As the name implies, a PN junction is formed when a section of P-type material is joined to a section of N-type material. This simple junction forms the basis of most electronic devices. We will first examine the PN junction in equilibrium, with no externally applied bias and no net current in the material. Then we will study what happens when equilibrium is disturbed due to reverse bias (\(V_N > V_P\)) and forward bias (\(V_P>V_N\)).
When N-type and P-type materials are joined together, their mobile charges tend to diffuse across the junction. Mobile \(e^-\) and \(h^+\) are attracted to each other, so they tend to meet and annihilate. As a result, the space near the junction becomes depleted of mobile carriers. This region is called a depletion region or space charge region. The depletion region extends a distance \(x_n\) into the N side, and \(x_p\) into the P side, with total width \(W=x_n+x_p\). The junction characteristics are depicted in the figure below.
Although the mobile carriers are stripped away in the depletion region, the immobile charges remain at dopant atom sites. On the P-type side, each acceptor atom retains a surplus \(e^-\) in its orbit, resulting in an immobile negative charge. On the N-type side, each donor ion retains an immobile \(h^+\). There is consequently a net negative charge \(Q_P\) on the P-type side and a net positive charge \(Q_N\) on the N-type side within the depletion region. At equilibrium, these charges cancel each other: \(Q_P = -Q_N\).
Since there is a non-zero net charge on each side of the junction, there is a built-in electric field \(\mathcal{E}\) oriented from the N side toward the P side. And since there is an \(\mathcal{E}\)-field present, there is also a built-in potential \(V_0\) across the junction.
The remaining material outside the depletion region is called the quasi-neutral region. This region behaves like a homogenous extrinsic material: mobile charges balance out the fixed dopant ions, so there is zero net charge. At equilibrium, there should be no \(\mathcal{E}\)-field and no potential drop across the quasi-neutral region. With no \(\mathcal{E}\)-field present, the drift and diffusion currents should also average to zero.
Because the material is in equilibrium, the Fermi level is well-defined and we can use energy band theory to predict \(V_0\). As indicated in the figure below, the conduction and valence band energies can “bend” in the depletion region. The Fermi level is a constant for the entire system. This means that \(E_C\) and \(E_V\) must curve in order to accommodate a constant \(E_F\).
On the P-side, \(E_F\) is below \(E_i\) by \(kT\ln(N_A/n_i)\). On the N-side, \(E_F\) is above \(E_i\) by \(kT\ln\left(N_D/n_i\right)\). In order for \(E_F\) to join on both sides, the bands have to bend by the sum of these differences:
\[\begin{aligned} \Delta E &= kT\ln\left(N_A/n_i\right) + kT\ln\left(N_D/n_i\right)\\ &= kT\ln\left(\frac{N_AN_D}{n_i^2}\right)\end{aligned}\]
This result has units of electron volts. We convert this to volts by dividing out the elementary charge \(q\), which gives us the built-in potential:
\[\begin{aligned} V_0 &= \frac{kT}{q}\ln\left(\frac{N_AN_D}{n_i^2}\right)\\ &= U_T\ln\left(\frac{N_AN_D}{n_i^2}\right)\end{aligned}\]
Since the junction has a built-in potential drop \(V_0\) across the junction, there must be a corresponding \(\mathcal{E}\)-field. By applying Maxwell’s equations in one dimension, we can determine the relationship between \(V_0\), \(\mathcal{E}(x)\) and the fixed charges \(Q_N\) and \(Q_P\). First, let’s state the boundary conditions for this problem:
The net charge density, \(\rho(x)\),[, is equal to zero in the quasi-neutral regions. In the depletion region there are two sides:
P side (\(-x_p < x < 0\)): \(\rho(x) = qN_A\)
N side (\(0 < x < x_n\)): \(\rho(x) = -qN_D\)
At equilibrium, the net charge must balance to zero, so the total charge on the N side must be opposite the total charge on the P side. Therefore \(Q_P = -Q_N\). Since the charge density is constant in the depletion region, this means that
\(Q_P = -x_p\times qN_A\)
\(Q_N = x_n\times qN_D\)
Therefore
The \(\mathcal{E}\)-field is only present in the depletion region where a net charge is present; in the quasi-neutral regions it must drop to zero.
Since the charge density is negative on the P side and positive on the N side, the \(\mathcal{E}\)-field is oriented N-to-P, and is maximum at the junction (\(x=0\)).
We define the quasi-neutral potential on the P side to be 0, and on the N side as \(-V_0\,\).
From these boundary conditions, we can apply Poisson’s equation to obtain the \(\mathcal{E}\)-field and potential difference across the junction:
\[\begin{aligned} \mathcal{E} &= \int_{-x_p}^x \frac{-\rho(x)}{\epsilon}dx & (x<0)\\ &= \mathcal{E}_{\rm max} - \int_{0}^x \frac{\rho(x)}{\epsilon}dx & (x>0)\\ V_0 &= \int_{-x_p}^{x_n} \mathcal{E}dx\end{aligned}\]
where \(\rho(x)\) is the charge density in the material and \(\epsilon\) is the material’s permittivity. The charge density is equal to the dopant concentration, which is a non-zero constant on each side of the depletion region:
After integrating these constants, the \(\mathcal{E}\) field takes a triangle shape:
Finally the potential is found as the area under the triangle:
\[\begin{aligned} \mathcal{E}_{\textrm{\small max}} &= \frac{qN_Ax_p}{\epsilon} \\ &= \frac{qN_Dx_n}{\epsilon}\\ V_0 &= \frac{1}{2}W\mathcal{E}_{\textrm{\small max}}, \end{aligned}\]
Combining the above expressions for \(\mathcal{E}_{\text{max}}\) and \(V_0\) gives us:
\[V_0 = \frac{W}{2}\left(\frac{qN_Dx_n}{\epsilon}\right)\]
Now recalling the charge balance between \(Q_N\) and \(Q_P\), we can further simplify the expression:
\[\begin{aligned} W &= x_n + x_p,\\ x_p &= x_n\frac{N_D}{N_A}\\ \Rightarrow W &= x_n\left(1+\frac{N_D}{N_A}\right)\\ \Rightarrow x_n &= \frac{WN_A}{N_A+N_D}\end{aligned}\]
Now we can substitute for \(x_n\) into the solution for \(V_0\):
\[\begin{aligned} V_0 &= \frac{qWN_D}{2\epsilon}\left(\frac{WN_A}{N_D + N_A}\right) \\ &= W^2\left(\frac{1}{2\epsilon}\right)\left(\frac{N_AN_D}{N_A+N_D}\right)\\ \Rightarrow W &= \sqrt{\frac{2\epsilon}{q}\left(\frac{1}{N_A} + \frac{1}{N_D}\right) V_0}\end{aligned}\]
Up to this point, we know \(V_0\) from energy band theory, and we know \(W\) from Maxwell’s equations. We will add to this discussion one important special case: if the doping on one side of the junction is much greater than the other, then the depletion region extends almost totally into the weakly doped side. This case is called a one-sided junction. Since Si wafers are commonly weakly doped p-type, junctions are formed on the surface with much stronger N-type doping. Then \(N_D \gg N_A\), and the depletion depths are:
\[\begin{aligned} x_n &= x_p\frac{N_D}{N_A}\\ x_p &= x_n\frac{N_A}{N_D}\end{aligned}\]
So if \(N_D\) is \(10^4\times\) greater than \(N_A\), we expect \(x_p\) to be \(10^4\times\) greater than \(x_n\), so in the common case of a one-sided junction on the P side:
\[W\approx x_p~\text{(one~sided,~}N_D\gg N_A\text{).}\]
The depletion region behaves like a parallel-plate capacitor with charge stored in the depletion region. The quasi-neutral regions act as plates on the P and N sides, separated by the depletion width \(W\). This concept is illustrated in the figure below.
At equilibrium, the total junction capacitance is equal to
\[C_{j0}=A\frac{\epsilon_0\epsilon_r}{W}\]
where \(\epsilon_0\) is the vacuum permittivity constant and \(\epsilon_r\) is the relative permittivity of the material.
Next let’s consider what happens when a positive bias \(V_R\) is applied to the N side relative to the P side. The applied voltage must be superimposed on top of \(V_0\). Since the quasi-neutral regions are essentially conductive, almost all of the voltage drop occurs in the depletion region. As a result, the \(\mathcal{E}\)-field strength must increase and the depletion region must grow wider:
\[\begin{aligned} V_0 + V_R &= \frac{1}{2}W\mathcal{E}_{\rm max}\\ \Rightarrow W &= \sqrt{\frac{2\epsilon}{q}\left(\frac{1}{N_A} + \frac{1}{N_D}\right) \left(V_0+V_R\right)}\\ \mathcal{E}_{\rm max} &= \frac{2\left(V_0+V_R\right)}{W}\end{aligned}\]
From this we can see that the \(\mathcal{E}\)-field strength increases with \(V_R^{1/2}\), which increases the barrier against current flow. The energy band interpretation is illustrated below. In order for current to flow in the N\(\rightarrow\)P direction, the \(e^-\) must obtain energy greater than \(q(V_0+V_R)\,\text{eV}\), so that they can move from the N-side conduction band into the P-side conduction band.
In reverse bias, the junction current is not precisely zero. Even though carriers are blocked from entering the depletion region, a few carriers are continually present due to spontaneous thermal generation in the depletion region. From time to time, an \(e^-/h^+\) pair spontaneously appears and the opposite charges are swept apart by the \(\mathcal{E}\)-field, allowing a small current to flow in the device.
At neutral bias, these charges simply generate and recombine in an equilibrium process. When a reverse bias is applied, the minority charges are attracted to the device’s terminals, which disturbs equilibrium and sets up a net current. Since the rate of generation and recombination is uniform throughout the material (including the quasi-neutral regions), the leakage current is approximately constant and usually far below a micro ampere. In practical diodes, the leakage current may show a slight dependence on \(V_R\) due to surface conduction, caused by material changes on the outer surfaces of the diode.
Since the reverse-bias potential alters the depletion width, it alters the separation between the two “plates” of the depletion capacitance, thereby changing the capacitance. The original expression for \(C_{j0}\) is modified by superimposing \(V_R\) onto \(V_0\):
\[\begin{aligned} W &= \sqrt{\frac{2\epsilon}{q}\left(\frac{1}{N_A} + \frac{1}{N_D}\right) \left(V_0 + V_R\right)}\\ \Rightarrow C_j &= A\frac{\epsilon}{W} = A\sqrt{\frac{q\epsilon}{2}\left(\frac{N_AN_D}{N_A+N_D}\right)\frac{1}{V_0+V_R}} &= C_{j0}\sqrt{\frac{V_0}{V_0+V_R}}\\ &= C_{j0}\left(\frac{1}{1+V_R/V_0}\right)^{1/2}\end{aligned}\]
In real junctions, the capacitance does not perfectly follow the above solution. A more accurate model is \(C_j = C_{j0}\left(\frac{1}{1 + \frac{V_R}{V_0}}\right)^m\)
where \(m\) is called the grading coefficient, and is usually close to 0.5. The grading coefficient accounts for junctions that are “graded,” meaning there is not a perfectly abrupt transition between the P and N material.
When the reverse bias potential \(V_R\) exceeds a critical threshold, commonly called the Zener Voltage \(V_Z\), the junction is said to break down and begins conducting current. There are two mechanisms behind this breakdown. The first mechanism is called the avalanche effect, in which electrons’ average kinetic energy exceeds the ionization energy within the depletion region, leading to a chain reaction causing a huge increase in mobile carriers. The second mechanism is quantum tunneling, also called the Zener effect after Clarence Zener. When a junction is designed as a “Zener diode,” it means it is intended to be used at or near its breakdown point.
Both avalanche and tunneling mechanisms can occur in the same device. If the breakdown voltage occurs below 5.6, the tunneling mechanism most likely applies. For \(V_Z\) beyond 5.6, the avalanche mechanism usually applies. If \(V_Z\) is close to 5.6, a mixture of these mechanisms will occur. Avalanche and tunneling mechanisms may also occur in other electronic structures and applications.
The avalanche breakdown mechanism is caused by electron collisions in the semiconductor lattice. If the electron is very energetic, then it may transfer enough impact energy in the collision to bridge the bandgap \(E_g\). This will occur when the maximum \(\mathcal{E}\)-field in the junction exceeds a critical strength, \(\mathcal{E}_{\rm crit}\). The \(\mathcal{E}\)-field is related to the junction potential by the expression we solved previously:
\[\begin{aligned} \mathcal{E}_{\rm crit} &= \frac{2\left(V_0+V_{BR}\right)}{W}\\ W &= \sqrt{\frac{2\epsilon}{q}\left(\frac{1}{N_A} + \frac{1}{N_D}\right) \left(V_0+V_{BR}\right)}\\ \Rightarrow \mathcal{E}_{\rm crit} &= \frac{2\left(V_0+V_{BR}\right)}{\sqrt{\frac{2\epsilon}{q}\left(\frac{1}{N_A} + \frac{1}{N_D}\right) \left(V_0+V_{BR}\right)}}\\ \Rightarrow V_{BR} &= \frac{\epsilon\mathcal{E}_{\rm crit}^2}{2q}\left(\frac{1}{N_A} +\frac{1}{N_D}\right) - V_0.\end{aligned}\]
This indicates that the breakdown voltage is inversely proportional to the weaker doping between \(N_A\) and \(N_D\). So to achieve a low breakdown voltage (as is usually the case in Zener diodes), relatively strong doping is needed on both sides of the junction.
When the doping concentration is especially high, the built-in potential increases and the depletion width decreases. This results in a steep energy band transition, as shown in the figure below. At the level where \(E_c\) aligns with \(E_v\), separated by a small distance \(L\), it is possible for electrons to undergo quantum mechanical tunneling across the barrier. Quantum theory reveals that the tunneling current density is given by
\[J_{\rm tunnel} = qv_R n \exp\left(-\frac{4L\sqrt{2m^*E_g}}{3\hbar}\right),\]
where \(m^*\) is the effective electron mass, and \(v_R\) is an average velocity constant known as the Richardson velocity. Since the current is exponential in \(L\), the diode switches on it a manner similar to the forward bias ON current.
In practical terms, the main difference between tunneling and avalanche mechanisms is in their response to temperature. In each case, temperature effects are measured via the temperature coefficient parameter, defined as
\[\text{TC} \triangleq \frac{dV_{Z}}{dT},\]
with \(T\) being the temperature in Kelvin. For Zener breakdown (tunneling), the temperature coefficient is negative, so \(V_{Z}\) tends to shift to lower values at higher temperatures. For avalanche breakdown, the temperature coefficient is positive, so \(V_{Z}\) tends to increase with rising temperature. These effects can have significant consequences for some sensitive circuits. For critical applications, the two mechanisms can be made to balance each other out when \(V_{Z}\) is around 5.6, so it is possible to obtain a Zener diode with nearly zero temperature coefficient at that voltage. Typical temperature coefficients for Si Zener diodes are shown below. The relationship between \(V_Z\) and temperature coefficients is also evident on the example Zener Diode Datasheet. On datasheets, the TC is usually given in units of %/oC, where the percentage is relative to the nominal value of \(V_Z\).
Overview of ESD protection use cases from Texas Instruments
Datasheet of TPD1E10B06 ESD protection diode
A diode is called a “Zener diode” when it is intentionally used in the reverse breakdown mode. Usually this means it has a fairly low breakdown voltage. For a randomly chosen example, the TPD1E10B06 ESD protection chip uses two Zener diodes with \(V_Z=6{\text{V}}\).
At last we can examine the junction’s behavior in forward bias. Since the full analysis is quite complex, in this section we will just summarize the relevant mechanisms that contribute to current flow in forward bias. Starting from zero bias, where there is zero net current, the junction is in equilibrium. In the equilibrium condition, the drift and diffusion currents balance perfectly at the edges of the depletion region. If a forward voltage \(V_F\) is applied across the junction, it is superimposed on the built-in potential as \(V_0-V_F\). This means the depletion width must shrink. In order for this to happen, extra mobile carriers must be generated at the edges of the depletion zone. Newly generated carriers appear as electron/hole pairs. On each side of the depletion zone, the majority carriers are rejected from the depletion region by the strong \(\mathcal{E}\)-field. But diffusing minority carriers are accelerated by the field and swept into the depletion zone, where they establish a current.
Based on this reasoning, we conclude that the forward-bias current is due to diffusion of minority carriers, electrons on the P side and holes on the N side. The full analysis is quite complex, but we can obtain an approximate solution by revising the Fermi level theory: On the P side, let \(n_{p0}\) be the minority carrier concentration at equilibrium (here the \(n\) refers to electrons, the subscript \(p\) refers to the P-type side, and the subscript 0 indicates equilibrium). Similarly on the N side, let \(p_{n0}\) be the minority carrier concentration at equilibrium. We know from before that
\[\begin{aligned} n_0 &= N_C \exp\left(\frac{E_F - E_C}{kT}\right)\\ p_0 &= N_V \exp\left(\frac{E_V - E_F}{kT}\right)\\ n_0 p_0 &= n_i^2\end{aligned}\]
When equilibrium is disturbed due to a forward voltage \(V_F\), the Fermi level is no longer well defined, however a new pair of quasi-Fermi levels can be defined separately for electrons and holes. The separation between the respective quasi-Fermi levels is \(E_F^{(n)} - E_F^{(p)} = qV_F\). Quasi-Fermi levels represent the disturbed balance between minority and majority carriers, and can vary at different positions in the material.
The quasi-Fermi levels also disturb the mass-action law, as governed by the so-called Law of the Junction:
\[np = n_i^2\exp\left(\frac{V_F}{U_T}\right).\]
Since the majority concentration is set by the dopant concentration, the increased concentration appears mainly in the minority carriers. As a rough approximation, we suppose those new carriers appear as an impulse function at the very edge of depletion, so the minority carrier gradient is equal to the impulse height. Minority carriers will diffuse due to this sharp gradient in a process called minority carrier injection. Then the diffusion current should be
\[I = I_S \exp\left(\frac{V_F}{U_T}\right),\]
where \(I_S\) is a scale constant.
A full analysis of generation and recombination reveals the scale current to be
\[I_S = qA_{CS}\left(\frac{D_n n_{p0}}{L_n} + \frac{D_p p_{n0}}{L_p}\right),\]
where \(L_n\) and \(L_p\) are the mean diffusion length parameters for electrons and holes, respectively. The diffusion length is the distance a particle diffuses, on average, before it recombines. The diffusion lengths are obtainable from the diffusivity and the average carrier lifetime \(\tau\):
\[\begin{aligned} L_n &= \sqrt{D_n\tau_n}\\ L_p &= \sqrt{D_p\tau_p},\end{aligned}\]
where \(\tau_p\) and \(\tau_n\) are the recombination lifetimes of holes and electrons, respectively. Calculating the lifetimes is not simple, and they are usually measured empirically. The concept of carrier lifetime is important for several effects and applications. Since the diffusing carriers last for some time in the material, there is a limit on how quickly the diode can be switched between OFF and ON states. The current does not immediately shutoff when the bias is reversed, there is a delay called the reverse recovery time. This limitation is linked to the scale current by the expression above: a larger scale current implies smaller lifetimes and therefore faster switching, but also implies higher leakage when the diode is OFF. This is also called a hysteresis effect and can pose significant limitations in radio-frequency (RF) circuits and other fast-switching applications.
Since diffusing minority carriers occupy the junction for some time, we can say they are stored in the junction during their lifetime \(\tau\). This results in a diffusion capacitance that is distinct from the reverse-bias depletion capacitance. For strong forward bias, the diffusion capacitance becomes larger than the depletion capacitance, and increases exponentially with the applied forward voltage. This capacitance is defined differentially as
\[\begin{aligned} C_{\rm diff} &= \frac{dQ}{dV} \\ &= \frac{d_I}{d_V}\tau\\ &= \frac{I_D}{U_T}\tau,\end{aligned}\]
hence the capacitance is directly proportional to the forward-bias current \(I_D\). Note that in a one-sided junction, where the depletion region extends mostly into the weakly doped side (usually that’s the P side), the characteristics are dominated by the weaker doping. In that case, the average lifetime is \(\tau=\tau_p\).
The forward-bias diode current contains \(U_T\) in its exponent, giving it a strong temperature dependence. We have also seen that mobility, diffusivity and the minority carrier concentrations are all sensitive to temperature, and they all contribute to the diode’s scale current, so \(U_T\) is does not give an accurate prediction of the device’s temperature response. In most datasheets, the scale current \(I_S\) is measured at room temperature \(T_0\). If the temperature (in ) is changed by \(\Delta T\), the resulting change in \(I_S\) is approximately given by
\[I_S\left(T_0+\Delta T\right) \approx I_S\left(T_0\right)\exp\left(x_{ti}\Delta T\right),\]
where \(x_{ti}\) is a parameter called the temperature exponent. For most Si diodes, \(x_{ti}=2.0\). Other types of diodes may have different exponents.
While the temperature exponent is applicable to SPICE simulation models, for hand analysis we usually appeal to characteristic curves showing measured data for a particular diode. The curves below are for a 1N4148 Switching Diode:
In this figure, if we pick a specific operating voltage, like \(V_F=0.7\)V for instance, then we can chart the temperature dependence by moving vertically from the \(-40^\circ\)C curve, to the \(25^\circ\)C curve, to the \(125^\circ\)C curve. We can see that the current increases from about \(0.3\)mA to \(6\)mA to over \(40\)mA. This predicts a problem known as thermal runaway: higher current in the device means higher power dissipation, which generates heat, which raises the temperature, which increases the current. This positive feedback cycle can lead to the well-known “magic smoke” effect, where a device spontaneously combusts. To avoid thermal runaway, datasheets will specify a number of maximum voltage and current ratings which ensure the device can dissipate heat faster than it is created.
When a junction has very high doping, so that the depletion region becomes very thin, there can be quantum tunneling during forward bias operation as well as in reverse breakdown. The concept is illustrated by the energy band diagrams in the figure below. The forward tunnel current comes to a maximum at the peak voltage \(V_P\), where the conduction and valence bands are aligned. This is because the allowed energy states are mostly found close to \(E_C\) and \(E_V\). Most of the mobile electrons are sitting at \(E_C\), and they need valid energy states near \(E_V\) in order to tunnel.
When the diode’s foward voltage is increased higher than \(V_P\), the bands are moved out of alignment so that the tunneling current actually decreases with increased forward bias. This causes a behavior called negative differential resistance – the slope \(di/dv\) swings negative. This means that in the small-signal equivalent circuit, the tunnel diode behaves like a negative resistor. Negative resistance has a variety of exotic applications, but the most important use is in high-frequency RF and microwave oscillators. These applications were important enough to earn Nobel prizes for the tunnel diode’s inventor, Leo Esaki (tunnel diodes are also called “Esaki diodes” in his honor).
The tunnel diode’s current consists of a primary tunneling current, superimposed with secondary or excess tunneling currents, and also superimposed with a normal exponential diode current. The tunneling current follows an exponential model around a peak voltage:
\[I_{\rm tun} = \frac{V_D}{R_0}\exp\left[-\left(\frac{V_D}{V_0}\right)^m\right],\]
where \(R_0\) is the linear resistance in the junction’s quasi-neutral regions, \(V_0\) is a parameter in the range 0.1V to 0.5V, and \(m\) is an empirical parameter in the range 1 to 3. From this expression we can show that the peak current is obtained when the voltage is
\[V_P = \left(\frac{1}{m}\right)^m V_0.\]
The excess tunneling current determines the “valley” and is given by a similar exponential expression:
\[I_{\rm ex} = \frac{V_D}{R_V}\exp\left[\frac{V_D-V_V}{V_{\rm ex}}\right] ,\]
where \(V_V\) is the valley voltage (i.e. where \(I_D\) is minimum), and \(R_V\) and \(V_{\rm ex}\) are empirical parameters.
