Gallium Arsenide
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The charge carrier density of GaAs at 31o C is 2.25x106 cm-3.
What is the charge carrier density (in cm-3) at 175oC?
(The bandgap of GaAs is 1.42 eV. )
I am trying with arrhenius relationship but get confused over "charge carrier density"
charge carrier density here means how many electrons per cm^3
will go into conductivity band
For temp1, 31C -> 304.2K
2.25*10^6 = exp( - 2.275*10^-19 / (1.381*10^-23 * 304.2))
For temp2, 175C -> 448.2K
x = exp( - 2.275*10^-19 / (1.381*10^-23 * 448.2))
put this into wolfram alpha
(2.25*10^6/x) = exp( - 2.275*10^-19 / (1.381*10^-23 * 304.2)) / exp( - 2.275*10^-19 / (1.381*10^-23 * 448.2))
=> 8.09*10^13
To determine the charge carrier density at 175°C, we can use the Arrhenius relationship, which relates the temperature dependence of a material's properties to the activation energy. However, it's important to note that the Arrhenius relationship primarily applies to the relationship between the rate of a chemical reaction and temperature.
In the context of charge carrier density, we need to consider the thermal generation and recombination processes that affect the number of charge carriers in a material. These processes can vary for different materials and may not solely depend on temperature.
To accurately determine the charge carrier density at different temperatures for GaAs, we need to consider its intrinsic properties, such as the bandgap and the effective masses of the charge carriers.
The charge carrier density in a semiconductor can be calculated using the following equation:
n = Nc * exp[-(Eg / 2kT) + (3/4) * ln(T / T0)]
Where:
n = Charge carrier density
Nc = Effective density of states in the conduction band
Eg = Bandgap energy
k = Boltzmann constant
T = Temperature
T0 = Reference temperature
In the provided question, the charge carrier density is given at 31°C (304K). To find the charge carrier density at 175°C (448K), we need to plug in the appropriate values into the equation.
Given:
Charge carrier density at 31°C = 2.25x10^6 cm^-3
Bandgap of GaAs = 1.42 eV
Temperature at 31°C = 304K
Temperature at 175°C = 448K
First, we need to convert the bandgap from eV to Joules:
Eg (J) = 1.42 eV * 1.6x10^-19 C/eV
Next, we can rearrange the equation to solve for Nc:
Nc = n * exp[(Eg / 2kT) - (3/4) * ln(T / T0)]
Finally, substitute the values into the equation and solve for the charge carrier density (n) at 175°C:
n = Nc * exp[-(Eg / 2kT) + (3/4) * ln(T / T0)]
It's important to note that this calculation assumes intrinsic properties of Gallium Arsenide and neglects any extrinsic effects such as doping or impurities that can significantly impact the charge carrier density.