Built-In Potential: Formula, Derivation & Example

The built-in potential is an important concept in semiconductor physics, particularly in understanding the operation of a PN junction diode. It is the internal voltage difference established across the depletion region when P-type and N-type semiconductor regions reach thermal equilibrium.

The built-in potential develops because electrons and holes diffuse across the junction and recombine, leaving fixed ionized atoms behind. These charges create an electric field that opposes further diffusion of majority carriers.

The built-in potential depends on the semiconductor material, doping concentrations, and absolute temperature. It exists even when no external voltage is applied to the PN junction.

Table of Contents

Built-In Potential Formula

The built-in potential of a PN junction is calculated using the following equation:

{V_{bi}=V_T\ln\left(\frac{N_A N_D}{n_i^2}\right)}

Since thermal voltage is $V_T=kT/q$ , the formula can also be written as:

{V_{bi}=\frac{kT}{q}\ln\left(\frac{N_A N_D}{n_i^2}\right)}

Where:

$V_{bi}$ = built-in potential in volts (V)
$V_T$ = thermal voltage in volts
$N_A$ = acceptor doping concentration of the P-type region
$N_D$ = donor doping concentration of the N-type region
$n_i$ = intrinsic carrier concentration of the semiconductor
$K$ = Boltzmann constant, $1.380649\times10^{-23}\text{ J/K}$
$T$ = absolute temperature in kelvin (K)
$q$ = magnitude of the electronic charge, $1.602176634\times10^{-19}\text{ C}$
$\ln$ = natural logarithm

The doping concentrations and intrinsic carrier concentration must use the same units, such as $\text{cm}^{-3}$ or ${m}^{-3}$ .

Derivation of the Built-In Potential Formula

The built-in potential formula can be derived by applying the equilibrium carrier-concentration relationship to a PN junction.

Step 1: Electron concentration in the N-type region

In a non-degenerate N-type semiconductor with complete dopant ionization, the electron concentration is approximately equal to the donor concentration:

n_n\approx N_D

Where $n_n$ is the electron concentration in the N-type region.

Step 2: Electron concentration in the P-type region

For a semiconductor at thermal equilibrium, the mass-action law gives:

np=n_i^2

In the P-type region, the hole concentration is approximately equal to the acceptor concentration:

p_p\approx N_A

Therefore, the minority electron concentration in the P-type region is:

n_p=\frac{n_i^2}{N_A}

Where $n_p$ represents the electron concentration in the P-type region.

Step 3: Apply the equilibrium carrier relationship

At thermal equilibrium, the built-in potential is related to the electron concentrations on the two sides of the junction by:

\frac{n_n}{n_p}=e^{V_{bi}/V_T}

Taking the natural logarithm of both sides:

\ln\left(\frac{n_n}{n_p}\right)=\frac{V_{bi}}{V_T}

Rearranging:

V_{bi}=V_T\ln\left(\frac{n_n}{n_p}\right)

Step 4: Substitute the carrier concentrations

Using $n_n\approx N_D$ and $n_p=n_i^2/N_A$ :

V_{bi}=V_T\ln\left(\frac{N_D}{n_i^2/N_A}\right)

Simplifying the expression gives:

V_{bi}=V_T\ln\left(\frac{N_A N_D}{n_i^2}\right)

Therefore, the built-in potential formula is:

{V_{bi}=\frac{kT}{q}\ln\left(\frac{N_A N_D}{n_i^2}\right)}

This equation applies to an ideal PN junction at thermal equilibrium, subject to the usual non-degenerate semiconductor and complete-ionization approximations.

Built-In Potential at Room Temperature

At room temperature, the absolute temperature is commonly taken as 300 K. The thermal voltage is then:

V_T=\frac{kT}{q}

V_T\approx25.85\text{ mV}

Substituting this value into the built-in potential formula gives:

V_{bi}=0.02585\ln\left(\frac{N_A N_D}{n_i^2}\right)

This simplified expression can be used when the semiconductor is at 300 K and all concentrations are expressed in consistent units.

The built-in potential is not a fixed value for every silicon diode. Its magnitude depends on the doping concentrations and the intrinsic carrier concentration at the specified temperature.

Solved Example: Calculate the Built-In Potential

Problem: A silicon PN junction has an acceptor concentration of $10^{16}\text{ cm}^{-3}$ and a donor concentration of $10^{16}\text{ cm}^{-3}$ . If the intrinsic carrier concentration is $10^{10}\text{ cm}^{-3}$ at 300 K, calculate the built-in potential.

Given:

N_A=N_D=10^{16}\text{ cm}^{-3}

n_i=10^{10}\text{ cm}^{-3}

V_T=0.02585\text{ V}

Using the formula:

V_{bi}=V_T\ln\left(\frac{N_A N_D}{n_i^2}\right)

Substituting the values:

V_{bi}=0.02585\ln\left( \frac{10^{16}\times10^{16}}{(10^{10})^2} \right)

Simplifying:

V_{bi}=0.02585\ln(10^{12})

Since $ln(10^{12})=27.631$

V_{bi}=0.02585\times27.631

{V_{bi}\approx0.714\text{ V}}

Answer: The built-in potential of the given silicon PN junction is approximately 0.714 V at 300 K.

Factors Affecting the Built-In Potential

The built-in potential depends on the following factors:

Doping concentration: Higher acceptor or donor concentrations generally increase the built-in potential, provided the other parameters remain unchanged and the assumptions of the formula remain valid.
Temperature: Temperature changes the thermal voltage and intrinsic carrier concentration, both of which influence the built-in potential.
Semiconductor material: Different semiconductor materials have different intrinsic carrier concentrations, affecting the calculated potential.
Intrinsic carrier concentration: The built-in potential depends on the square of the intrinsic carrier concentration in the denominator of the formula.

Built-In Potential vs. Forward Voltage Drop

The built-in potential and forward voltage drop are related to PN junction behavior, but they are not the same quantity.

Built-in Potential	Forward Voltage Drop
Exists internally at thermal equilibrium.	Occurs across a diode carrying forward current.
Depends on doping, material, and temperature.	Depends on current, temperature, material, and diode construction.
Does not represent a directly measurable terminal voltage of an unbiased diode.	Can be measured across the diode during forward conduction.
Describes the equilibrium potential barrier.	Describes the terminal voltage under operating conditions.

Conclusion

The built-in potential is the internal voltage difference established across the depletion region of a PN junction at thermal equilibrium. It is calculated using $V_{bi}=V_T\ln(N_A N_D/n_i^2$ and depends on doping concentrations, semiconductor properties, and temperature. Understanding this potential is essential for analyzing the behavior of PN junction diodes.

Read Next:

Built-In Potential: Definition, Formula, and Derivation