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 Incoming links: Effective mass for exponentially decaying tight-binding hopping

Past exam of the quantum condensed matter course of the University of Cambridge / 2021 / Section B / 4 / e / Solution 2026-10-18

Taking

E_{1} > 0

so that the zone centre is a minimum, the effective mass for exponentially decaying tight-binding hopping is

m^{*} (a) = \frac{ℏ ^{2}}{4 E _{1} a ^{2}} e^{a / a_{1}} .

Its logarithmic derivative is

\frac{1}{m ^{*}} \frac{d m ^{*}}{d a} = \frac{1}{a _{1}} - \frac{2}{a} .

The curve decreases for

a < 2 a_{1}

, has a unique minimum at

a = 2 a_{1}

, and increases for

a > 2 a_{1}

:

m_{m i n}^{*} = \frac{ℏ ^{2} exp ( 2 )}{16 E _{1} a _{1}^{2}} .

The formal limits are an

a^{- 2}

a \to 0

and an exponential divergence as

a \to \infty

. The small-separation extrapolation eventually invalidates the localized-orbital tight-binding model; the sketch describes the specified formula.

Figure 1.
Effective mass versus atomic spacing for exponentially decaying nearest-neighbour hopping
. Original plot of the specified tight-binding model, normalized by its minimum. The minimum occurs at $a = 2 a_{1}$ , where the logarithmic derivative of the hopping magnitude balances the explicit $a^{2}$ band-curvature factor. Details.

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