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Electron bands

An isolated atom has discrete electron energies. When many atoms form a crystal, overlap and the Pauli exclusion principle split corresponding atomic states into enormous numbers of closely spaced levels. In a macroscopic solid, these groups are treated as nearly continuous energy bands, separated in some materials by ranges with no allowed bulk states.

A band diagram is not a path traced by one electron. It reports allowed one-electron energies as functions of crystal wave vector, along with assumptions about the lattice and interactions.

Periodicity and Bloch states

In an ideal crystal, the potential repeats under a lattice translation \(\mathbf R\):

\[V(\mathbf r+\mathbf R)=V(\mathbf r)\]

Bloch's theorem says a one-electron eigenstate can be written

\[\psi_{n\mathbf k}(\mathbf r)=e^{i\mathbf k\cdot\mathbf r}u_{n\mathbf k}(\mathbf r)\]

where \(u_{n\mathbf k}(\mathbf r)\) has the lattice periodicity. The index \(n\) labels a band, and \(\mathbf k\) is crystal wave vector.

Crystal momentum \(\hbar\mathbf k\) is not always identical to an electron's ordinary mechanical momentum. It is a quantity associated with translation symmetry and is conserved modulo a reciprocal-lattice vector in ideal crystal processes.

Brillouin zones

States whose wave vectors differ by a reciprocal-lattice vector describe equivalent crystal periodicity. It is sufficient to label states within one primitive region of reciprocal space, commonly the first Brillouin zone. For a one-dimensional lattice with spacing \(a\), one convenient choice is

\[-\frac{\pi}{a}\leq k<\frac{\pi}{a}\]

Band energies are periodic in reciprocal space:

\[E_n(\mathbf k+\mathbf G)=E_n(\mathbf k)\]

where \(\mathbf G\) is any reciprocal-lattice vector.

How gaps appear

For a free electron,

\[E=\frac{\hbar^2k^2}{2m}\]

Periodic lattice scattering couples wave components separated by reciprocal lattice vectors. Near Brillouin-zone boundaries, Bragg reflection mixes otherwise degenerate waves and can open an energy gap. Whether the Fermi level lies within a band or a gap strongly affects electrical behavior.

The simple picture assumes an ideal infinite crystal. Real disorder, surfaces, finite size, phonons, and electron interactions broaden or modify the states.

A one-dimensional tight-binding band

A minimal model with one orbital per site and nearest-neighbor hopping is

\[E(k)=E_0-2t\cos(ka)\]

Here \(E_0\) is the on-site energy and \(t\) is a hopping-energy parameter. Because cosine ranges from negative one to positive one,

\[E_{\min}=E_0-2|t|,\qquad E_{\max}=E_0+2|t|,\qquad W=4|t|\]

The bandwidth \(W\) doubles if the hopping magnitude doubles. The sign of \(t\) changes which wave vector contains the minimum, but not the width.

import numpy as np

phase = np.linspace(-np.pi, np.pi, 1001)
on_site, hopping = 0.0, 1.0  # eV.
band = on_site - 2 * hopping * np.cos(phase)
bandwidth = np.ptp(band)
assert np.isclose(bandwidth, 4 * abs(hopping))
print("Bandwidth (eV):", bandwidth)

This model illustrates periodicity and dispersion. It is not a prediction for a named material without justified orbitals and fitted or calculated parameters.

Filling bands

Electrons are fermions, so each one-electron state can accept limited occupancy according to spin and the Pauli principle. At thermal equilibrium, the mean occupation of a state with energy \(E\) is given by the Fermi–Dirac function

\[f(E)=\frac{1}{e^{(E-\mu)/(k_{\mathrm B}T)}+1}\]

The chemical potential is \(\mu\). At zero temperature, states below the Fermi energy are filled and those above it are empty in the independent-electron picture. At nonzero temperature, occupation changes smoothly over an energy range of order \(k_{\mathrm B}T\).

An energy band and its occupation are separate data. A plotted band can exist without being occupied, and temperature can alter occupation without greatly changing the band energies in a simple fixed-band model.

Metals, insulators, and semiconductors

The basic classification is:

A completely filled band does not normally carry net current in equilibrium. For every occupied state with one group velocity, symmetry supplies an occupied state with the opposite velocity. Current requires an asymmetric occupation or partially available states.

This band classification is more informative than saying conductors merely have “loose electrons.” Scattering, carrier density, and band curvature are also needed to predict conductivity.

Read Figure 1 from the Fermi-level line outward. In the metal, that line cuts an allowed band: occupied blue states have nearby empty gray states in the same band. In the other two panels, the line lies in a white gap with no allowed bulk states. The gray conduction band still exists even though it is empty at zero temperature; gray indicates unoccupied states, whereas white indicates the energy gap.

Band gaps

The vertical gap arrows in Figure 1 measure the separation between the valence-band top and conduction-band bottom. The figure summarizes energies and occupations only; it contains no wave-vector axis, so it cannot tell us whether either gap is direct or indirect. That distinction requires locating the extrema in reciprocal space.

For a valence band and conduction band,

\[E_g=E_{\mathrm C,min}-E_{\mathrm V,max}\]

The extrema must be searched across the entire Brillouin zone. If they occur at the same \(\mathbf k\), the material has a direct gap. If they occur at different wave vectors, the gap is indirect.

An optical transition must conserve crystal momentum as well as energy. Photon momentum is often small on the Brillouin-zone scale, so a direct gap supports efficient near-vertical optical transitions. An indirect transition commonly needs a phonon to supply or remove crystal momentum.

valence = -1.0 - 0.5 * (1 - np.cos(phase))
conduction = 1.0 + 0.5 * (1 - np.cos(phase - np.pi / 2))
valence_index, conduction_index = np.argmax(valence), np.argmin(conduction)
gap = conduction[conduction_index] - valence[valence_index]
kind = (
    "direct on this grid"
    if valence_index == conduction_index
    else "indirect on this grid"
)
print("Gap (eV):", gap, kind)

Exact index equality is suitable only when both extrema lie on the same sampled grid. A real calculation compares wave vectors with symmetry and numerical tolerances and refines around candidate extrema.

Group velocity and effective mass

A semiclassical wave packet in band \(n\) has group velocity

\[\mathbf v_n(\mathbf k)=\frac{1}{\hbar}\nabla_{\mathbf k}E_n(\mathbf k)\]

Band curvature controls acceleration under an applied force. In one dimension, the effective mass is defined locally by

\[\frac{1}{m^*}=\frac{1}{\hbar^2}\frac{d^2E}{dk^2}\]

Near a conduction-band minimum, curvature is often positive and an electron-like positive effective mass is useful. Near a valence-band maximum, curvature is negative. Describing missing occupation as positively charged holes gives a more convenient account of transport than tracking nearly every electron in the filled band.

Effective mass is a response parameter determined by the band curvature. It is not a change in the electron's invariant rest mass, and in three dimensions it is generally a tensor.

Density of states

Band dispersion alone does not say how many states lie in an energy interval. The density of states \(g(E)\) counts available states per energy and volume. Carrier number combines states and occupation:

\[n=\int g(E)f(E)\,dE\]

The density of states depends on dimensionality and band shape. A visually similar band edge can therefore yield different carrier behavior in one, two, and three dimensions.

Numerical and interpretive checks

For a calculated band structure:

Sorting eigenvalues independently at each wave vector can swap labels at band crossings. Use symmetry labels, eigenvector overlaps, or subspace tracking when band identity matters.

The independent-electron band picture omits or approximates many-body correlation. Strongly correlated materials can be insulating despite a simple partially filled-band prediction. Defects create localized levels, and surfaces have states absent from an infinite bulk calculation. Band theory is powerful, but its approximation level must accompany any material claim.

A worked Python notebook

Inspect a one-dimensional band with zero on-site energy and hopping energy one electronvolt. The question for this calculation is precise: how does tight-binding band energy change when we change crystal momentum phase ka, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

The one-orbital nearest-neighbour model is a deliberately limited band description. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from -3.14159 to 3.14159 rad. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for crystal momentum phase ka, expressed in rad. The output is tight-binding band energy, expressed in eV. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = - 2 \cos{\left(q \right)}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 1.0472 rad, the expression gives -1 eV. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("-2*cos(q)", locals={"q": q})
reference_input = 1.0471975511965979
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "rad")
print("Predicted tight-binding band energy:", reference_output, "eV")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 2 eV. The midpoint, at 0 rad, gives -2 eV. The upper endpoint gives 2 eV. Read these together with the assumptions: they describe different choices of crystal momentum phase ka within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([-3.141592653589793, 0.0, 3.141592653589793])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} rad -> {output_value:.6g} eV")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows crystal momentum phase ka; the vertical axis shows tight-binding band energy. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(-3.141592653589793, 3.141592653589793, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Crystal momentum phase ka (rad)")
ax.set_ylabel("Tight-binding band energy (eV)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does tight-binding band energy change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 1.0472 rad is 1.73205 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 2 \sin{\left(q \right)}\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 1.0786134777324958
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "eV")
print("Full expression:", full_prediction, "eV")
print("Approximation difference:", full_prediction - linear_prediction, "eV")

Here the changed input is 1.07861 rad. The tangent estimate is -0.945586 eV, while the full expression gives -0.945102 eV. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 6.283185307179586
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of crystal momentum phase ka inside the stated interval, predict whether tight-binding band energy will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: The one-orbital nearest-neighbour model is a deliberately limited band description. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of eV: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(3.141592653589793) - evaluate(-3.141592653589793))
for points in [101, 201, 401]:
    grid = np.linspace(-3.141592653589793, 3.141592653589793, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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