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 $1mm^{2}$

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

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

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

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 L = 5 mm
• Width W = 4 mm
• Thickness H = 2 mm
• Current I = 40 mA
• 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

Determine

1. 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^{-11}m^2}{0.002m} \;} 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 \times 10^{-7} \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}{5 \times 10^{-7} \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 = 2 \times 10^{6} \tfrac{S}{m} \;}

1. 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{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}{-1.602 \times 10^{-19}C} \cdot \frac{0.015V}{\left (0.04A \right ) \cdot \tfrac{\left (0.1T \right )}{\left (0.002m \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 = 1.170 \times 10^{22} m^{-3} \;}

1. Mobility

1. 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.