PN-Junction Diode

The following table contains the model parameters for the pn-junction diode model.

Name	Symbol	Description	Unit	Default
Is	$I_{S}$	saturation current	$\ampere$	$10^{-14}$
N	$N$	emission coefficient		$1.0$
Isr	$I_{SR}$	recombination current parameter	$\ampere$	$0.0$
Nr	$N_{R}$	emission coefficient for Isr		$2.0$
Rs	$R_{S}$	ohmic resistance	$\ohm$	$0.0$
Cj0	$C_{j0}$	zero-bias junction capacitance	$\farad$	$0.0$
M	$M$	grading coefficient		$0.5$
Vj	$V_{j}$	junction potential	$\volt$	$0.7$
Fc	$F_{c}$	forward-bias depletion capacitance coefficient		$0.5$
Cp	$C_{p}$	linear capacitance	$\farad$	$0.0$
Tt	$\tau$	transit time	$\second$	$0.0$
Bv	$B_v$	reverse breakdown voltage	$\volt$	$\infty$
Ibv	$I_{Bv}$	current at reverse breakdown voltage	$\ampere$	$0.001$
Kf	$K_F$	flicker noise coefficient		$0.0$
Af	$A_F$	flicker noise exponent		$1.0$
Ffe	$F_{FE}$	flicker noise frequency exponent		$1.0$
Temp	$T$	device temperature	$\degree \mathrm{C}$	$26.85$
Xti	$X_{TI}$	saturation current exponent		$3.0$
Eg	$E_G$	energy bandgap	eV	$1.11$
Tbv	$T_{Bv}$	Bv linear temperature coefficient	$1/\degree \mathrm{C}$	$0.0$
Trs	$T_{RS}$	Rs linear temperature coefficient	$1/\degree \mathrm{C}$	$0.0$
Ttt1	$T_{\tau 1}$	Tt linear temperature coefficient	$1/\degree \mathrm{C}$	$0.0$
Ttt2	$T_{\tau 2}$	Tt quadratic temperature coefficient	$1/\degree \mathrm{C}^2$	$0.0$
Tm1	$T_{M1}$	M linear temperature coefficient	$1/\degree \mathrm{C}$	$0.0$
Tm2	$T_{M2}$	M quadratic temperature coefficient	$1/\degree \mathrm{C}^2$	$0.0$
Tnom	$T_{NOM}$	temperature at which parameters were extracted	$\degree \mathrm{C}$	$26.85$
Area	$A$	default area for diode		$1.0$

Large signal model

Figure 10.2: pn-junction diode symbol and large signal model

The current equation of the diode and its derivative writes as follows:

$$I_{d} = I_{S}\cdot \left(e^{\frac{V_{d}}{N\cdot V_{T}}} - 1\right) + I_{SR}\cdot \left(e^{\frac{V_{d}}{N_R\cdot V_{T}}} - 1\right)$$ (10.10)

$$g_{d} = \dfrac{\partial I_{d}}{\partial V_{d}} = \dfrac{I_{S}}{N\cdot V... ... V_{T}}} + \dfrac{I_{SR}}{N_R\cdot V_{T}}\cdot e^{\frac{V_{d}}{N_R\cdot V_{T}}}$$ (10.11)

Figure 10.3: accompanied DC model of intrinsic diode

The complete MNA matrix entries are:

$$\begin{bmatrix}g_{d} & -g_{d}\\ -g_{d} & g_{d}\\ \end{bmatrix} \c... ...n{bmatrix}+I_{d} - g_{d}\cdot V_{d}\\ -I_{d} + g_{d}\cdot V_{d}\\ \end{bmatrix}$$ (10.12)

Small signal model

Figure 10.4: small signal model of intrinsic diode

The voltage dependent capacitance consists of a diffusion capacitance, a junction capacitance and an additional linear capacitance which is usually modeled by the following equations.

$$C_{d} = C_p + \tau \cdot g_{d} + \begin{cases}\begin{array}{ll} C... ..._c\right)}\right) & \textrm{ for } V_{d} > F_c\cdot V_j \end{array} \end{cases}$$ (10.13)

The S-parameters of the passive circuit shown in fig. 10.4 can be written as

$$S_{11} = S_{22} = \dfrac{1}{1 + 2\cdot y}$$ (10.14)

$$S_{12} = S_{21} = 1 - S_{11} = \dfrac{2\cdot y}{1 + 2\cdot y}$$ (10.15)

with

$$y = Z_{0}\cdot \left(g_{d} + j\omega C_{d}\right)$$ (10.16)

Noise model

The thermal noise generated by the series resistor is characterized by the following spectral density.

$$\dfrac{\overline{i_{R_S}^2}}{\Delta f} = \dfrac{4 k_B T}{R_S}$$ (10.17)

Figure 10.5: noise model of intrinsic diode

The shot noise and flicker noise generated by the DC current flow through the diode is characterized by the following spectral density.

$$\dfrac{\overline{i_{d}^2}}{\Delta f} = 2e I_d + K_F \dfrac{I_d^{A_F}}{f^{F_{FE}}}$$ (10.18)

Thus the noise current correlation matrix can be formed. This matrix can be easily converted to the noise wave correlation matrix representation using the formulas given in section 2.4.2 on page .

$$\underline{C}_Y = \Delta f \begin{bmatrix}+\overline{i_{d}^2} & -\overline{i_{d}^2}\\ -\overline{i_{d}^2} & +\overline{i_{d}^2}\\ \end{bmatrix}$$ (10.19)

An ideal diode (pn- or schottky-diode) generates shot noise. Both types of current (field and diffusion) contribute independently to it. That is, even though the two currents flow in different directions ("minus" in dc current equation), they have to be added in the noise equation (current is proportional to noise power spectral density). Taking into account the dynamic conductance $g_d$ in parallel to the noise current source, the noise wave correlation matrix writes as follows.

$$\begin{split}(\underline{C}) = \left\vert \frac{0.5\cdot Y_0}... ...cdot \begin{pmatrix}1 & -1\\ -1 & 1\\ \end{pmatrix} \end{split}$$ (10.20)

Where $e$ is charge of an electron, $V_T$ the temperature voltage, $g_d$ the (dynamic) conductance of the diode and $C_d$ its junction capacitance.

To be very precise, the equation above only holds for diodes whose field and diffusion current dominate absolutely (diffusion limited diode), i.e. $N=1$. Many diodes also generate a generation/recombination current ( $N\approx 2$), which produces shot noise, too. But depending on where and how the charge carriers generate or recombine, their effective charge is somewhat smaller than $e$. To take this into account, one needs a further factor $K$. Several opinions exist according the value of $K$. Some say 1 and 2/3 are common values, others say $K=1/N$ with $K$ and $N$ being bias dependent. Altogether it is:

$$\begin{split}(\underline{C}) = 2\cdot e\cdot Z_0\cdot K\cdot ... ...{pmatrix}\\ \text{with}\qquad\frac{1}{2}\le K \le 1 \end{split}$$ (10.21)

Remark: Believing the diode equation $I_D = I_S\cdot (\exp(V/(N\cdot V_T)) - 1)$ is the whole truth, it is logical to define $K=1/N$, because at $V=0$ the conductance $g_d$ of the diode must create thermal noise.

Some special diodes have additional current or noise components (tunnel diodes, avalanche diodes etc.). All these mechanisms are not taken into account in equation (10.21).

The parasitic ohmic resistance in a non-ideal diode, of course, creates thermal noise.

Noise current correlation matrix (for details on the parameters see above):

$$(\underline{C}_Y) = 2\cdot e\cdot K\cdot \left(I_{d} + 2\cdot I_{S}\right)\cdot \begin{pmatrix}1 & -1\\ -1 & 1\\ \end{pmatrix}\\$$ (10.22)

Temperature model

This section mathematically describes the dependencies of the diode characterictics on temperature. For a junction diode a typical value for $X_{TI}$ is $3.0$, for a Schottky barrier diode it is $2.0$. The energy band gap at zero temperature $E_G$ is by default $1.11$eV. For other materials than Si, $0.69$eV (for a Schottky barrier diode), $0.67$eV (for Ge) and $1.43$eV (for GaAs) should be used.

$$n_i^2\left(T\right) = B\cdot T^3 \cdot e^{-E_G\left(T\right)/k_B T}$$ (10.23)

$$n_i\left(T\right) = 1.45\cdot 10^{10}\cdot \left(\dfrac{T}{300K}\right)^{1.5}\cdot\... ...ot k_B\cdot 300K} - \dfrac{e\cdot E_G\left(T\right)}{2\cdot k_B\cdot T}\right)}$$ (10.24)

$$E_G\left(T\right) = E_G - \dfrac{\alpha\cdot T^2}{\beta + T}$$ (10.25)

with experimental values for Si given by

$$\alpha = 7.02\cdot 10^{-4}$$

$$\beta = 1108$$

$$E_G = 1.16eV$$

The following equations show the temperature dependencies of the diode parameters. The reference temperature $T_1$ in these equations denotes the nominal temperature $T_{NOM}$ specified by the diode model.

$$I_S\left(T_2\right) = I_S\left(T_1\right)\cdot \left(\dfrac{T_2}{T_1}\right)^{X_{TI} ... ...00K\right)}{N\cdot k_B\cdot T_2}\cdot \left(1 - \dfrac{T_2}{T_1}\right)\right]}$$ (10.26)

$$V_j\left(T_2\right) = \dfrac{T_2}{T_1}\cdot V_j\left(T_1\right) + \dfrac{2\cdot k_B\c... ..._2}{e} \cdot \ln{\left(\dfrac{n_i\left(T_1\right)}{n_i\left(T_2\right)}\right)}$$ (10.27)

$$= \dfrac{T_2}{T_1}\cdot V_j\left(T_1\right) - \dfrac{2\cdot k_B\c... ...- \left(\dfrac{T_2}{T_1} \cdot E_G\left(T_1\right) - E_G\left(T_2\right)\right)$$ (10.28)

$$C_{j0}\left(T_2\right) = C_{j0}\left(T_1\right)\cdot\left(1 + M\cdot\left(400\cdot 10^{-... ...c{V_j\left(T_2\right) - V_j\left(T_1\right)}{V_j\left(T_1\right)}\right)\right)$$ (10.29)

Some additionial temperature coefficients determine the temperature dependence of even more model parameters.

$$B_{v}\left(T_2\right) = B_{v}\left(T_1\right) - T_{Bv}\cdot\left(T_2 - T_1\right)$$ (10.30)

$$\tau\left(T_2\right) = \tau\left(T_1\right)\cdot\left(1 + T_{\tau 1}\cdot\left(T_2 - T_1\right) + T_{\tau 2}\cdot\left(T_2 - T_1\right)^2\right)$$ (10.31)

$$M\left(T_2\right) = M\left(T_1\right)\cdot\left(1 + T_{M1}\cdot\left(T_2 - T_1\right) + T_{M2}\cdot\left(T_2 - T_1\right)^2\right)$$ (10.32)

$$R_S\left(T_2\right) = R_S\left(T_1\right)\cdot\left(1 + T_{RS}\cdot\left(T_2 - T_1\right)\right)$$ (10.33)

Area dependence of the model

The area factor $A$ used in the diode model determines the number of equivalent parallel devices of the specified model. The diode model parameters affected by the $A$ factor are:

$$I_S\left(A\right) = I_S\cdot A$$ (10.34)

$$C_{j0}\left(A\right) = C_{j0}\cdot A$$ (10.35)

$$R_S\left(A\right) = \dfrac{R_S}{A}$$ (10.36)