Solution to Set 6

Problem 1.

Given

Aluminum(Al) is trivalent with

• atomic mass $m_{a}=27$ amu
• density $n=2.7g/cm^{3}$
• room temperature $T=293-296.5K$
• mean free time between electron collisions $t_{avg}=4{\rm {x}}10^{-14}$ s.

(a) Resistivity

Calculate the resistivity $\rho$ of aluminum(Al) at room temperature.

$\rho ={\frac {1}{\sigma }}\;$ $={\frac {1}{37.8\times 10^{6}Sm^{-1}}}\;$ $=2.82\times 10^{-8}\;$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 $1mm^{2}$

$R={\frac {\ell \cdot \rho }{A}}\;$ $={\frac {\left(10m\right)\cdot \left(2.646\times 10^{-8}\Omega \cdot m\right)}{\left(10^{-6}m\right)}}\;$ $=0.2646\Omega \;$

What is the current flowing through it?

$I={\frac {V}{R}}={\frac {2V}{0.2646\Omega }}=7.559A\;$

Problem 2

The resistivity of a certain material at room temperature is 0.02 Wm and the Hall coefficient is $5{\rm {x}}10^{-4}$ $m^{3}/C$. An electric field of 1 V/m is applied across it. Deduce all the information you can think of about this material.

Given

• Temperature $T=20^{\circ }C\;$
• Resistivity $\rho =0.02\Omega \cdot m\;$
• Hall coefficient $R_{H}=5{\rm {x}}10^{-4}{\tfrac {m^{3}}{C}}\;$
• Electric field $E=1{\tfrac {V}{m}}\;$

Deduction

• Conductivity

$\sigma ={\frac {1}{\rho }}={\frac {1}{0.02\Omega \cdot m}}=50{\tfrac {S}{m}}\;$

• Current Density

$\mathbf {J} ={\frac {E}{\rho }}={\frac {1{\tfrac {V}{m}}}{0.02\Omega \cdot m}}=50{\tfrac {A}{m^{2}}}\;$

• Magnetic Field

$R_{H}={\frac {E}{\mathbf {J} \cdot B}}\;$

$B={\frac {E}{\mathbf {J} \cdot R_{H}}}\;$ $={\frac {\left(1{\tfrac {V}{m}}\right)}{\left(50{\tfrac {A}{m^{2}}}\right)\left(5\times 10^{-4}m^{3}\right)}}\;$ 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 ( 5 \Omega \cdot m \right )}{\left ( 0.002m \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 = 2.247 \times 10^6 \tfrac{N}{C} \;}

• Fermi velocity

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.

(b) Fermi Energy & Velocity of 2D Gas

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