An approximate one-dimensional quantum well can be formed by surrounding a layer of GaAs with layers of \(\mathrm{Al}_{x} \mathrm{Ga}_{1-x}\) As. The GaAs layers can be fabricated in thicknesses that are integral multiples of the single-layer thickness, \(0.28 \mathrm{nm}\). Some electrons in the GaAs layer behave as if they were trapped in a box. For simplicity, treat the box as an infinite one-dimensional well and ignore the interactions between the electrons and the Ga and As atoms (such interactions are often accounted for by replacing the actual electron mass with an effective electron mass). Calculate the energy of the ground state in this well for these cases: a) 2 GaAs layers b) 5 GaAs layers

Step by step solution

01

Calculate the size of the well for both cases

Since the GaAs layers can be fabricated at thicknesses that are integral multiples of the single-layer thickness (\(0.28nm\)), we calculate the size of the well for both cases: a) \(L_a = 2 \times 0.28 nm = 0.56 nm\) b) \(L_b = 5 \times 0.28 nm = 1.4 nm\) Convert these lengths to meters: a) \(L_a = 0.56 \times 10^{-9} m\) b) \(L_b = 1.4 \times 10^{-9} m\)

02

Calculate the effective mass of the electron in GaAs

To find out the effective mass of the electron in GaAs, we need to multiply the effective mass ratio (\(m^* \approx 0.067 m_e\)) by the electron mass (\(m_e \approx 9.11 \times 10^{-31} kg\)): \(m^* = 0.067 \times 9.11 \times 10^{-31} kg \approx 6.1 \times 10^{-32} kg\)

03

Calculate the ground state energy for both cases using the formula

Now we are ready to use the formula for the energy levels of an infinite potential well to calculate the ground state energy (\(n = 1\)) for both cases: a) \(E_{1a} = \frac{1^2 \pi^2 \hbar^2}{2 m^* L_a^2}\) b) \(E_{1b} = \frac{1^2 \pi^2 \hbar^2}{2 m^* L_b^2}\) a) \(E_{1a} = \frac{1^2 \pi^2 (6.58 \times 10^{-16} eV \cdot s)^2}{2 (6.1 \times 10^{-32} kg) (0.56 \times 10^{-9} m)^2} \approx 15.94 eV\) b) \(E_{1b} = \frac{1^2 \pi^2 (6.58 \times 10^{-16} eV \cdot s)^2}{2 (6.1 \times 10^{-32} kg) (1.4 \times 10^{-9} m)^2} \approx 2.26 eV\) Thus, the ground state energy for a GaAs well surrounded by two GaAs layers is around \(15.94 eV\), and for the well surrounded by five GaAs layers, it is around \(2.26 eV\).

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