Solution to Set 6

Problem 1

Given

Aluminum(Al) is trivalent with

• atomic mass Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_{a} = 27} amu
• density Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = 2.7 g/cm^3}
• room temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T = 293 - 296.5 K}
• mean free time between electron collisions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_{avg} = 4{\rm x}10^{-14}} s.

(a) Resistivity

Calculate the resistivity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho} of aluminum(Al) at room temperature.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = \frac{1}{\sigma} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{1}{ 37.8 \times 10^{6} S m^{-1}} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 2.82 \times 10^{-8} \;} Ω·m

(b) Current

If a 2-V voltage is applied to the ends of an aluminum wire 10 m long and with a cross- sectional area of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 mm^2}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R = \frac{\ell \cdot \rho}{A} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{\left (10m \right ) \cdot \left (2.646 \times 10^{-8} \Omega \cdot m \right )}{\left (10^{-6}m \right )} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 0.2646 \Omega \;}

What is the current flowing through it?

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I = \frac{V}{R} = \frac{2V}{0.2646 \Omega} = 7.559 A \;}

Problem 2

The resistivity of a certain material at room temperature is 0.02 Wm and the Hall coefficient is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 5{\rm x}10^{-4}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m^3 /C} . An electric field of 1 V/m is applied across it. Deduce all the information you can think of about this material.

Given

• Temperature Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T = 20^{\circ} C \;}
• Resistivity Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = 0.02 \Omega \cdot m \;}
• Hall coefficient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{H} = 5{\rm x}10^{-4} \tfrac{m^{3}}{C} \;}
• Electric field Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E = 1 \tfrac{V}{m} \;}

Deduction

• Conductivity

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \frac{1}{\rho} = \frac{1}{0.02 \Omega \cdot m} = 50 \tfrac{S}{m} \;}

• Current Density

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{J} = \frac{E}{\rho} = \frac{1 \tfrac{V}{m}}{0.02 \Omega \cdot m} = 50 \tfrac {A}{m^{2}} \;}

• Magnetic Field

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_{H} = \frac{E}{\mathbf{J} \cdot B} \;}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B = \frac{E}{\mathbf{J} \cdot R_{H}} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{\left (1 \tfrac{V}{m} \right )}{\left ( 50 \tfrac{A}{m^{2}} \right ) \left ( 5 \times 10^{-4} m^3 \right )} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 40 T \;}

Problem 3.

(a) Hall Effect Sketch

Sketch a setup used to measure the Hall effect. Label each part.

(b) Semiconductor Crystal

A semiconductor crystal is 5 mm long, 4 mm wide, and 2 mm thick. A 40mA current flows across the length of the sample after a 2-V battery is connected to the ends. When a 0.1T magnetic field is applied perpendicular to the large surface of the specimen, a Hall voltage of 15mV develops across the width of the sample.

Given

• Length Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \ell} = 5 mm = 0.005 m
• Width W = 4 mm = 0.004 m
• Thickness H = 2 mm = 0.002 m
• Current I = 40 mA = 0.04 A
• Voltage V = 2 V
• Mag Field B = 0.1 T
• Hall Volt Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_H} = 15 mV = 0.015 V

You can deduce that:

• Area Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = L \times W = 2.0 \times 10^{-5} m^2}

Determine

• Conductivity

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho = R \cdot \frac {A}{\ell} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{V}{I} \cdot \frac {A}{\ell} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{2V}{0.04A} \cdot \frac{2 \times 10^{-5}m^2}{0.005m} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 0.2 \Omega \cdot m \;}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma = \frac{1}{\rho} = \frac{1}{0.2 \Omega \cdot m} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 5 \tfrac{S}{m} \;}

• Carrier density

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_H = \frac{E_y}{\mathbf{J_x} \cdot B} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{V_H}{I \cdot \tfrac{B}{\ell}} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = -\frac{1}{ne} \;}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n = -\frac{1}{e} \cdot \frac{I \cdot \tfrac{B}{\ell}}{V_H} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = - \frac{1}{-1.602 \times 10^{-19}C} \cdot \frac{\left (0.04A \right ) \cdot \tfrac{\left (0.1T \right )}{\left (0.005m \right )}}{0.015V} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 3.329 \times 10^{20} m^{-3} \;}

• Mobility

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{E} = \frac{1}{4 \pi \varepsilon_0 } \int \frac{\rho}{r^2} \mathbf{\hat{r}} dV \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{1}{4 \pi \left (8.854 \times 10^{-12} \tfrac{C^2}{N \cdot m^2} \right )} \int \frac{\left ( 0.2 \Omega \cdot m \right )}{\left ( 0.005m \right )^2} dV \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 3.595 \times 10^{11} \tfrac{N}{C} \;}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{F} = q \cdot \mathbf{E} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \left ( 1.602 \times 10^{-19} C \right ) \left ( 3.595 \times 10^{11} \tfrac{N}{C} \right ) \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 5.759 \times 10^{-8} N \;}

• Fermi velocity

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E_f = \frac{\hbar^2 \pi^2}{2 m_e \ell^2} n_f^2 \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \frac{\left (1.055 \times 10^{-34} J \cdot s \right )^2 \pi^2}{2 \cdot \left ( 9.109 \times 10^{-31} kg \right ) \left ( 0.005m \right )^2} \;} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = 2.412 \times 10^{-33} J \;}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V_f = \sqrt{\frac{2 E_f}{m_e}} \;} $={\sqrt {\frac {2\left(2.412\times 10^{-33}J\right)}{\left(9.109\times 10^{-31}kg\right)}}}\;$ $=0.0727715501{\tfrac {m}{s}}\;$

Problem 4

(a) Fermi Derivations

Derive the expressions for the Fermi energy, Fermi velocity, and electronic density of states for a two-dimensional free electron gas.

Fermi Energy

$E_{f}={\frac {\hbar ^{2}\pi ^{2}}{2m_{e}\ell ^{2}}}n_{f}^{2}\;$

Fermi Velocity

Fermi energy is related to fermi velocity such that:

$E_{F}={\tfrac {1}{2}}m_{e}v_{F}^{2}\;$

Solve for velocity

$v_{F}={\sqrt {\frac {2E_{F}}{m_{e}}}}\;$

Electronic Density

In 1 dimension

$E(k)={\frac {\hbar ^{2}k^{2}}{2m}}$

$k={\frac {2\pi m}{L}}\;$; $m=0,\pm 1,\pm 2,...$

$n={\frac {N}{L}}={\frac {2}{L}}\sum _{k<|k_{F}|}={\frac {2}{2\pi }}\sum _{k<|k_{F}|}{\frac {2\pi }{L}}={\frac {2}{2\pi }}2\int _{0}^{k_{F}}dk={\frac {2k_{F}}{\pi }}.$

Here the extra factor of 2 is due to spin

The other factor of two comes from the addition of the negative values for k

$E_{F}={\frac {\hbar ^{2}\pi ^{2}n^{2}}{8m}}.$

(b) Fermi Energy & Velocity of 2D Gas

A 2D electron gas formed in a GaAs/AlGaAs quantum well has a density of $2{\rm {x}}10^{11}cm^{-2}$. Assuming that the electrons there have the free electron mass, calculate the Fermi energy and Fermi velocity.