SaM - Instrinsic Semiconductor • Density of Carriers • Energy Gap and Carriers Dependance on Temperature

Index

• Fermi Energy
• Electron and Hole Density of States Functions
• Density of Carriers
• Density of Carriers in Different Medium
• The Fermi-Dirac Dependence on Temperature
• The Fermi’s Energy Dependence on Temperature
• Energy Gap and Carriers Dependance on Temperature
• Instrinsic Mass Action Law

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Fermi Energy

We started by saying that at $0$ Kelvin all the electrons are here in the Valence Band, and no electrons are in the Conduction Band. What happens when we go to a temperature which is different from $0$ Kelvin? To understand this, we have to look at the probability that an electron occupies a given energy state (or level).

==The Fermi-Dirac distribution $f_e(E)$ gives the probability that given a certain energy state with an energy level value $E$, this value $E$ is occupied by the electron==, and it is defined as follows:$f_e(E)=\frac{1}{1+e^{\frac{E-E_f}{kT}}}$Where:

• $E$ is the energy that the electron takes.
• $k$ is the Boltzmann constant $=1.380649\cdot 10^{-23}\frac{\text{J}}{\text{K}}$.
• $T$ is the temperature in Kelvin.
• Whereas $E_F$ is a special energy level, which is called Fermi’s energy.

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Electron and Hole Density of States Functions

Then we have to remeber the density of states functions, these represent the density of states in the conduction band (at a certain energy level $E$): $$N_C(E) = \gamma\left(\frac{m_e^{*}}{m_0}\right)^{\frac{3}{2}}\sqrt{E-E_C}$$$N_C(E)$quantifieshowmany\ast \ast electronenergystates\ast \ast areavailableintheconductionbandofamaterialataparticularenergylevel$E$. It tells you how many electrons can potentially occupy a specific energy level in a material.

The density of states in the valance band (at a certain energy level $E$):$N_V(E)=\gamma {\left(\frac{m_h^{\ast }}{m_0}\right)}^{\frac{3}{2}}\sqrt{E_V-E}$Where:

• $\gamma =\frac{\sqrt{2}{M}_c}{{\hslash }^3{\pi }^2}{m}_0^{\frac{3}{2}}$: a really small value that represents the “effective density of states”.
  • $\hslash$ : planck constant $=\frac{h}{2\pi }\approx 1.0545718\times 10^{-34}\text{J}\cdot \text{s}$
• $m_e^{\ast }$ : effective mass of the electron (dependent on the material)
• $m_h^{\ast }$ : effective mass of the hole (dependent on the material)
• $m_0$ : mass of the electron $=9.10938356\cdot 10^{-31}\text{kg}$
• $E_C$ : Conduction band lowest energy, (rember the CB in defined by $E_C$).
• $E_V$ : Valance band highest energy, (rember the VB in defined by $E_V$).
• This functions can be represented as:
  

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Density of Carriers

Now if we wish to know the exact density of free electrons and free holes, so of carriers (free electrons and free holes will aid to conduction), we can integrate the density of states of the Conduction Band with the fermi energy like so, the density of electrons in the Conduction Band is:$n_i={\int }_{E_C}^{\infty }{N}_C(E)f_e(E)dE$And, the density of holes in the Valence Band is:$p_i={\int }_{-\infty }^{E_V}{N}_V(E)\left(1-f_e(E)\right)dE$Where:

• $\left(1-f_e(E)\right)$: is the probability of finding a hole at a certain energy $E$.

NOTE: We don’t care about the electrons in the valance band and of holes in the conduction band, since they are not carriers.

NOTE: We are talking about $n_i$ and $p_i$, the suffix ${}_i$ stands for instrinsic. They are the density of carriers present in an intrinsic semiconductor, we will use them only for Silicon, but this formulas are general for every intrisic semiconductor.

We will find that:$n_i=N_C{e}^{-\frac{E_F-E_C}{kT}}$And:$p_i=N_V{e}^{-\frac{E_V-E_F}{kT}}$Where, this time:

• $N_C$ and $N_V$ do not depend on $E$, we can think of them as constants depending on the material and temperature.
  For Silicon at room temperautre ($300°\text{K}$), we have that:
  • $N_C\simeq 2.8\cdot 10^{19}{\text{cm}}^{-3}$
  • $N_V\simeq 1.83\cdot 10^{19}{\text{cm}}^{-3}$
  • Also we have that $n_i\simeq 0.881\cdot 10^{10}{\text{cm}}^{-3}$

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Density of Carriers in Different Medium

So for intrinsic Silicon at room temperautre $n_i$ assumes the value that represent (approximately) an insulator.

Instead we have that, more generarly:

• For metals (conductors) we have that: $n_i\sim 10^{22}{\text{cm}}^{-3}$
• For insulators we have that: $n_i\sim 10{\text{cm}}^{-3}$

Silicon at room temperature is about in the middle, but closer to an insulator than a metal.

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The Fermi-Dirac Dependence on Temperature

We can see that the Fermi-Dirac Distribution depends on temperature looking at its formula:

• If we work at $0°\text{K}$ the formula assumes a steep curve, in fact at $0°\text{K}$ we have all electrons in the valance band (under the fermi energy).
• However at higher temperatures the curve gets more smooth.

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The Fermi’s Energy Dependence on Temperature

Not only the Fermi-Dirac probability ($f_e(E)$) depends on temperature, but also the Fermi’s Energy ($E_F$).

If we have an intrinsic semiconductor, at $0°\text{K}$ the Fermi’s Energy is defined like so:$E_F=E_i=\frac{E_C-E_V}{2}$Where:

• $E_i$ is called the Intrinsic Fermi’s Energy.
• $E_C-E_V=E_{gap}$, for silicon the gap spans for a total of $\simeq 1.12\text{eV}$.
• For Intrinsic Silicon $E_i\simeq 0.56\text{eV}$

While more in general the Fermi’s Energy is given by:$E_F=\frac{E_C-E_V}{2}+\frac{3}{4}kT\ln \left(\frac{m_e^{\ast }}{m_0}\right)$Although for intrinsic Silicon the ratio between $m_e^{\ast }$ and $m_0$ is usually very close to $1$, so $E_F$ will be almost equal to $E_g$, however $E_g$ too depends on temperature.

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Energy Gap and Carriers Dependance on Temperature

We have that:$n_i\propto T^{\frac{3}{2}}\cdot e^{-\frac{E_g}{2kT}}$While we can see the energy gap dependence on temperature as:$E_g(T)=E_g(0)+\frac{\alpha T^2}{T+\beta }$ So, since $n_i$ depends on $E_g$ we can draw this graph:

• We have that both $n_i$ and $E_g$ depend on temperature, so if we change the temperature too much we will have a change of behaviours, from insulator to metal (conductor) behaviour.

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Instrinsic Mass Action Law

In intrinsic Semiconductors we have that: $n_i^2=p_i^2$So we can simplify and say:$n_i=p_i=\sqrt{N_C{N}_V}{e}^{-\frac{E_g}{2kT}}$Where:

• $E_g$ is the energy gap, defined as $E_C-E_V$, for intrinsic silicon $\simeq 1.12\text{eV}$.

NOTE: In intrinsic semiconducotrs it is always true that:$n_i=p_i$

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Complete Notes

• The integral for $p_i$ is wrong:$N_V={\int }_{-\infty }^{E_V}{N}_V(E)[1-f_e(E)]dE$