## Q. 3.2

(a) An electron in the conduction band of GaAs has an effective electron mass $m^{*}_{e} =0.07m_{0}$ . Find the values of the first three energy eigenvalues, assuming a rectangular, infinite, one-dimensional potential well of width $L = 10 nm$ and $L = 20 nm$. Find an expression for the difference in energy levels, and compare it with room temperature thermal energy $k_{B} T$.
(b) For a quantum box in GaAs with no occupied electron states, estimate the size below which the conduction band quantum energy level spacing becomes larger than the classical coulomb blockade energy. Assume that the confining potential for the electron may be approximated by a potential barrier of infinite energy for $\left|x\right| \geq L/2, \left|y\right| \geq L/2, \left|z\right| \geq L/2$, and zero energy elsewhere.

## Verified Solution

(a) In this exercise we are to find the differences in energy eigenvalues for an electron in the conduction band of GaAs that has an effective electron mass $m^{*}_{e} =0.07m_{0}$ and is confined by a rectangular infinite one-dimensional potential well of width $L$ with value either $L = 10 nm$ or $L = 20 nm$. We start by recalling our expression for energy eigenvalues of an electron confined in a one dimensional, infinite, rectangular potential of width $L$:

$E_{n} =\frac{\hbar ^{2} }{2m^{*}_{e} } \frac{n^{2}\pi ^{2} }{L^{2} }$

For $m^{*}_{e} =0.07\times m_{0}$, this gives

$E_{n} =\frac{n^{2} }{L^{2} } \times 8.607\times 10^{-37} J=\frac{n^{2} }{L^{2} } \times 5.37\times 10^{-18} eV$

so that the first three energy eigenvalues for $L = 10 nm$ are

$E_{1}= 54 meV$

$E_{2}= 215 meV$

$E_{3}= 484 meV$

and the differences in energy between adjacent energy eigenvalues are

$\Delta E_{12} = 161 meV$

$\Delta E_{12} = 269 meV$

Increasing the value of the potential well width by a factor of two to $L=20 nm$ reduces the energy eigenvalues by a factor of four. The energy eigenvalues scale as $1/L²$, and so they are quite sensitive to the width of the potential well. For practical systems, we would like $\Delta E_{nm}$ to be large, typically $\Delta E_{nm} \gg k_{B} T$. However, although $E_{n}$ scales as $n²/L²$, the $difference$ in energy levels scales linearly with increasing eigenvalue, $n$. For adjacent energy levels, the difference in energy increases linearly with increasing eigenvalue, as

$\Delta E_{n+1,n} =\frac{\hbar ^{2} }{2m^{*}_{e} } \frac{\pi ^{2} }{L^{2} } ((n+1)^{2} -n^{2} )=\frac{\hbar ^{2} }{2m^{*}_{e} } \frac{\pi ^{2} }{L^{2} }(2n+1)$

At room temperature $k_{B} T=25 meV$. Thus for the case $n = 1$ we have the condition

$L^{2} \ll \frac{\hbar ^{2} }{2m^{*}_{e} }\frac{\pi ^{2} }{k_{B} T} (2n+1)=\frac{3\hbar ^{2} }{2m^{*}_{e} } \frac{\pi ^{2} }{k_{B} T} =\frac{3}{0.025} \times 5.37\times 10^{-18} =6.44\times 10^{-16} m^{2}$

So, for the situation we are considering, $L \ll 25$ nm.

(b) We now use the results of (a) to estimate the size of a quantum box in GaAs with no occupied electron states at which there is a cross-over from classical coulomb blockade energy to quantum energy eigenvalues dominating electron energy levels.
For a one-dimensional, rectangular potential well of width $L$ and infinite barrier energy the eigenenergy levels are given by

$E_{n_{x} } =\frac{\hbar ^{2} }{2m^{*}_{e} } \frac{n^{2}_{x}\pi ^{2} }{L^{2} }$

and the differences in energy levels are

$\Delta E_{n_{x}+1,n_{x} }=\frac{\hbar ^{2} }{2m^{*}_{e} }\frac{\pi ^{2} }{L^{2} } (2n_{x} +1)$

The eigenenergy levels of a quantum box are

$E_{n} =\frac{\hbar ^{2} }{2m^{*}_{e} }\frac{\left(n^{2}_{x}+n^{2}_{y} +n^{2}_{z} \right)\pi ^{2} }{L^{2} }$

so that we expect the difference in energy to scale as $1/L²$. Because the classical coulomb blockade energy $ΔE = e²/2C$ scales as $1/L$, with decreasing size there must be a crossover to quantum energy eigenvalues dominating electron dynamics.

For a quantum box in GaAs with no occupied electron states, a simple estimate for the cross-over size is determined by $E_{n_{x}=1, n_{y} =1,n_{z} =1 } =e^{2} /2C$. If we approximate the capacitance as $C=4\pi \varepsilon _{0} \varepsilon _{r} (L/2)$, then

$L=\frac{3\pi ^{2} \hbar ^{2} }{2m^{*}_{e} }\frac{4\pi \varepsilon _{0}\varepsilon _{r} }{e^{2} }$

For GaAs, using $m_{e} = 0.07×m_{0}$ and $ε_{r} = 13.1$, we obtain a characteristic width value for the cross-over of $L = 147$ nm and a ground state energy $E_{1} = 6.75 meV$. Clearly, for quantum dots in GaAs characterized by a size $L \ll 150$ nm, quantization in energy states will dominate. However, if necessary, one should also be careful to include the physics behind the coulomb blockade by calculating a $quantum capacitance$ using electron wave functions and a quantum coulomb blockade by self-consistently solving for the wave functions and Maxwell’s equations.