Diode modelling#Ideality factor

The Shockley diode equation relates the diode current $I$ of a p-n junction diode to the diode voltage $V\_D$. This relationship is the diode I-V characteristic:

::$I\; =\; I\_S\backslash left(e^\backslash frac\{V\_D\}\{nV\_\backslash text\{T\}\}\; -\; 1\backslash right)$,

where $I\_S$ is the saturation current or scale current of the diode (the magnitude of the current that flows for negative $V\_D$ in excess of a few $V\_\backslash text\{T\}$, typically 10⁻¹²{{nbsp}}A). The scale current is proportional to the cross-sectional area of the diode. Continuing with the symbols: $V\_\backslash text\{T\}$ is the thermal voltage ($kT/q$, about 26 mV at normal temperatures), {{anchor|Ideality factor}}and $n$ is known as the diode ideality factor (for silicon diodes $n$ is approximately 1 to 2).

When $V\_D\backslash gg\; nV\_\backslash text\{T\}$ the formula can be simplified to:

::$I\; \backslash approx\; I\_S\; \backslash cdot\; e^\backslash frac\{V\_D\}\{nV\_\backslash text\{T\}\}$.

This expression is, however, only an approximation of a more complex I-V characteristic. Its applicability is particularly limited in case of ultra-shallow junctions, for which better analytical models exist..

{{cite journal

|doi=10.1109/TED.2008.2009028

|last1=Popadic

|first1=Miloš

|last2=Lorito

|first2=Gianpaolo

|last3=Nanver

|first3=Lis K.

|title=Analytical Model of I – V Characteristics of Arbitrarily Shallow p-n Junctions

|year= 2009

|pages=116–125

|volume=56

|journal=IEEE Transactions on Electron Devices

|issue=1

|bibcode=2009ITED...56..116P

|url=http://resolver.tudelft.nl/uuid:5f28db46-5b7b-4444-89b0-1b84db666fa9

}}

To illustrate the complications in using this law, consider the problem of finding the voltage across the diode in Figure 1.

File:Diode Modelling Image2.svg

Because the current flowing through the diode is the same as the current throughout the entire circuit, we can lay down another equation. By Kirchhoff's laws, the current flowing in the circuit is

::$I\; =\; \backslash frac\{V\_S\; -\; V\_D\}\{R\}$.

These two equations determine the diode current and the diode voltage. To solve these two equations, we could substitute the current $I$ from the second equation into the first equation, and then try to rearrange the resulting equation to get $V\_D$ in terms of $V\_S$. A difficulty with this method is that the diode law is nonlinear. Nonetheless, a formula expressing $I$ directly in terms of $V\_S$ without involving $V\_D$ can be obtained using the Lambert W-function, which is the inverse function of $f(w)\; =\; we^w$, that is, $w\; =\; W(f)$. This solution is discussed next.

An explicit expression for the diode current can be obtained in terms of the Lambert W-function (also called the Omega function).{{Cite journal|title = Exact analytical solution for current flow through diode with series resistance|journal = Electronics Letters|volume = 36|issue = 4|pages = 291|doi = 10.1049/el:20000301|first1 = T.C.|last1 = Banwell|first2 = A.|last2 = Jayakumar|year = 2000| bibcode=2000ElL....36..291B }} A guide to these manipulations follows. A new variable $w$ is introduced as

::$w\; =\; \backslash frac\{I\_SR\}\{nV\_\backslash text\{T\}\}\; \backslash left(\backslash frac\; \{I\}\{I\_S\}\; +\; 1\backslash right)$.

Following the substitutions $I/I\_S\; =\; e^\{V\_D/nV\_\backslash text\{T\}\}\; -\; 1$:

::$w\; e^w\; =\; \backslash frac\{I\_SR\}\{nV\_\backslash text\{T\}\}\; e^\backslash frac\{V\_D\}\{nV\_\backslash text\{T\}\}\; e^\{\backslash frac\{I\_SR\}\{nV\_\backslash text\{T\}\}\; \backslash left(\backslash frac\{I\}\{I\_S\}\; +\; 1\; \backslash right)\}$

and $V\_D\; =\; V\_S\; -\; IR$:

::$w\; e^w\; =\; \backslash frac\{I\_SR\}\{nV\_\backslash text\{T\}\}\; e^\backslash frac\{V\_S\}\{nV\_\backslash text\{T\}\}\; e^\{\backslash frac\{-IR\}\{nV\_\backslash text\{T\}\}\}\; e^\{\backslash frac\{IR\; I\_S\}\{nV\_\backslash text\{T\}\; I\_S\}\}\; e^\backslash frac\{I\_SR\}\{nV\_\backslash text\{T\}\}$

rearrangement of the diode law in terms of w becomes:

::$w\; e^w\; =\backslash frac\{I\_SR\}\{nV\_\backslash text\{T\}\}\; e^\backslash frac\{V\_s\; +\; I\_sR\}\{nV\_\backslash text\{T\}\}$,

which using the Lambert $W$-function becomes

::$w\; =\; W\backslash left(\; \backslash frac\{I\_SR\}\{nV\_\backslash text\{T\}\}\; e^\backslash frac\{V\_s\; +\; I\_sR\}\{nV\_\backslash text\{T\}\}\backslash right)$.

The final explicit solution being

::$I\; =\; \backslash frac\{n\; V\_\backslash text\{T\}\}\{R\}\; W\backslash left(\; \backslash frac\{I\_SR\}\{nV\_\backslash text\{T\}\}\; e^\backslash frac\{V\_s\; +\; I\_sR\}\{nV\_\backslash text\{T\}\}\backslash right)\; -I\_S$.

With the approximations (valid for the most common values of the parameters) $I\_sR\; \backslash ll\; V\_S$ and $I/I\_S\; \backslash gg\; 1$, this solution becomes

::$I\; \backslash approx\; \backslash frac\{n\; V\_\backslash text\{T\}\}\{R\}\; W\; \backslash left(\backslash frac\{I\_S\; R\}\{n\; V\_\backslash text\{T\}\}\; e^\backslash frac\{V\_s\}\{n\; V\_\backslash text\{T\}\}\backslash right)$.

Once the current is determined, the diode voltage can be found using either of the other equations.

For large x, $W(x)$ can be approximated by $W(x)\; =\; \backslash ln\; x\; -\; \backslash ln\backslash ln\; x\; +\; o(1)$. For common physical parameters and resistances, $\backslash frac\{I\_S\; R\}\{n\; V\_\backslash text\{T\}\}\; e^\backslash frac\{V\_s\}\{n\; V\_\backslash text\{T\}\}$ will be on the order of 10⁴⁰.

The diode voltage $V\_D$ can be found in terms of $V\_S$ for any particular set of values by an iterative method using a calculator or computer..

{{cite book

|author=A.S. Sedra and K.C. Smith

|title=Microelectronic Circuits

|year= 2004

|edition=Fifth

|publisher=Oxford

|location=New York

|isbn=978-0-19-514251-8

|pages=Example 3.4 p. 154

|url=http://worldcat.org/isbn/0195142519

|no-pp=true

}} The diode law is rearranged by dividing by $I\_S$, and adding 1. The diode law becomes

::$e^\backslash frac\{V\_D\}\{nV\_\backslash text\{T\}\}\; =\; \backslash frac\{I\}\{I\_S\}\; +\; 1$.

By taking natural logarithms of both sides the exponential is removed, and the equation becomes

::$\backslash frac\{V\_D\}\{nV\_\backslash text\{T\}\}\; =\; \backslash ln\; \backslash left(\backslash frac\{I\}\{I\_S\}\; +\; 1\backslash right)$.

For any $I$, this equation determines $V\_D$. However, $I$ also must satisfy the Kirchhoff's law equation, given above. This expression is substituted for $I$ to obtain

::$\backslash frac\{V\_D\}\{nV\_\backslash text\{T\}\}\; =\; \backslash ln\; \backslash left(\backslash frac\; \{V\_S\; -\; V\_D\}\{R\; I\_S\}\; +\; 1\backslash right)$,

or

::$V\_D\; =\; nV\_\backslash text\{T\}\; \backslash ln\; \backslash left(\backslash frac\{V\_S\; -\; V\_D\}\{R\; I\_S\}\; +\; 1\backslash right)$.

The voltage of the source $V\_S$ is a known given value, but $V\_D$ is on both sides of the equation, which forces an iterative solution: a starting value for $V\_D$ is guessed and put into the right side of the equation. Carrying out the various operations on the right side, we come up with a new value for $V\_D$. This new value now is substituted on the right side, and so forth. If this iteration converges the values of $V\_D$ become closer and closer together as the process continues, and we can stop iteration when the accuracy is sufficient. Once $V\_D$ is found, $I$ can be found from the Kirchhoff's law equation.

Sometimes an iterative procedure depends critically on the first guess. In this example, almost any first guess will do, say $V\_D\; =\; 600\backslash ,\backslash text\{mV\}$. Sometimes an iterative procedure does not converge at all: in this problem an iteration based on the exponential function does not converge, and that is why the equations were rearranged to use a logarithm. Finding a convergent iterative formulation is an art, and every problem is different.

File:Diode Modelling Image3.jpg

Graphical analysis is a simple way to derive a numerical solution to the transcendental equations describing the diode. As with most graphical methods, it has the advantage of easy visualization. By plotting the I-V curves, it is possible to obtain an approximate solution to any arbitrary degree of accuracy. This process is the graphical equivalent of the two previous approaches, which are more amenable to computer implementation.

This method plots the two current-voltage equations on a graph and the point of intersection of the two curves satisfies both equations, giving the value of the current flowing through the circuit and the voltage across the diode. The figure illustrates such method.

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File:Diode Modelling Image10.jpg

In practice, the graphical method is complicated and impractical for complex circuits. Another method of modelling a diode is called piecewise linear (PWL) modelling. In mathematics, this means taking a function and breaking it down into several linear segments. This method is used to approximate the diode characteristic curve as a series of linear segments. The real diode is modelled as 3 components in series: an ideal diode, a voltage source and a resistor.

The figure shows a real diode I-V curve being approximated by a two-segment piecewise linear model. Typically the sloped line segment would be chosen tangent to the diode curve at the Q-point. Then the slope of this line is given by the reciprocal of the small-signal resistance of the diode at the Q-point.

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File:Diode Modelling Image5.png

Firstly, consider a mathematically idealized diode. In such an ideal diode, if the diode is reverse biased, the current flowing through it is zero. This ideal diode starts conducting at 0 V and for any positive voltage an infinite current flows and the diode acts like a short circuit. The I-V characteristics of an ideal diode are shown below:

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Now consider the case when we add a voltage source in series with the diode in the form shown below:

File:Diode Modelling Image6.jpg

When forward biased, the ideal diode is simply a short circuit and when reverse biased, an open circuit.

If the anode of the diode is connected to 0{{nbsp}}V, the voltage at the cathode will be at Vt and so the potential at the cathode will be greater than the potential at the anode and the diode will be reverse biased. In order to get the diode to conduct, the voltage at the anode will need to be taken to Vt. This circuit approximates the cut-in voltage present in real diodes. The combined I-V characteristic of this circuit is shown below:

File:Diode Modelling Image8.png

The Shockley diode model can be used to predict the approximate value of $V\_t$.

::$\backslash begin\{align\}$

&I = I_S \left( e^\frac{V_D}{n \cdot V_\text{T}} - 1 \right) \\

\Leftrightarrow {} &\ln \left( 1 + \frac{I}{I_S} \right) = \frac{V_D}{n \cdot V_\text{T}} \\

\Leftrightarrow {} &V_D = n \cdot V_\text{T} \ln\left(1+\frac{I}{I_S}\right) \approx n \cdot V_\text{T} \ln \left( \frac{I}{I_S} \right) \\

\Leftrightarrow {} &V_D \approx n \cdot V_\text{T} \cdot \ln{10} \cdot \log_{10}{\left( \frac{I}{I_S} \right)}

\end{align}

Using $n\; =\; 1$ and $T\; =\; 25\backslash ,\backslash text\{°C\}$:

::$V\_D\; \backslash approx\; 0.05916\; \backslash cdot\; \backslash log\_\{10\}\{\backslash left(\; \backslash frac\{I\}\{I\_S\}\; \backslash right)\}$

Typical values of the saturation current at room temperature are:

• $I\_S\; =\; 10^\{-12\}\; \backslash text\{A\}$ for silicon diodes;
• $I\_S\; =\; 10^\{-6\}\; \backslash text\{A\}$ for germanium diodes.

As the variation of $V\_D$ goes with the logarithm of the ratio $\backslash frac\{I\}\{I\_S\}$, its value varies very little for a big variation of the ratio. The use of base 10 logarithms makes it easier to

think in orders of magnitude.

For a current of 1.0{{nbsp}}mA:

• $V\_D\; \backslash approx\; 0.53\backslash ,\backslash text\{V\}$ for silicon diodes (9 orders of magnitude);
• $V\_D\; \backslash approx\; 0.18\backslash ,\backslash text\{V\}$ for germanium diodes (3 orders of magnitude).

For a current of 100{{nbsp}}mA:

• $V\_D\; \backslash approx\; 0.65\backslash ,\backslash text\{V\}$ for silicon diodes (11 orders of magnitude);
• $V\_D\; \backslash approx\; 0.30\backslash ,\backslash text\{V\}$ for germanium diodes (5 orders of magnitude).

Values of 0.6 or 0.7 volts are commonly used for silicon diodes.

{{cite book

| last = Kal

| first = Santiram

| title = Basic Electronics: Devices, Circuits and IT Fundamentals

| chapter = Chapter 2

| year = 2004

| publisher = Prentice-Hall of India Pvt.Ltd

| isbn = 978-81-203-1952-3

| edition = Section 2.5: Circuit Model of a P-N Junction Diode

}}

The last thing needed is a resistor to limit the current, as shown below:

File:Diode Modelling Image9.jpg

The I-V characteristic of the final circuit looks like this:

File:Diode Modelling Image11.png

The real diode now can be replaced with the combined ideal diode, voltage source and resistor and the circuit then is modelled using just linear elements. If the sloped-line segment is tangent to the real diode curve at the Q-point, this approximate circuit has the same small-signal circuit at the Q-point as the real diode.

File:PWL diode characteristic.PNG

When more accuracy is desired in modelling the diode's turn-on characteristic, the model can be enhanced by doubling-up the standard PWL-model. This model uses two piecewise-linear diodes in parallel, as a way to model a single diode more accurately.

File:Diode 2piece linear model.png

File:Piecewise diode vs shockley1.PNG

{{clear}}

Using the Shockley equation, the small-signal diode resistance $r\_D$ of the diode can be derived about some operating point (Q-point) where the DC bias current is $I\_Q$ and the Q-point applied voltage is $V\_Q$. To begin, the diode small-signal conductance $g\_D$ is found, that is, the change in current in the diode caused by a small change in voltage across the diode, divided by this voltage change, namely:

::$g\_D\; =\; \backslash left.\backslash frac\{dI\}\{dV\}\backslash right|\_Q\; =\; \backslash frac\{I\_s\}\{n\; \backslash cdot\; V\_\backslash text\{T\}\}\; e^\backslash frac\{V\_Q\}\{n\; \backslash cdot\; V\_\backslash text\{T\}\}\; \backslash approx\; \backslash frac\{I\_Q\}\{n\; \backslash cdot\; V\_\backslash text\{T\}\}$.

The latter approximation assumes that the bias current $I\_Q$ is large enough so that the factor of 1 in the parentheses of the Shockley diode equation can be ignored. This approximation is accurate even at rather small voltages, because the thermal voltage $V\_\backslash text\{T\}\; \backslash approx\; 25\backslash ,\backslash text\{mV\}$ at 300{{nbsp}}K, so $V\_Q/V\_\backslash text\{T\}$ tends to be large, meaning that the exponential is very large.

Noting that the small-signal resistance $r\_D$ is the reciprocal of the small-signal conductance just found, the diode resistance is independent of the ac current, but depends on the dc current, and is given as

::$r\_D\; =\; \backslash frac\{n\; \backslash cdot\; V\_\backslash text\{T\}\}\{I\_Q\}$.

The charge in the diode carrying current $I\_Q$ is known to be

::$Q\; =\; I\_Q\backslash tau\_F\; +\; Q\_J$,

where $\backslash tau\_F$ is the forward transit time of charge carriers:{{cite book | title = Microelectronic Circuit Design | edition= second | author = R.C. Jaeger and T.N. Blalock | publisher = McGraw-Hill | year = 2004 | isbn = 978-0-07-232099-2 | url = https://books.google.com/books?id=u6vH4Gsrlf0C&dq=%22Microelectronic+Circuit+Design%22+inauthor:Jaeger+small-signal+diode&pg=PA883 }} The first term in the charge is the charge in transit across the diode when the current $I\_Q$ flows. The second term is the charge stored in the junction itself when it is viewed as a simple capacitor; that is, as a pair of electrodes with opposite charges on them. It is the charge stored on the diode by virtue of simply having a voltage across it, regardless of any current it conducts.

In a similar fashion as before, the diode capacitance is the change in diode charge with diode voltage:

::$C\_D\; =\; \backslash frac\{dQ\}\{dV\_Q\}\; =\; \backslash frac\{dI\_Q\}\{dV\_Q\}\; \backslash tau\_F\; +\; \backslash frac\{dQ\_J\}\{dV\_Q\}\; \backslash approx\; \backslash frac\{I\_Q\}\{V\_\backslash text\{T\}\}\; \backslash tau\_F\; +\; C\_J$,

where $C\_J\; =\; \backslash frac\{dQ\_J\}\{dV\_Q\}$ is the junction capacitance and the first term is called the diffusion capacitance, because it is related to the current diffusing through the junction.

The Shockley diode equation has an exponential of $V\_D/(kT/q)$, which would lead one to expect that the forward-voltage increases with temperature. In fact, this is generally not the case: as temperature rises, the saturation current $I\_S$ rises, and this effect dominates. So as the diode becomes hotter, the forward-voltage (for a given current) decreases.

Here is some detailed experimental data,{{cite web |url=http://www.cliftonlaboratories.com/1n400x_diode_family_forward_voltage.htm#Forward_Drop_versus_Temperature_and_Current |title=1n400x diode family forward voltage |author= |date=Apr 14, 2009 |website=www.cliftonlaboratories.com |publisher=Clifton Laboratories |access-date=2019-02-10 |archive-url=https://archive.today/20130906112448/http://www.cliftonlaboratories.com/1n400x_diode_family_forward_voltage.htm%23Forward_Drop_versus_Temperature_and_Current#Forward_Drop_versus_Temperature_and_Current |archive-date=September 6, 2013 |url-status=dead }} which shows this for a 1N4005 silicon diode. In fact, some silicon diodes are used as temperature sensors; for example, the CY7 series from OMEGA has a forward voltage of 1.02{{nbsp}}V in liquid nitrogen (77{{nbsp}}K), 0.54{{nbsp}}V at room temperature, and 0.29{{nbsp}}V at 100 °C.http://www.omega.com/Temperature/pdf/CY7.pdf datasheet

In addition, there is a small change of the material parameter bandgap with temperature. For LEDs, this bandgap change also shifts their colour: they move towards the blue end of the spectrum when cooled.

Since the diode forward-voltage drops as its temperature rises, this can lead to thermal runaway due to current hogging when paralleled in bipolar-transistor circuits (since the base-emitter junction of a BJT acts as a diode), where a reduction in the base-emitter forward voltage leads to an increase in collector power-dissipation, which in turn reduces the required base-emitter forward voltage even further.

• Bipolar junction transistor
• Semiconductor device modelling

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Category:Electronic device modeling