Varactors: Capacitance - Voltage properties of a pn junction







PHT.301 Physics of Semiconductor Devices

Varactors: Capacitance - Voltage properties of a pn junction

A varactor - variable reactor - is a diode with a pn-junction in reverse bias, where the capacitance can be varied by changing the applied voltage. The capacitance changes because the depletion width changes with voltage. Varactors are typically used as voltage-controlled capacitors and have largely replaced rotary capacitors. They are widely used in amplifiers, for frequency selection in radio receivers, for frequency modulation, in frequency multipliers, in filters and for oscillator circuits. This page numerically calculates the C-V relationship for an arbitrary doping profile.

There are analytic formulas for the relationship between the applied voltage and the capacitance of an abrupt junction, a linearly graded junction, and a junction where the charge density in the depletion region has the form $\rho \propto \mathrm{sgn}(x)\mathrm{abs}(x)^m$. A value of $m=-1$ results in a linear relation between capacitance and voltage. Since a completely abrupt junction is impossible to fabricate, a $\tanh(x)$ function can be used to smooth out the doping profiles.

Constant source diffusion

When a uniformly doped substrate is put in an oven with a gas that contains a dopant of the opposite polarity, a pn junction is formed with a constant source diffusion doping profile. The dopant that diffuses in has a concentration profile of the form $C_0\mathrm{erfc}\left(\frac{x}{\sqrt{4Dt}}\right)$. Here, $\mathrm{erfc}(x)$ is the complementary error function, $D$ is the diffusion constant of the dopant, and $t$ is the time that the diffusion occurs.

Limited source diffusion

Sometimes a thin glass layer containing dopants is spun on a uniformly doped substrate. The glass layer is patterned to determine where the dopants should enter the substrate, and then it is put into an oven to let the dopants diffuse. This results in a limited source diffusion doping profile. The dopant that diffuses in has a concentration profile of the form $C_0\exp\left(\frac{-x^2}{4Dt}\right)/\sqrt{4\pi Dt}$. Here, $D$ is the diffusion constant of the dopant, and $t$ is the time that the diffusion occurs.

PIN junctions

In a PIN diode, an undoped intrinsic region separates the p- and n-doped regions of the diode. The capacitance of a PIN diode is similar to a pn-diode in reverse bias, but the diode does not conduct in forward bias due to the intrinsic region.

Enter the doping profiles of the donors $N_D(x)$ and the acceptors $N_A(x)$ as a function of $x$ in microns in the form below. First, the program searches the interval between $x_1$ and $x_2$ for the condition $N_D(x)=N_A(x)$. The point where the two doping concentrations are equal is the position of the pn-junction. If there are more than one junction in the interval, adjust $x_1$ and $x_2$ so that only one junction appears.

$N_A(x)=$ 1E15*exp(-x) [1/cm³]
$N_D(x)=$ 1E15 [1/cm³]
$D_c$ = eV^-3/2cm^-3 $D_v$ = eV^-3/2cm^-3 $E_g =$ eV $\epsilon_r = $ $T = $ K
in the range from $x_1=$ [μm] to $x_2=$ [μm].

Semiconductors: the semiconductors are modeled using the Boltzmann approximation.
Diode types:

The bandgap is $E_g=$ eV.

$\log (N_D)$ $\log (N_A)$ [1/cm³]
	$x$ [μm]

The pn-junction is at $x_{pn}=$ μm.

The charge density is numerically integrated to determine the electric field and the electric field is integrated to determine the electrostatic potential under the assumption that the entire region from $x_1$ to $x_2$ is depleted.

$x$ [m] $\large \frac{\rho(x)}{\epsilon_r\epsilon_0}$ [V/m²]

$\large \frac{\rho(x)}{\epsilon_r\epsilon_0}$ [V/m²]
	$x$ [μm]

The electric field is the integral of the charge density,

$\large E(x)=\frac{1}{\epsilon_r\epsilon_0}\int\limits_{x_1}^{x}\rho(x')dx'$.

The integral is calculated numerically using a method called Simpson's rule.

$x$ [m] $E(x)$ [V/m]

$E(x)$ [V/m]
	$x$ [μm]

The electrostatic potential is minus the integral of the electric field,

$\large \varphi(x) = -\int \limits_{x_1}^{x} E(x')dx'.$

$x$ [m] $\varphi(x)$ [V]

$\varphi$ [V]
	$x$ [μm]

The electric field must go to zero at edges of the depletion width so the program draws horizontal lines on the electric field plot. Where a horizontal line cuts the curve for the electric field determines the positions of the edges of the depletion width. The horzontal line is translated to $E=0$ and the the $\varphi(x)$ plot is used to determine the electrostatic potential across the junction for that depletion width. This gives the black curve for the depletion width $W$ as a function of $V_{bi}-V$. The built-in voltage is then determined from the band diagram. The doping is calculated at the edges of the depletion width for every depletion width $W$ and the built-in voltage is calculated from $eV_{bi} = E_g - (E_c-E_F)_n - (E_F-E_v)_p$. Where the red line crosses the black line is $V_{bi}$ at $V=0$.

$V_{bi}-V$ [V] $W$ [μm]

$W$ [μm]
	$V_{bi} -V$

Here $V_{bi}=$ V is the built-in voltage and $V$ is the forward bias voltage (p-side is more positive than the n-side). Below, the capacitance is plotted against the applied reverse bias voltage.

$V_R$ [V] $C$ [nF/cm²]

$C$ [nF/cm²]
	$V_R$

$V_R$ is the applied reverse bias voltage (positive $V_R$ means diode is in reverse bias with the n-side is more positive than the p-side).