QMMS Lecture #2 (Venables/Heggie)

Notes for QMMS Lecture 2 (Venables/Heggie)

Lecture notes by John A. Venables. Latest version 23rd March 2008, ex 25 Aug 2005.

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2. Particles in boxes: free and nearly free electrons

2.1 Non-interacting states in the 1D square well

The starting point here is the construction of a square well, where the potential is zero inside (-a < x < a), and infinite outside (|x| > a). Then the states inside are free-electron like, and the wavefunction must go to zero at the edge of the well, as illustrated in many places (e.g. Gasiorowicz, 2nd edition fig. 4.2 page 60, 3rd edition fig. 3.3 page 50; Liboff, 3rd edition also figure 4.2 page 93).

For students learning this material in my Quantum Physics course, there has often been confusion between the above formulae, and those for a well which goes from 0 < x < a, where the solution always has the sine form, and the normalisation constant is (2/a)1/2. Factors of 4 easily creep into the energy expression for the same reason.

The sin-cos form resulting from a symmetrically placed well is more instructive, because it emphasises the difference between solutions with even parity (cosine-like) and those with odd parity (sine-like). Because of this, the physics is much more easily extended to unbound and bound states in finite wells (e.g. Gasiorowicz, 2nd edition: fig. 5.2 page 78 and fig. 5.10 page 90; 3rd edition: fig 4.2, page 69 and fig 4.7, page 77; or Liboff, 3rd edition fig. 7.27 page 244 and fig. 8.1, page 290). Chemists typically don't need convincing on this point, because they meet states which are labelled g (gerade, even parity) and u (ungerade, odd parity) at an early stage of their undergraduate careers.

In a finite well, the wavefunctions are not zero in the classically forbidden regions, but they decay exponentially as

Thus the less strongly bound states have extensive wavefunctions; there are many aspects of solids which can be understood qualitatively on this basis alone. For example, when many wells are put close together, valence electrons interact strongly and form bands, whereas core electrons largely retain their atomic-like character. The subtle problems in electronic structure arise when different atomic states give rise to overlapping bands, as in the sp3 bands in diamond structure semiconductors, or the overlap of broad s and narrow d bands in the transition metals.

2.2 Free electron densities of states in 1, 2 and 3D

The electron states inside a potential 'box' of the type discussed in section 2.1 are defined by their wave vector, k. Only particular values of k are allowed, such that the wavefunctions match the boundary conditions at the edges of the box. For an infinitely deep, square well potential of width L = 2a, the allowed values are given simply by L = nl, with the electron wavelength l = 2p/k.

Thus the allowed k-values are equally spaced, with k = 2pn/L, and so each state occupies a volume element of k-space, dk = (2p/L). This result generalises easily to 2D and 3D, provided we use the vector k, or its components kx, ky, kz, as is most easily seen by using periodic boundary conditions exp(ik.r) = 1, with r = iLx + jLy + kLz. The volume element of k-space in 3D is now dk = (8p3/LxLy Lz) = (2p)3/V, where V is the volume of the box. The corresponding formula for 2D is clearly (2p)2/A, with A the area of the 2D box. Sometimes these formulae are written as (2p)d/Vd, where the d-dimensional volume Vd = L, A or V for d = 1, 2 or 3.

This k-space volume element contains two electrons, when spin is taken in account, but beware that different authors introduce this point at different stages of the argument. Note also, in passing, that we have L = nl, not nl/2, which we might have deduced from the standing wave solutions sin() and cos(); these trigonometric forms just differ by a phase factor when periodic boundary conditions are used. Moreover, these formula are used to describe the density of states (DOS) per unit energy, called D(E) by Sutton or g(E) by Liboff, where for free electrons, E = (hk)2/2m, so that the allowed values of nx, ny, nz are all positive. This gives much scope for mistakes of factors of 2, 4, 8, so considerable numerical care needs to be taken at each stage of the argument.

Differentiating E = (hk)2/2m gives dE = h2kdk/m in one dimension (1D). The number of states this corresponds to, excluding spin but including negative values of k, is dN = 2dk/(2p/L). So the number of states per unit energy interval in 1D is D(E) = dN/dE = (L/p).(m/h 2k) = 0.5(E1E)-1/2. In the last equality, the form given by Liboff, E1 is the ground state energy of an electron in a box of width L, i.e. E1 = h2/(8mL2). But the most important result is that D(E) is proportional to E-1/2, and so peaks at low energies (Sutton figure 7.2 page 138, or Liboff section 8.8, page 349-50 in the third edition).

In the case of 2D and 3D, we need to include the degeneracy caused by different directions of k having the same energy, and to apply the derivatives and signs consistently. For example, Liboff's derivations use n, which is always positive, but only a quadrant in 2D (factor 1/4) or an octant in 3D (factor 1/8) is used in the summation; Sutton's derivation using k has no such restriction. The results are simply expressed in terms of E1 in the following Table. In 2D D(E) has no energy dependence, and in 3D D(E) is proportional to E1/2.

Table 2.1 Density of states in 1, 2 and 3 dimensions
with E1 = h2/(8mL2), after Sutton chapter 7 and Liboff section 8.8.
Larger figure
Larger figure
Larger figure
 1D: D(E) = 0.5(E1E)-1/2
 2D: D(E) = 0.25p/E1
 3D: D(E) = 0.25p E1-3/2E1/2

These differences in dimensionality are important for discussions of electron states in low dimensional structures, which form a large class of solid state devices. Some of the subtleties can be explored, at this or a later stage, via problem 2.4.3.

2.3 Introducing the lattice: Bloch's theorem and energy gaps

Once one introduces the lattice, which has a particular period in one or more dimensions, the electron density, being real, has to be periodic with the same period. However, the wavefunction, which in general is complex, can have an extra phase factor. This is the main result of Bloch's theorem, which states that the wavefunction for state k, namely yk is given by <r + T|yk > = exp(ik.T) <r|yk>. Here T is a translation vector of the lattice, and so can be expressed in terms of the real lattice coordinates [uvw]. As we will see in the next lecture, this theorem is closely related to Fourier analysis of the lattice potential and various diffraction ideas. You can get into the above equation via Sutton's chapter 4, page 74-78, once you are clear about the notation, but we will come at these topics more gently, taking into account individual backgrounds.

When the lattice potential has non-zero Fourier components, energy gaps open up in the electron energy spectrum at particular k-values, and may find their way into gaps in the density of states D(E). Whether or not there are gaps at the Fermi energy forms the basis of the elementary classification of solids into metals, semiconductors and insulators. Metals have no gap at EF, insulators a large gap, and most interestingly semiconductors have an intermediate size gap. It is this feature which allows semiconductors to form the basis of devices. By manipulating the Fermi level via temperature, light or defects, the electronic structures of semiconductors can be externally controlled, in ways which are not in general possible for either metals or insulators.

2.4 Problems relating to this topic

2.4.1: Bound states in single and double well potentials

The problems discussed in section 2.1 are probably familiar to all physics-trained graduate students, but not necessarily to chemists. They can be attacked directly as wave-matching problems, using the references given or more amusingly followed as computational demonstrations as in the CUPS Quantum Mechanics Simulations programs (Hiller et al., Wiley, 1995) or other programs on the web, accessed though my Web-based education pages. Some of these problems will be done in the computation sessions as the term proceeds.

2.4.2: Energies and states of the infinite potential well

A problem sheet will be handed out concerning matrix solutions of the infinite potential well using 1-dimensional states described by polynomials. This problem allows one to explore ideas of normalisation and othogonality, and the evaluation of integrals invoking symmetry properties. The result that the energy eigenvalues are relatively insensitive to the choice of wavefunction is an important element in both perturbation theory and the variational principle.

2.4.3: Density of states in low dimensional solids

This problem can be used to make the discussion of section 2.2 concrete. Calculate and draw the energy spectrum (density of states) for free electrons in:
a) a rectangular box with sides (a,L,L) with a = 1nm << L = 1 micron. This is a first approximation to a quantum well.
b) a rectangular box with sides (a,a,L) with a = 1nm << L = 1 micron; a quantum wire.
c) a rectangular box with sides (a,a,a) with a = 1nm; a quantum dot.

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