6  Dynamics in bands

How does an electron in a crystal respond to an external force? What defines the conductive or insulating nature of a crystal? These are the main topics of this chapter.

6.1 The acceleration theorem

Starting from the definition of group velocity for an electron in a band with dispersion \(E(\mathbf{k})\),

\[\mathbf{v} = \frac{1}{\hbar}\nabla_\mathbf{k}E,\]

basic mechanics indicates that, given an applied force \(\mathbf{F}\), the electron energy satisfies \(dE = \mathbf{F}\cdot \mathbf{v}\,dt\). If we make the subtle assumption that the electron remains in a given band \(E(\mathbf{k})\), the derivation via the chain rule gives

\[\frac{dE}{dt} = \frac{d\mathbf{k}}{dt}\cdot\nabla_\mathbf{k}E = \mathbf{F}\cdot\frac{1}{\hbar}\nabla_\mathbf{k}E,\]

finally implying \(\hbar\dot{\mathbf{k}} = \mathbf{F}\) and confirming that “quasimomentum” is a well-deserved name. This can be verified on a more rigorous footing and in a quantum mechanical framework by the following theorem.

Theorem 6.1: Acceleration theorem

Let the initial state \(\psi(\mathbf{r},t=0) = \psi_{\mathbf{k}_0}(\mathbf{r})\) be a Bloch state¹ with initial quasi-momentum \(\mathbf{k}_0\). Assuming the electron is subject to a periodic Hamiltonian \(\mathcal{H}_0\) plus a constant force \(\mathbf{F}\), then during its time evolution the electron remains a Bloch state with a crystal momentum obeying the equation

\[\mathbf{k}(t) = \mathbf{k}_0 + \frac{\mathbf{F}t}{\hbar},\]

or equivalently \(\hbar\dot{\mathbf{k}} = \mathbf{F}\). As a subtle but important difference from the semiclassical derivation, here \(\psi(\mathbf{r},t)\) is never assumed to be an eigenstate nor to belong to a specific band dispersion \(E_n(\mathbf{k})\). If we start from one band, we might or might not remain in that band depending on the details of the force and of the band structure.

Show proof

By hypothesis, the Hamiltonian has the form

\[\mathcal{H}(\mathbf{r}) = \mathcal{H}_0 - \mathbf{F}\cdot\mathbf{r},\]

where \(\mathcal{H}_0=\mathbf{p}^2/2m + V(\mathbf{r})\) is periodic with \(V(\mathbf{r}+\mathbf{R}) = V(\mathbf{r})\), for any \(\mathbf{R}\in\mathcal{BL}\). Since \(\mathcal{H}(\mathbf{r})\) does not depend on time, time evolution can be simply calculated as

\[\psi(\mathbf{r},t) = \exp\left[-\frac{i}{\hbar}\mathcal{H}(\mathbf{r})t\right]\psi(\mathbf{r},0).\]

By hypothesis, the initial state \(\psi(\mathbf{r},0)\) is a Bloch state with quasimomentum \(\mathbf{k}_0\), so we can simply verify whether its time evolution \(\psi(\mathbf{r},t)\) is still a Bloch state or not, i.e. whether it satisfies the Bloch condition:

\[ \begin{aligned} \psi(\mathbf{r}+\mathbf{R},t) &= \exp\left[-\frac{i}{\hbar}\mathcal{H}(\mathbf{r}+\mathbf{R})\,t\right]\,\psi(\mathbf{r}+\mathbf{R},0)\\ &= \exp\left[-\frac{i}{\hbar}\mathcal{H}(\mathbf{r})t+\frac{i}{\hbar}\left(\mathbf{F}\cdot\mathbf{R}\right)t\right]\,e^{i\mathbf{k}_0\cdot\mathbf{R}}\psi(\mathbf{r},0)\\ &= \exp\left[i\left(\mathbf{k}_0+\frac{\mathbf{F}t}{\hbar}\right)\cdot\mathbf{R}\right]\exp\left[-\frac{i}{\hbar}\mathcal{H}(\mathbf{r})t\right]\psi(\mathbf{r},0)\\ &= \exp\left[i\left(\mathbf{k}_0+\frac{\mathbf{F}t}{\hbar}\right)\cdot\mathbf{R}\right]\psi(\mathbf{r},t). \end{aligned} \]

Indeed \(\psi(\mathbf{r},t)\) is a Bloch state with \(\mathbf{k}(t)=\mathbf{k}_0 + \mathbf{F}t/\hbar\) and \(\hbar\dot{\mathbf{k}} = \mathbf{F}\). \(\blacksquare\)

CautionParametric hamiltonians and adiabatic evolution

Time evolution of a Bloch configuration according to Theorem 6.1 is, de facto, an evolution under the time-dependent \(\mathcal{H}\left[\mathbf{k}(t)\right]\) acting on the discrete spectrum of periodic functions \(u(\mathbf{r},t)\). From the theorem, we know that the solution can take the functional form \(\psi(\mathbf{r},t)=e^{i\mathbf{k}(t)\cdot\mathbf{r}}u(\mathbf{r},t)\) with \(\hbar\dot{\mathbf{k}} = \mathbf{F}(t)\) and a periodic function \(u(\mathbf{r},t)\) that has to be determined. The Schrödinger equation with \(\mathcal{H}(t)=\mathcal{H}_0-\mathbf{F}(t)\cdot\mathbf{r}\) is

\[ \begin{aligned} i\hbar\,\frac{\partial \psi}{\partial t} &= \left(-\hbar\dot{\mathbf{k}}\cdot\mathbf{r}\right)\,e^{i\mathbf{k}(t)\cdot\mathbf{r}}u(\mathbf{r},t) +i\hbar\,e^{i\mathbf{k}(t)\cdot\mathbf{r}}\frac{\partial u}{\partial t} \\ &= \left[\mathcal{H}_0 -\mathbf{F}(t)\cdot\mathbf{r}\right]\,e^{i\mathbf{k}(t)\cdot\mathbf{r}}u(\mathbf{r},t). \end{aligned} \tag{6.1}\]

where terms proportional to \(-\hbar\dot{\mathbf{k}}\) and \(-\mathbf{F}(t)\) cancel each other. As already discussed in Section 3.1, we recall that the commutation relation \([\mathbf{p},e^{i\mathbf{k}\cdot\mathbf{r}}] = \hbar\mathbf{k}\,e^{i\mathbf{k}\cdot\mathbf{r}}\) implies

\[\mathcal{H}_0\left[e^{i\mathbf{k}\cdot\mathbf{r}}\,u(\mathbf{r},t)\right] = e^{i\mathbf{k}\cdot\mathbf{r}}\left[\frac{(\mathbf{p}+\hbar \mathbf{k})^2}{2m} + V(\mathbf{r})\right]\,u(\mathbf{r},t) = e^{i\mathbf{k}\cdot\mathbf{r}}\,\mathcal{H}(\mathbf{k})\,u(\mathbf{r},t)\]

and, after removing the common phase factor, we finally obtain a simple equation

\[i\hbar\,\frac{\partial u}{\partial t} = \mathcal{H}\left[\mathbf{k}(t)\right]\,u(\mathbf{r},t),\]

where the external force \(\mathbf{F}(t)\) only enters through the time evolution of the parameter \(\mathbf{k}(t)\). Given the discrete nature of the spectrum of \(\mathcal{H}(\mathbf{k})\) and the fact that we only need to solve \(u(\mathbf{r},t)\) on a single cell with periodic conditions, this is a greatly simplified formulation.

The connection with the adiabatic theorem is also immediate: in the limit of slow evolution (i.e. weak fields) and in the absence of degeneracies, the electron will track its instantaneous eigenvalue \(E_n\left[\mathbf{k}(t)\right]\) and thus remain in the same band, as naively assumed in the initial semiclassical derivation of the acceleration theorem. This will be further elaborated in the next sections.

6.2 SBA and EMA

As further discussed in Section 6.4, in the limit of “weak” fields the electron can be expected to remain in the same band \(E_n(\mathbf{k})\) while \(\mathbf{k}\) evolves. This is called single-band approximation (SBA).

A useful concept arises in this limit: the acceleration \(\mathbf{a}\) caused by a force \(\mathbf{F}\)

\[a_i = \frac{dv_i}{dt} = \frac{d}{dt}\left(\frac{1}{\hbar}\,\frac{\partial E}{\partial k_i}\right) = \frac{1}{\hbar}\,\frac{\partial^2E}{\partial k_i\partial k_j}\,\frac{dk_j}{dt} = \frac{1}{\hbar^2}\,\frac{\partial^2E}{\partial k_i\partial k_j}\,F_j=\left[\frac{1}{m^*}\right]_{ij}F_j,\]

can be described in terms of an effective mass \(m^*\) parameter, which is proportional to the inverse of the Hessian of the band dispersion

\[\left[\frac{1}{m^*}\right]_{ij} \equiv \frac{1}{\hbar^2}\frac{\partial^2 E}{\partial k_i\,\partial k_j} \tag{6.2}\]

This allows writing the time evolution as a Newton-like equation \(\mathbf{F} = m^*\mathbf{a}\). Note that while in the case of an isotropic band \(m^*\) is a multiple of the identity matrix and can be practically replaced by a scalar, in general it has to be treated as a tensor.

The value of \(m^*\) will change as a function of \(\mathbf{k}\), and thus of time. However, as long as the electron remains close to a band extremum, as it often happens for a semiconductor, the dispersion is likely parabolic and \(m^*\) can be assumed to be a constant. This is called effective mass approximation (EMA): in this limit, we can forget about the crystal and simply treat the electron as a free particle with a modified mass \(m^*\).

A systematic perturbative route to the band curvature around a known point — the \(\mathbf{k}\cdot\mathbf{p}\) method, which expresses the effective mass tensor through interband momentum matrix elements — is developed in the semiconductor chapter (Section 8.1.2), where it is put to work on real band edges.

6.3 Bloch oscillations

The Theorem 6.1 has counterintuitive consequences. Let us assume we have a 1D crystal with lattice spacing \(a\) and one electron prepared at \(t=0\) in the Bloch state at \(k_0=0\). Time evolution under a force \(F=-eE>0\) dictates that the electron will follow the band dispersion according to

\[k(t) = - \frac{eE}{\hbar}\,t\]

and reach \(k(T_B)=2\pi/a\), which is exactly the same initial state since this quasimomentum is equivalent to \(k=0\). This will happen after a time \(T_B = 2\pi\hbar/e|E|a\) and, during its time evolution, the electron will acquire positive and negative velocities according to the sign of \(dE_n/dk\). While strictly speaking a Bloch state is completely delocalized, we can argue that the electron will also go back to its initial position in the real space; in fact

\[\Delta x = \int_0^{T_B} v(t)\,dt = \frac{1}{\hbar}\int_0^{T_B} \frac{dE_n}{dk}\,dt = -\frac{1}{eE}\int_{k(0)}^{k(T_B)} \frac{dE_n}{dk}\,dk = 0.\]

This dynamics, illustrated in Figure 6.1, is called a Bloch oscillation (BO) and occurs at the Bloch frequency \(\omega_B\) satisfying

\[\hbar\omega_B = e|E|a. \tag{6.3}\]

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Figure 6.1: Semiclassical picture of a Bloch oscillation in reciprocal space. The blue dot represents the electron state, evolving according to Theorem 6.1. The evolution is depicted in a repeated zone scheme, which is redundant: \(k\) values outside the FBZ have a different shade to clearly highlight that they just represent a copy of the Bloch state indicated by the point in the FBZ.

CautionBO waveform in the real space

BOs in the real space can be easily integrated in 1D. In fact, the group velocity is given by

\[v = \frac{1}{\hbar}\,\frac{dE}{dk},\]

and the real-space trajectory can be obtained as

\[\begin{align} x(t) &= x_0+\frac{1}{\hbar}\int_0^t \frac{dE}{dk}\,dt' = x_0-\frac{1}{eE}\int_{k(0)}^{k(t)} \frac{dE}{dk}\,dk\\ &= \frac{1}{F}\,E_n\left(k_0+\frac{Ft}{\hbar}\right) + \text{const.} \end{align} \]

where \(F=-eE\) and we assumed a starting position \(x_0\) and momentum \(k_0\). Finally, the real space trajectory reproduces the band dispersion \(E_n(k)\), scaled by the applied force. The amplitude of the oscillation depends on the band width and on the electric field strength. Clearly, it is impossible to fully specify both position and quasi-momentum in a quantum mechanical limit: this is further analyzed in the next optional pill.

CautionQuantum visualization of BOs and the Wannier–Stark ladder

Figure 6.2 shows a fully-quantum real-space simulation of a BO. The chosen periodic Hamiltonian implements a Mathieu potential plus a constant force \(\mathcal{H}(x) = p^2/2m - V_0\cos(2\pi x/a)-xF\). The dimensionless constant \(mV_0a^2/\hbar^2=3\) is exactly the one already set in Figure 6.1, opening a sizeable gap between the first two bands. In the current interactive figure, the force \(F\) can be controlled by a slider.

The initial state is prepared as a gaussian superposition of Bloch states \(\psi_k(x)\) belonging to the first band dispersion \(E_1(k)\)

\[\psi(x,0) \propto \sum_k \exp\left[-\frac{(k-k_0)^2}{2\sigma_k^2}-ikx_0+i\theta(k)\right]\psi_k(x),\]

where \(\sigma_k\) controls the quasi-momentum spread, \(k_0\) the initial phase of the orbit \(\varphi_0=k_0a\), \(x_0\) the initial position. The orbit phase

\[\theta(k) = -\int^k\frac{E_1(k')-\overline{E_1}}{F}\,dk'\]

places every \(k\)-component on the correct classical orbit \(x(k) = [E_1(k)-\overline{E_1}]/F\) and is crucial to obtain a compact wavepacket in the real space. The choice of \(\sigma_k\) is a trade-off: the corresponding spread of the wavepacket in the real space depends also on the propagating velocity and generally the wavepacket will “breathe” due to its propagation. The chosen \(\sigma^2_k=|F|/2\hbar v_B\), with \(v_B\) equal to the maximal group velocity during the oscillation, keeps the effect small while maintaining a compact wavepacket.

Note that the wavepacket extends outside the band regions allowed in the “classical” limit, and that a minor tunneling is observed at the highest end of the \(F\) slider. Note also that wave-function lobes confined in the potential minima are superimposed in phase at the left end of the oscillation (typically expected at a band minimum) while they are in antiphase at the right end (typically expected at a band maximum). In heuristic terms this can be rationalized as follows: the electron wavepacket accelerates and consequently increases the density of its oscillations; this proceeds until they match the crystal periodicity (i.e. the wavepacket wavelength is double the crystal periodicity), constructive interference from backscattering becomes more and more effective, and the electron is thus “repelled” back by Bragg reflection.

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Figure 6.2: Time-dependent split-step Fourier simulation of BOs. Top: \(|\psi|\), \(\mathrm{Re}\,\psi\) and \(\mathrm{Im}\,\psi\) in the washboard potential; the gray stripes are the field-tilted allowed bands (the gaps are left unshaded). Bottom: the folded quasimomentum density superimposed over the three lowest band dispersions.

In the limit of a very wide spread in the \(k\) space, and assuming that SBA is exactly satisfied, we can obtain an eigenstate localized in space

\[(\mathcal{H}_0+eEx)\psi_0(x) = E_0\psi_0(x),\]

which represents a quantized Bloch oscillation, while the simulation in Figure 6.2 can rather be considered a coherent-state limit. Given \(\psi_0(x)\) we can easily construct many more solutions as follows:

\[\mathcal{H}(x)\psi_0(x-a) = \left[\mathcal{H}(x-a)+eEa\right]\psi_0(x-a) = (E_0+eEa)\psi_0(x-a),\]

thus we have a sequence of eigenstates \(\psi_n(x) = \psi_0(x-na)\) with energies \(E_n = E_0 + neEa\), which can be called a Wannier-Stark ladder. The energy spacing between the states is exactly \(\hbar\omega_B = eEa\), and the time evolution of any superposition of Wannier-Stark states can be expected to be periodic, with the BO oscillation period \(T_B\).

CautionWhat about momentum conservation?

A possible residual doubt concerns the conservation of momentum during BOs: the electron is continuously pushed by the force \(-eE\), but still it oscillates back to its initial configuration after every Bloch period \(T_B\). What happens to the transferred momentum? Is it lost and are we breaking momentum conservation?

Any answer must involve the role of the crystal, which also interacts with the electron and carries a linear momentum. Indeed, the electron responds to the sum of the external force plus the crystal reaction. Thus, when the electron is close to the top of the band, the mass is negative and the electron slows down despite being pushed by the external force: it is the crystal that takes up the missing linear momentum. If we imagine the velocity reversal to occur impulsively at \(k=\pi/a\) when the Bragg condition is met (not really an impulsive exchange in reality, but this might help intuition), the crystal takes a discrete momentum kick \(h/a\) caused by the “bouncing” electron, every \(T_B\). The crystal is thus the “momentum reservoir” that allows the electron to oscillate back and forth without violating momentum conservation. Since the mass of the crystal is much larger than that of the electron, this is completely irrelevant and normally not even mentioned.

As a further subtlety, charge neutrality of the overall system implies that while the negative electron is accelerated in one direction, the positively charged crystal is accelerated in the opposite one, thus the net momentum transfer is null. The electron-crystal exchange still balances out during oscillations as mentioned above, but in addition the overall system does not accumulate any net total momentum. This detail is again largely irrelevant due to the mass of the crystal.

6.3.1 BOs cannot be observed in a natural crystal

So far we assumed that electrons just follow the acceleration theorem, but in any real system dynamics is dominated by scattering, which tends to relax the electrons to their equilibrium configuration within a timescale \(\tau\), which is typically of the order of tens to hundreds of femtoseconds in any natural crystal.

The observation of BOs requires \(\omega_B\tau \gg 1\) and thus

\[|E| \gg \frac{\hbar}{ea\tau},\]

which is basically impossible to achieve in a real crystal without breaking SBA. This is the topic of the next sections.

6.4 Zener tunnelling

SBA breaks down when the electric field is strong enough to drive transitions between bands; here we want to make this statement more quantitative.

While semiclassically electrons cannot enter an energetically forbidden region, this process can typically be described in terms of evanescent waves in the quantum limit. This is also true for electrons in a crystal: in the band gap it is still possible to have states satisfying

\[\psi(x+a) = e^{ika}\psi(x),\]

but \(k\) is no longer a simple real number and the phase factor \(e^{ika}\) becomes an extinction/enhancement factor \(\pm e^{\pm\beta a}\). This is visible in the left panel of Figure 6.3, where the standard dispersion curve (blue) continues in the gaps with imaginary \(k\) solutions (red). The red portions are discarded when looking for proper eigenstates, since they always lead to a divergence for either positive or negative \(x\). Nevertheless, such states can be used to describe evanescent waves and tunneling through a finite-size barrier. The band configuration in the presence of an electric field is reported on the right side of Figure 6.3: electrons have to cross a gapped region (white band, with energy extent equal to \(E_g\)) to tunnel from one band to the next one. In doing so, the electron energy resides in the gap over a spatial width \(d = E_g/e|E|\), thus the tunneling probability can be estimated using a semiclassical Wentzel-Kramers-Brillouin (WKB) approach

\[P \approx \exp\left[-2\int_{-d/2}^{+d/2} |\beta(x)|\,dx\right].\]

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Figure 6.3: Left. Band dispersions \(E_n(k)\) in blue can be extended into the energy gaps if \(k\) is allowed to be complex (red curves). While these solutions are not suitable to define an eigenstate, they highlight the size of the extinction factor in the energy gap and its typical dependence on the energy position. Right. In the presence of an electric field, an electron has to cross a gap region with a size \(d = E_g/e|E|\), where it is exponentially damped by the imaginary quasi-momentum \(\beta(E)\).

The exact magnitude of the extinction coefficient \(\beta(x)\) has to be calculated according to the energy position in the gap, but here we only want to estimate its order of magnitude and its functional dependence on relevant physical parameters. To this end, we can take advantage of EMA: as the energy \(E(k)\) and the value of \(k\) for energies above a band minimum at \(k=0\) are linked by the relation

\[E(k) = E_{min} + \frac{\hbar^2 k^2}{2m^*}\]

this can be extended to energies below the minimum and in the gap using \(k=i\beta\) and

\[E(\beta) = E_{min} - \frac{\hbar^2 \beta^2}{2m^*}.\]

Clearly, the energy scale of the problem is \(E_g\) and thus an (overestimated) order of magnitude for the extinction coefficient is \[\beta_\text{max} = \frac{\sqrt{2m^* E_g}}{2\hbar}.\]

This leads to the following functional dependence of the Zener tunneling probability

\[P \approx \exp\left[-c\frac{\sqrt{2m^* E^3_g}}{\hbar e|E|}\right],\]

where \(c\) is a numerical factor taking into account the many approximations behind the present rough estimate. A close inspection of the WKB integral gives \(c = \pi/4\). Most importantly, the tunneling probability is exponentially suppressed for small fields \(\mathbf{E}\) and large energy gaps \(E_g\). It is interesting to consider the typical size of the critical field \(E_c\) for which the tunneling probability is not negligible, i.e. when the exponent is of order 1:

CautionLandau–Zener approach to tunneling

While the WKB approach is the one universally used to describe Zener tunneling in any solid state physics textbook, one formal dissatisfaction might emerge from the fact that the potential \(\mathcal{H}_0+e\mathbf{E}\cdot\mathbf{r}\) is not periodic, thus any calculation involving Bloch states and a “local” extinction factor \(\beta\) only makes sense in a not very well defined weak-field limit. A more rigorous approach relies on the parametric Hamiltonian approach and on the exact result of the Landau–Zener (LZ) formula. For purely historical reasons, this is the formalism normally used to describe inter-band tunneling in optical lattices in cold-atom systems, but it can be equally used for Zener tunneling of electrons in crystals.

The LZ formula analytically quantifies the tunneling probability for a linear crossing of two levels in the presence of an off-diagonal coupling, described by the Hamiltonian

\[\mathcal{H}(t) = \begin{pmatrix} +\alpha t/2 & \Delta \\ \Delta^* & -\alpha t/2 \end{pmatrix},\]

where the diabatic energies \(\pm\alpha t/2\) cross linearly in time — \(\alpha = d(E_1-E_2)/dt\) is the sweep rate of their splitting — while the (generally complex) coupling \(\Delta\) turns the crossing into an avoided crossing with minimum splitting \(2|\Delta|\). The exact LZ prediction for the probability of a diabatic jump (the system “ignores” the avoided crossing and lands on the other adiabatic branch) is

\[P=\exp\left(-2\pi\Gamma\right),\qquad\Gamma = \frac{|\Delta|^2}{\hbar \alpha}.\]

The historical derivations of this result are notoriously laborious; a remarkably compact modern one, based on the Markov approximation, can be found in Glasbrenner and Schleich (2023).

Interestingly, the dimensionless control parameter \(\Gamma\) is directly linked to the Rabi frequency of the anticrossing, \(2|\Delta|/\hbar\), and the timescale required to go through the crossing, \(|\Delta|/\alpha\). In very broad terms, it quantifies how many oscillations the state is able to perform between the top and bottom branch while traversing the anticrossing: many oscillations average out and the evolution stays adiabatic, while \(\Gamma \ll 1\) leaves no time to oscillate and the state jumps diabatically.

Derivation at the first NFE anticrossing. In the parametric evolution \(\mathcal{H}[k(t)]\) with \(\hbar\dot{k} = eE\), the anticrossing at \(k = \pi/a\) of the nearly-free electron model is exactly a LZ problem. The two folded free-electron branches cross with diabatic slopes \(\pm\hbar v_B\), where \(v_B = \hbar\pi/ma\) is the free velocity at the zone boundary, and are coupled by the first harmonic of the potential: \(\Delta = V_1\), i.e. \(|\Delta| = |V_1| = E_g/2\) (the NFE gap). The splitting is therefore swept at \(\alpha = 2\hbar v_B|\dot k| = 2v_B\,eE\), and

\[2\pi\Gamma \;=\; \frac{2\pi\,(E_g/2)^2}{\hbar\; 2v_B\,eE} \;=\; \frac{\pi E_g^2}{4\hbar v_B\, eE}.\]

The same NFE \(2\times2\) block also supplies the band-edge mass: expanding Equation 3.18 around the edge, \(\hbar^2/2m^* = (\hbar^2/2m)\left(1+2E_\pi/|V_1|\right)\) with \(E_\pi = \hbar^2\pi^2/2ma^2\) the free-electron energy at the crossing, so for a weak potential \(m^* \approx m E_g/(4E_\pi)\). Since \(v_B^2 = 2E_\pi/m\), this is precisely the identity \(v_B = \sqrt{E_g/2m^*}\), and eliminating \(v_B\) from the exponent gives

\[T_\text{Zener} = \exp\!\left[-\frac{\pi}{4}\,\frac{\sqrt{2m^*\,E_g^3}}{\hbar\,eE}\right], \tag{6.4}\]

recovering the WKB estimate with the announced factor \(c = \pi/4\) — but now with every ingredient (\(E_g\), \(m^*\)) supplied by the band structure itself. The corresponding critical field is \(E_c = \pi\sqrt{2m^* E_g^3}/(4\hbar e)\).

This approach is also more suitable to see the obvious links with the adiabatic theorem: the SBA corresponds to the adiabatic limit of the parametric evolution, which is realized in the weak-field limit; differently, Zener tunneling becomes more and more important as we move towards the impulsive limit.

Consistency with WKB. Carrying out the WKB integral exactly for the two-band model (gap \(E_g = 2E_0\)) leads to the same transmission Equation 6.4. In the gap region the imaginary quasi-momentum is \(|\beta(E)| = \frac{1}{|\gamma|a}\sqrt{E_0^2 - E^2}\). The WKB transmission factor is

\[T = \exp\!\left[-2\int_{x_1}^{x_2}|\beta(E(x))|\,dx\right],\]

where \(x_{1,2} = \mp E_0/(eE)\) are the classical turning points. Substituting \(E(x) = eEx\) and evaluating the integral:

\[\int_{-E_0/(eE)}^{E_0/(eE)} \frac{1}{|\gamma|a}\sqrt{E_0^2 - e^2E^2 x^2}\,dx = \frac{\pi E_0^2}{2|\gamma|eEa}.\]

Re-expressing in terms of the gap \(E_g = 2E_0\) and effective mass \(m^* = \hbar^2 E_g/(2\gamma^2 a^2)\) gives Equation 6.4.

6.5 Full bands: metals and insulators

A key fact underpins the whole response of a crystal to an external low-frequency electric field.

TheoremFull band \(\implies\) zero net charge current

For any electron in a band with quasi-momentum \(\mathbf{k}\) and finite group velocity \(\propto\nabla_\mathbf{k}E\), Kramers’ degeneracy warrants that another one exist at \(-\mathbf{k}\) with exactly the opposite group velocity. Thus the net current carried by this pair is null, thus the total current carried by a full band is also null.

This fact remains true also in the presence of a field: the acceleration theorem shifts all the occupied \(\mathbf{k}\) rigidly, and a full band maps onto itself under any rigid shift (thanks to FBZ periodicity). A filled band is therefore dynamically inert and much as an empty one — electrons accelerates, Bloch-oscillates and yet the band carries exactly zero current at all times. Of course here we are neglecting two important facts:

• when the field is so large to induce Zener tunneling, then electrons can overcome Bragg reflection by atoms and conduct electricity. This is what leads to the so-called dielectric breakdown.
• when electric field oscillates very fast, optical transition allow promoting charges between different bands: this will be discussed in later chapters devoted to optical response of crystals.

These observations leads to a broad classification of crystalline solids

Definition 6.1: Metals vs insulators

A metal is a crystal where at least one band is partially filled and electrons can easily rearrange by occupying the empty band states. This will lead to good electrical conduction properties.

An insulator is a crystal where every band is either completely filled or completely empty. An insulator is charaterized by a gap \(E_g\) between the highest filled band (valence band or VB) and the lowest empty band (conduction band or CB). In this case the electron system is “rigid” and we will get no electrical conduction.

Two qualifications are usually appended to this binary picture. A semiconductor is structurally an insulator, but its relatively small gap² and the existence of suitable dopants allow achieving a “tunable conductor”; this is the cornerstone of modern electronics. A semimetal sits at the opposite end: structurally a metal, but due to small band overlap it only has a vanishingly small density of states at the highest occupied electronic configurations. Semimetals therefore conduct, but poorly compared to a normal metal.

The Fermi energy and the geometry of Fermi surfaces — including the instructive Ca/Al case study — were introduced in Section 5.4; here we only need the filling dichotomy above.

6.6 Electron-hole duality

Empty and full bands are similarly unresponsive to low-frequency and not-too-strong fields. This symmetry can be pushed further by introducing the concept of hole. Before thinking about holes as positive quasi-particles magically manifesting themselves inside a solid, it is important to start from the existence of a so-called hole picture when describing the behavior of a band.

6.6.1 The hole picture

Given the fact that the current carried by a completely filled band is null, current can be equivalently calculated taking into account filled states, or empty states, which we will call “holes”

\[\mathbf{J} = -e\sum_{filled} \mathbf{v}(\mathbf{k}) = -e\left[\sum_{all} \mathbf{v}(\mathbf{k})-\sum_{empty} \mathbf{v}(\mathbf{k})\right] = +e\sum_{empty} \mathbf{v}(\mathbf{k})\]

where we assumed to have either fully occupied states or completely empty ones. Later on the discussion will be easily extended to generic occupation numbers. Given that the group velocity of a state is the same regardless of whether it is occupied or not, missing electrons behave as if they were positively charged objects with \(q=+e\), at least for what concerns the resulting current density \(\mathbf{J}\). A true mapping to a hole picture, however, should also deal with the response to external forces. For instance, the acceleration theorem tells that the quasi-momentum of a Bloch state evolves according to

\[\hbar\dot{\mathbf{k}} = -e\mathbf{E}=\mathbf{F}, \tag{6.5}\]

again this is a behavior of the Bloch state, which holds regardless of whether the state is occupied or not. Coherently with the picture of a positively-charged particle, however, we would like the hole response to be connected to a force \(\mathbf{F}=+e\mathbf{E}\): this is at odds with Equation 6.5. This inconsistency can be solved if, in the hole picture we are building, we assign the hole a quasi-momentum \(\mathbf{k}_h = -\mathbf{k}\), so that we can rewrite the acceleration theorem as

\[\hbar\dot{\mathbf{k}}_h = +e\mathbf{E}.\]

A quick inspection on energy conservation, as well as building on the observation that the group velocity \(\mathbf{v}\propto\nabla_\mathbf{k}E\) of a state has to be the same regardless of whether it is occupied or not, indicates that a consistent hole picture finally requires the energy of the hole to take a minus sign with respect to the original one of the electron, i.e. \(E_h=-E\). This also implies, as anticipated, \(\mathbf{v}_h=\mathbf{v}\) and, in addition, \(m^*_h=-m^*\). Finally, we have the full mapping summarized in the following box.

Definition 6.2: The electron and hole pictures

The behavior of a band can be equivalently described in terms of electrons populating a band, or taking into account the unoccupied states, which will be called “holes”. The following mapping has to be used when describing the properties of a hole.

Table 6.1: Mapping between the electron and hole pictures.

Property	Electron	Hole
Charge	\(-e\)	\(+e\)
Quasi-momentum	\(\mathbf{k}_e\)	\(\mathbf{k}_h = -\mathbf{k}_e\)
Velocity	\(\mathbf{v}_e\)	\(\mathbf{v}_h = \mathbf{v}_e\)
Effective mass	\(m^*_e\)	\(m^*_h = -m^*_e\)
Energy	\(E_e\)	\(E_h = -E_e\)

6.6.2 Which picture, then?!

So far the electron and hole pictures appeared as two completely legitimate and alternative ways to describe the same thing: a partially filled band. While they are truly equivalent, depending on the filling of the band one of the two points of view might be more or less suitable. Consider for example a valence band with one missing electron: we have two (mathematically equivalent) options.

• Track \(\sim 10^{23}\) electrons, with all sort of group velocities and effective mass signs, and sum their collective behavior to calculate the response of the band to an external force;
• Move to the hole picture and track one single hole, with a positive charge \(+e\), a well-defined group velocity and effective mass (which will be probably positive since an empty state will typically reside at the top of the band, where \(m^*\) for the electrons is counter-intuitively negative).

The second picture is obviously immensely more intuitive to picture, and is in fact used whenever we have to deal with an almost completely occupied band. Differently, the electron picture will be typically chosen to describe a conduction band populated by only a few electrons at its bottom. The concept behind the two dual pictures is illustrated in Figure 6.4.

import { renderElectronHole } from "./_interactive/ch06-electron-hole.js"
{
  const W = Math.min(width, 740);
  const container = document.createElement("div");
  container.style.cssText = "width:100%; max-width:" + W + "px; margin:0 auto;";
  yield container;
  renderElectronHole(container);
}

Figure 6.4: n-type (left) and p-type (right) bands at \(T=0\). Blue overlay: occupied states (electron picture); red overlay, after the flip: empty states (hole picture) — same physics, alternative bookkeeping. The field buttons rigidly shift the occupation in \(\mathbf{k}\); \(J\) reports the sign of the resulting current.

Any situation in between, including in particular the case of metals, will be typically described in terms of electrons… but it might well be described (probably at no gain) in terms of holes. In general, both pictures will probably be complicated, with non-trivial Fermi surfaces, \(\mathbf{k}\)-dependent effective masses and intricate band dispersions. For instance, the Hall coefficient of Al can be positive or negative depending on the magnitude of the magnetic field.

CautionWhat about hopping bonds in Silicon?

Introductory textbooks on semiconductors often discuss “holes” as localized “missing bonds” in the Silicon crystal network, often marking a difference with “free electrons” residing in the conduction band and romaing around freely. In these cartoons, sketched in Figure 6.5, the fact that “holes” move in the real space in the opposite direction with respect to “electrons” is typically presented as due to the sequential hopping of electrons into the spaces left behind by missing charges.

import { renderHoleCartoons } from "./_interactive/ch06-hole-cartoons.js"
{
  const W = Math.min(width, 740);
  const container = document.createElement("div");
  container.style.cssText = "width:100%; max-width:" + W + "px; margin:0 auto;";
  yield container;
  renderHoleCartoons(container);
}

Figure 6.5: The classic textbook cartoons for holes. (a) Balls (electrons) on bonding sites: under the field, a neighboring ball hops into the vacancy and thus the vacancy drifts in the other direction. (b) Broken-bond picture in the Si network: a free electron (dot) leaves behind a hole (open circle) in the bond grid.

While these cartoons have the merit of helping intuition of students still missing a proficient knowledge of quantum mechanics, they might lead to some misconceptions, which it is useful to clarify:

• Localized holes vs free electrons. Neither electrons nor holes are free in a crystal, nor are they in a qualitatively different degree of interaction with whatever surrounds them: both are associated with delocalized Bloch waves due to the presence of the periodic potential of the crystal.
• Hole “popagating” thanks to sequential hopping of bound electrons. Both electrons and holes are associated with Bloch waves (occupied or not) that propagate in a completely equivalent way; again, there is no other qualitative difference between the dynamics of the two, no sequential hopping, no fixed positions.
• Hole moving in the same direction as \(\mathbf{E}\) versus electron moving in the opposite one. This is not due to the mechanism depicted in panel (a): low-energy holes accelerate in the direction of the field (as much as any electron residing in the same states at the top of a band would do) because of the negative effective mass in that region of the band dispersion.

Hopefully, the section above showed that a completely rigorous mathematical description of holes is perfectly possible and not even particularly complicated. Electrons and holes behave in completely equivalent ways and their dynamics is the one of Bloch states, which does not depend on such states being occupied or not.

In conclusion…

ImportantTake home messages…

At the end of this chapter you should know…

• Semiclassical dynamics. Acceleration theorem and wavepacket velocity: Newton’s law inside a band, and key approximations (SBA, EMA).
• Dynamics. Bloch oscillation and its breakdown, Zener tunnelling: how do BOs work, what are the limits to their observability? what rules Zener tunnelling? How good is SBA?
• Conductors and insulators. Zero current from a full band. Definition of Fermi energy and surface, connection with a crystal being a conductor or insulator.
• Hole picture. Definition of electron-hole duality, why do we use one or the other picture?

Glasbrenner, Eric P., and Wolfgang P. Schleich. 2023. “The Landau–Zener Formula Made Simple.” Journal of Physics B: Atomic, Molecular and Optical Physics 56 (10): 104001. https://doi.org/10.1088/1361-6455/acc774.

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1. Please note this is one of the hypotheses of the theorem: generally speaking, an electron at \(t=0\) could be in any state, not necessarily a Bloch state.

2. Exactly how small is somewhat undefined, and depends on many factors that will be discussed later on. As a basic rule of thumb, THE semiconductor Silicon has a gap of about \(1.12\,{\mathrm{eV}}\).