias. We can think of the constriction as a scattering centre or barrier.

For electrons incident from the left we have the possibility of transmission or reflection at the constriction. To be sufficiently general we allow that the leads may have different geometries so that the confinement and DOS can be different between the input lead and the output lead. The number of propagating states in each is denoted by and . Each lead has several subbands arising from the transverse states. The transverse potential is taken to be constant along the entire length of the lead so that the subbands do not change along the length of the lead.

The wave function within a lead with current propagating in the z-direction is given by:

The sum is over all transverse states in the -direction (vertical confinement being in the -direction), each with a wavefunction and energy . The term represents the particle flux per each state. The total energy of each state is:

States propagate only if (i.e. they have positive rightward kinetic energy). If we inject a wave from the left purely in mode , the scattering center mixes modes and so the outgoing wave has contributions from all outgoing modes on both sides:

The total conductance for this single incident mode is given by:

The total conductance is found by summing over all input and output modes:

Note that since the sum is over intensities rather than amplitudes it is assumed that there is no phase coherence between electrons injected in different modes.

For surface gates that are short in length, the potential looks like a saddle and the potential varies smoothly so that the adiabatic approximation can be used. Also if the scattering between modes can be neglected, then amplitudes can be calculated independently for each mode and we thus ignore interactions between them.

Transverse energy of each subband varies with distance through the constriction as shown in the ﬁgure. For the case shown only electrons in subband with lie below the Fermi level throughout so only electrons in that subband can propagate through the constriction without a high

72 probability of being reflected. Electrons in higher subbands have high probability of being reflected. This occurs because we have for high :

Hence high means imaginary which therefore do not propagate across the barrier. Thus, when we consider the double sum over modes, only those modes that lie below the Fermi level contribute, and each of these coefficients is close to unity. Hence we have the approximate relation:

Where refers to the number of modes below the Fermi level.

Changing the gate voltage will change the constriction and can allow higher order modes to be transmitted. As applied bias is varied in general one extra subband at a time can be let through. This gives rise to the characteristic quantisation of conductance.

Exam Notes Crystal types

Basic math

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Phonon formulae

 To get DOS use

  (Debye approx, used for acoustic modes)  (Einstein approx, used for optical modes) 

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 Equations of motion for unequal mass atoms in lattice:

Band structure formulae

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 Measure in units of ,

computing only the end-point energies and then filling in a curve between these  Bloch theorem: (plane wave times periodic function)  Average kinetic energy of conduction electrons:

Semiconductor notes

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 two dimensional electron gas

 Always  For an intrinsic semiconductor,  Quantum point contact is analogous to diffraction limit on lens resolution

Low dimensional structures

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  Canonical momentum:  Landau gauge:

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There are always 2N orbitals available in each Brillouin cell, for N the number of unit cells in the lattice . These always fill from lowest energy up, so the highest energy orbitals are the last to fill. These will therefore be the orbitals right at the band gap between Brillouin zones. For a single-atom basis lattice with one free electron per atom, only N of these orbitals will be filled, meaning the Fermi level is halfway in the band and so the material is a conductor. If instead there are two free electrons per atom, or two atoms per lattice point, then 2N orbitals will be filled, and the material will be an insulator.