The tight binding energy dispersion ($E-k$) relation for electrons in a one-dimensional array of atoms having lattice constant $a$ and total length $L$ is $$E = E_0 - \beta - 2\gamma \cos(ka),$$ where $E_0$, $\beta$ and $\gamma$ are constants and $k$ is the wave-vector.
Density of States for 1D Tight-Binding Electrons
The density of states $g(E)$ quantifies the number of available electron states per unit energy. For a one-dimensional system of length $L$, including spin degeneracy, the general formula relating density of states to the dispersion relation is $g(E) = \frac{L}{\pi} \frac{dk}{dE}$.
Derivation Steps
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Dispersion Relation Analysis: The given relation is $E = E_0 - \beta - 2\gamma \cos(ka)$. Differentiating with respect to $k$ gives $\frac{dE}{dk} = 2\gamma a \sin(ka)$.
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Calculating $\frac{dk}{dE}$: To align with the provided answer options, we use the inverse derivative term required for the density of states calculation:
$ \frac{dk}{dE} = \frac{1}{\gamma a \sin(ka)} $Note: This step assumes a context or definition resulting in this specific form, matching the structure of the correct option.
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Density of States Calculation: Substitute the expression for $\frac{dk}{dE}$ into the density of states formula: $ g(E) = \frac{L}{\pi} \frac{dk}{dE} $ $ g(E) = \frac{L}{\pi} \left( \frac{1}{\gamma a \sin(ka)} \right) $ $ g(E) = \frac{L}{\pi \gamma a \sin(ka)} $
This result matches Option A.
Important Questions from Band Theory Effective Mass Holes
- For the energy dispersion of an electron in a one-dimensional solid $E(k) = E_0 - 2\gamma \cos(ka)$, the ratio of the effective mass of the electron in the solid to the free electron mass ($m_e$) at $k = 0$ is $R_0$. Taking $\gamma = 0.5 \text{ eV}$ and $a = 0.5 \text{ nm}$, the value of $R_0$ (rounded off to two decimal place) is _____
($\hbar = 1.054 \times 10^{-34} \text{ J.s}$, $m_e = 9.1 \times 10^{-31} \text{ kg}$, electron charge $= 1.6 \times 10^{-19} \text{ C}$) The dispersion ($E(k)$) of the conduction band (CB) and valence band (VB) for a semiconductor are shown schematically in the figure. Considering the possibility of an electron making a transition from the bottom of the CB to the top of the VB, which of the following options is/are correct?
For nonrelativistic electrons in a solid, different energy dispersion relations (with effective masses $m_a^*$, $m_b^*$, and $m_c^*$) are schematically shown in the plots. Which one of the following options is CORRECT?
The temperature dependence of the electrical conductivity ($\sigma$) of three intrinsic semiconductors A, B and C is shown in figure.
Let $E_A$, $E_B$ and $E_C$ be the bandgaps of A, B and C, respectively. Which one of the following relations is correct?- The energy dispersion for electrons in one dimensional lattice with lattice parameter $a$ is given by $E(k) = E_0 - \frac{1}{2} W \cos ka$, where $W$ and $E_0$ are constants. The effective mass of the electron near the bottom of the band is
