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Semiconductor Detector Systems$

Helmuth Spieler

Print publication date: 2005

Print ISBN-13: 9780198527848

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198527848.001.0001

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(p.447) APPENDIX E THE DIODE EQUATION

(p.447) APPENDIX E THE DIODE EQUATION

Source:
Semiconductor Detector Systems
Publisher:
Oxford University Press

Before entering a quantitative analysis of the diode characteristics, it is useful to review the relationships that determine carrier concentrations. To avoid confusion with exponential functions the symbol q e will be used for the electronic charge instead of e.

E.1 Carrier concentrations in pure semiconductors

In thermal equilibrium the probability that an electron state in the conduction band is filled is given by the Fermi–Dirac distribution

(E.1)
f e ( E ) = 1 e ( E E F ) / k T + 1 .
The parameter E F is the Fermi level (or chemical potential). Figure E.1 shows the distribution for several temperatures. The density of atoms in a Si or Ge crystal is about 5 · 1022 atoms/cm3. Since the minimum carrier density of interest in practical devices is of order 1010 to 1011 cm−3, very small ocupancies are quite important.

In silicon the bandgap is 1.12 eV. If the Fermi level is at midgap, the band-edges will be 0.56 eV above and below E F. As is apparent from Figure E.1, relatively large deviations from the Fermi level, i.e. extremely small occupancies, will still yield significant carrier densities.

APPENDIX E THE DIODE EQUATION

Fig. E.1 The Fermi–Dirac distribution plotted for various temperatures.

(p.448)

APPENDIX E THE DIODE EQUATION

Fig. E.2 Comparison between the Fermi–Dirac and Boltzmann distributions.

The number of occupied electron states N e is determined by summing over all available states multiplied by the occupation probability for each individual state

(E.2)
N e = i m i f ( E i )
Since the density of states near the band edge tends to be quite high, this can be written as an integral
(E.3)
N e = E c f ( E ) g ( E ) d E ,
where g(E) is the density of states. Solution of this integral requires knowledge of the density of states. Fortuitously, to a good approximation the density of states near the band edge has a parabolic distribution
(E.4)
g ( E ) d E ( E E c ) 1 / 2 .
As the energy increases beyond the band edge, the distribution will deviate from the simple parabolic form, but since the probability function decreases very rapidly, the integral will hardly be affected.

The second obstacle to a simple analytical solution of the integral is the intractability of integrating over the Fermi distribution. Fortunately, if EE F is at least several times kT, the Fermi distribution can be approximated by a Boltzmann distribution, as shown in Figure E.2. (p.449)

(E.5)
1 + e ( E E F ) / k T e ( E E F ) / k T f ( E ) e ( E E F ) / k T .

At energies 2.3 kT beyond the Fermi level the difference between the Boltzmann approximation and the Fermi distribution is < 10%, for energies > 4.5 kT it is less than 1%.

Applying the approximation to the occupancy of hole states, the probability of a hole state being occupied, i.e. a valence state being empty, is

(E.6)
f h ( E ) = 1 f e ( E ) = 1 e ( E F E ) / k T + 1 e ( E F E ) / k T .
The conditions for the Boltzmann approximation are fulfilled for excitation across the bandgap, as the bandgap is of order 1 eV and kT at room temperature is 0.026 eV.

With these simplifications the number of electrons in the conduction band in thermal equilibrium

(E.7)
n ( k T ) 3 / 2 e ( E c E F ) / k T
or
(E.8)
n = N c e ( E c E F ) / k T ,
where N c is the effective density of states at the band edge. Correspondingly, the hole concentration
(E.9)
p = N v e ( E F E v ) / k T .
In an ideal semiconductor the only source of mobile carriers is thermal excitation across the bandgap (additional impurity atoms or crystal imperfections that would allow other excitation mechanisms are absent), so the concentrations of electrons and holes are equal
(E.10)
n = p = n i .
n i is called the intrinsic carrier concentration. In silicon (E g = 1.12 eV) the intrinsic concentration n i = 1.45 · 1010 cm−3 at 300 K and in germanium (E g = 0.66 eV), n i = 2.4 · 1013 cm−3. For comparison, the purest semiconductor material that has been fabricated is Ge with active impurity levels of about 3 · 1010 cm−3.

Using the above results

(E.11)
n i = N c e ( E c E F ) / k T = N v e ( E F E v ) / k T ,
which one can solve to obtain
(E.12)
E F = E i = E c + E v 2 k T 2 log ( N c / N v ) .
If the band structure is symmetrical (N c = N v), the intrinsic energy level E i lies near the middle of the bandgap. Even rather substantial deviations from (p.450) a symmetrical band structure will not affect this result significantly, as N c/N v enters logarithmically and kT is much smaller than the bandgap.

A remarkable result is that the product of the electron and hole concentrations

(E.13)
n p = n i 2 = N c N v e ( E c E v ) / k T = N c N v e E g / k T
depends only on the bandgap E g and not on the Fermi level.

This result, the law of mass action, is very useful in semiconductor device analysis. It requires only that the Boltzmann approximation holds. Qualitatively, it says that if one carrier type exceeds this equilibrium concentration, recombination will decrease the concentrations of both electrons and holes to maintain n p = n i 2 , a relationship that also holds in doped crystals.

E.2 Carrier concentrations in doped crystals

The equality n = p only holds for pure crystals, where all of the electrons in the conduction band have been thermally excited from the valence band. In practical semiconductors the presence of impurities tips the balance towards either electrons or holes.

Impurities are an unavoidable byproduct of the crystal growth process, although special techniques can achieve astounding results. For example, as noted above, in the purest semiconductor crystals – “ultrapure” Ge – the net impurity concentration is about 3 · 1010 cm−3.

In semiconductor device technology impurities are introduced intentionally to control the conductivity of the semiconductor. Let N d + be the concentration of ionized donors and N a the concentration of ionized acceptors. Overall charge neutrality is preserved, as each ionized dopant introduces a charged carrier and an oppositely charged atom, but the net carrier concentration is now

(E.14)
Δ n = n p = N d + N a
or
(E.15)
p + N d + = n + N a .
Assume that the activation energy of the donors and acceptors is sufficiently small so that they are fully ionized. Then N d + = N d and N a = N a , so
(E.16)
p + N d = n + N a ,
which, using n p = n i 2 , becomes
(E.17)
p + N d = n i 2 p + N a p N a + N d N a = n i p n i N a + 1 .
If the acceptor concentration N aN d and N an i, the hole and electron concentrations (p.451)
(E.18)
p N a and  n n i 2 N a N a ,
i.e. the conductivity is dominated by holes. Conversely, if the donor concentration N dN a and N dn i the conductivity is dominated by electrons.

If the conductivity is dominated by only one type of carrier, the Fermi level is easy to determine. If, for example, np then eqn E.16 can be written as

(E.19)
n = N d N a N c e ( E c E F ) / k T = N d N a ,
yielding
(E.20)
E c E F k B T = log ( N c N d N a ) .
If N dN a, then E cE F must be small, i.e. the Fermi level lies close to the conduction band edge.

In reality the impurity levels of common dopants are not close enough to the band edge for the Boltzmann approximation to hold, so the calculation must use the Fermi distribution and solve numerically for E F. Nevertheless, the qualitative conclusions derived here still apply.

It is often convenient to refer all of these quantities to the intrinsic level E i, as it accounts for both E c and E v. Then

(E.21)
n = N c e ( E c E F ) / k T = n i e ( E F E i ) / k T p = N v e ( E F E v ) / k T = n i e ( E i E F ) / k T
and the Fermi level
(E.22)
E F E i = k B T log N a N d n i .

E.3 pn-junctions

A pn-junction is formed at the interface of a p- and an n-type region. Since the electron concentration in the n-region is greater than in the p-region, electrons will diffuse into the p-region. Correspondingly, holes will diffuse into the n-region. As electrons and holes diffuse across the junction, a space charge due to the ionized donor and acceptor atoms builds up. The field due to this space charge is directed to impede the flow of electrons and holes.

The situation is dynamic. The concentration gradient causes a continuous diffusion current to flow, whereas the field due to the space charge drives a drift current in the opposite direction. Equilibrium is attained when the two currents are equal, i.e. the sum of the diffusion and drift currents is zero. The net hole current density is

(E.23)
J p = q e D p d p d x + q e p μ p E p ,
where D p is the diffusion constant for holes and E p is the electric field in the p-region.

(p.452) To solve this equation we make use of the following relationships: The hole concentration is

(E.24)
p = n i e ( E i E F ) / k T ,
so its derivative
(E.25)
d p d x = p k T ( d E i d x d E F d x ) .
Since the force on a charge q e due to an electric field E is equal to the negative gradient of the potential energy,
(E.26)
q e E = d E c d x = d E v d x = d E i d x .
As only the gradient is of interest and E c, E v, and E i differ only by a constant offset, any of these three measures can be used. We'll use the intrinsic Fermi level E i, since it applies throughout the sample.

The remaining ingredient is the Einstein relationship, which relates the mobility to the diffusion constant

(E.27)
μ p = q e D p k T .
Using these relationships the net hole current becomes
(E.28)
J p = q e p D p k T d E F d x = μ p p d E F d x .
Accordingly, the net electron current
(E.29)
J n = q e n D n k T d E F d x = μ n n d E F d x .
Since, individually, the net hole and electron currents in equilibrium must be zero, the derivative of the Fermi level
(E.30)
d E F d x = 0 .
In thermal equilibrium the Fermi level must be constant throughout the junction region.

For the Fermi level to be flat, the band structure must adapt, since on the p-side the Fermi level is near the valence band, whereas on the n-side it is near the conduction band (Figure 2.19). If we assume that the dopants are exclusively donors on the n-side and acceptors on the p-side, the difference in the respective Fermi levels is

(E.31)
Δ E F = k T log N a N d n i 2 .
This corresponds to an electric potential
(E.32)
Δ V F = 1 q e Δ E F V b i ,
often referred to as the “built-in” voltage of the junction.

(p.453) As either N a or N d increases relative to n i, the respective Fermi level moves closer to the band edge, increasing the built-in voltage. With increasing doping levels the built-in voltage approaches the equivalent potential of the bandgap E g/q e.

E.4 The forward-biased pn-junction

Applying an external bias leads to a condition that deviates from thermal equilibrium, i.e. the Fermi level is no longer constant throughout the junction. If a positive voltage is applied to the p-electrode relative to the n-electrode, the total variation of the electric potential across the junction will decrease (Figure 2.20). Since this reduces the electric field across the junction, the drift component of the junction current will decrease. Since the concentration gradient is unchanged, the diffusion current will exceed the drift current and a net current will flow.

This net current leads to an excess of electrons in the p-region and an excess of holes in the n-region. This “injection” condition leads to a local deviation from equilibrium, i.e. p n > n i 2 . Equilibrium will be restored by recombination.

Note that a depletion region exists even under forward bias, although its width is decreased. The electric field due to the space charge opposes the flow of charge, but the large concentration gradient overrides the field.

Consider holes flowing into the n-region. They will flow through the depletion region with small losses due to recombination, as the electron concentration is small compared with the bulk. When holes reach the n-side boundary of the depletion region the concentration of electrons available for recombination increases and the concentration of holes will decrease with distance, depending on the cross-section for recombination, expressed as a diffusion length. Ultimately, all holes will have recombined with electrons. The required electrons are furnished through the external contact from the power supply.

On the p-side, electrons undergo a similar process. The holes required to sustain recombination are formed at the external contact to the p-region by electron flow toward the power supply, equal to the electron flow toward the n-contact. The following derivation follows the discussions by Shockley (1949, 1950) and Grove (1967).

The steady-state distribution of charge is determined by solving the diffusion equation,

(E.33)
D n d 2 n p d x 2 n p n p 0 τ n = 0 .
Electrons flowing into the p-region give rise to a local concentration n p in excess of the equilibrium concentration n p0. This excess will decay with a recombination time τn, corresponding to a diffusion length L n.

The first boundary condition required for the solution of the diffusion equation is that the excess concentration of electrons vanish at large distances x,

(E.34)
n p ( ) = n p 0 .
(p.454) The second boundary condition is that the carriers are injected at the origin of the space charge region x = 0 with a concentration n p(0). This yields the solution
(E.35)
n p ( x ) = n p 0 + ( n p ( 0 ) n p 0 ) e x / L n .
From this we obtain the electron current entering the p-region
(E.36)
J n p = q e D n d n p d x | x = 0 = q e D n n p ( 0 ) n p 0 L n .
This says that the electron current is limited by the concentration gradient determined by the carrier density at the depletion edge n p(0) and the equilibrium minority carrier density n p0. Determining the equilibrium density n p0 is easy,
(E.37)
n p 0 = n i 2 / N a .
The problem is that n p(0) is established in a non-equilibrium state, where the previously employed results do not apply.

To analyze the regions with non-equilibrium carrier concentrations Shockley introduced a simplifying assumption by postulating that the product pn is constant. In this specific quasi-equilibrium state this constant will be larger than n i 2 , the pn-product in thermal equilibrium. In analogy to thermal equilibrium, this quasi-equilibrium state is expressed in terms of a “quasi-Fermi level”, which is the quantity used in place of E F that gives the carrier concentration under non-equilibrium conditions.

The postulate pn = const is equivalent to stating that the non-equilibrium carrier concentrations are given by a Boltzmann distribution, so the concentration of electrons is

(E.38)
n = n i e ( E F n E i ) / k T ,
where E Fn is the quasi-Fermi level for electrons, and
(E.39)
p = n i e ( E i E F p ) / k T ,
where E Fp is the quasi-Fermi level for holes. The product of the two carrier concentrations in non-equilibrium is
(E.40)
p n = n i 2 e ( E F n E F p ) / k T .
If pn is constant throughout the space-charge region, then E FnE Fp must also remain constant.

Using the quasi-Fermi level and the Einstein relationship, the electron current entering the p-region becomes

(E.41)
J n p = q e D n d n p d x | x = 0 = q e D n d d x ( n i e ( E F n E i ) / k B T ) = μ n n d E F n d x .

These relationships describe the behavior of the quasi-Fermi level in the depletion region. How does this connect to the neutral region?

(p.455) In the neutral regions the majority carrier motion is dominated by drift (in contrast to the injected minority carrier current, which is determined by diffusion). Consider the n-type region. Here the bulk electron current that provides the junction current

(E.42)
J n n = μ n n d E i d x .
Since the two electron currents must be equal
(E.43)
J n n = J n p ,
it follows that
(E.44)
d E F n d x = d E i d x ,
i.e. the quasi-Fermi level follows the energy band variation. Thus, in a neutral region, the quasi-Fermi level for the majority carriers is the same as the Fermi level in equilibrium. At current densities small enough not to cause significant voltage drops in the neutral regions, the band diagram is flat, and hence the quasi-Fermi level is flat.

In the space charge region, pn is constant, so the quasi-Fermi levels for holes and electrons must be parallel, i.e. both will remain constant at their respective majority carrier equilibrium levels in the neutral regions.

If an external bias V is applied, the equilibrium Fermi levels are offset by V, so it follows that the quasi-Fermi levels are also offset by V,

(E.45)
E F n E F p = q e V .
Consequently, the pn-product in non-equilibrium
(E.46)
p n = n i 2 e ( E F n E F p ) / k T = n i 2 e q e V / k T .

If the majority carrier concentration is much greater than the concentration due to minority carrier injection (“low-level injection”), the hole concentration at the edge of the p-region remains essentially at the equilibrium value. Consequently, the enhanced pn-product increases the electron concentration.

(E.47)
n p ( 0 ) = n p 0 e q e V / k T .
Correspondingly, the hole concentration in the n-region at the edge of the depletion zone becomes
(E.48)
p n ( 0 ) = p n 0 e q e V / k T .
Since the equilibrium concentrations
(E.49)
n p 0 = n i 2 N a and  p n 0 = n i 2 N d ,
the components of the diffusion current due to holes and electrons are (p.456)
(E.50)
J n = q e D n n i 2 N a L n ( e q e V / k T 1 ) J p = q e D p n i 2 N d L p ( e q e V / k T 1 ) .
The total current is the sum of the electron and hole components
(E.51)
J = J n + J p = J 0 ( e q e V / k T 1 ) ,
where
(E.52)
J 0 = q e n i 2 ( D n N a L n + D p N d L p ) .

This is the diode equation (or Shockley equation), which describes the current–voltage characteristic both under forward and reverse bias. Under forward bias the current indreases exponentially. Under reverse bias (negative V), the exponential term vanishes when the bias exceeds several kT/q e and the current becomes the reverse saturation current J = −J 0. For a uniform junction cross-section the current densities J n, J p, and J 0 can be replaced by their respective currents.

Note that in the diode equation:

  1. 1. The bandgap does not appear explicitly (only implicitly in J 0 via n i).

  2. 2. The total current has two distinct components, due to electrons and holes.

  3. 3. The electron and hole currents are generally not equal. The ratio

    (E.53)
    J n J p = N d N a if  D n L n = D p L p .

  4. 4. Current flows for all values of V. However, when plotted on a linear scale, the exponential appears to have a knee, often referred to as the “turn-on” voltage.

  5. 5. The magnitude of the turn-on voltage is determined by J 0. Diodes with different bandgaps will show the same behavior if J 0 is the same.

Figure E.3 shows measured IV curves for commercial Si and Ge junction diodes (1N4148 and 1N34A). On a linear scale the Ge diode “turns on” at 200 – 300 mV, whereas the Si diode has a threshold of 500 – 600 mV. However, on a logarithmic scale it becomes apparent that both diodes pass current at all voltages > 0.

The reverse current (Figure E.4) shows why the Ge diode shows greater sensitivity at low voltages. The smaller bandgap leads to increased n i. The Si diode shows a “textbook” exponential forward characteristic at currents > 10 nA, whereas the Ge diode exhibits a more complex structure.

The discrepancies in the forward current between the measured results and the simple theory require the analysis of all processes in the depletion zone.

  1. 1. Generation-recombination in the depletion region (see Appendix F).

  2. 2. Diffusion current (as just calculated for the ideal diode).

  3. (p.457)
    APPENDIX E THE DIODE EQUATION

    Fig. E.3 Current vs. voltage for forward biased Si and Ge diodes.

  4. 3. High-injection region where the injected carrier concentration affects the potentials in the neutral regions.

  5. 4. Voltage drop due to bulk series resistance.

For a discussion of these effects see Sze (1981).

The reverse current is increased due to generation and recombination currents in the depletion zone, as discussed in Appendix F. In optimized photodiodes reverse bias currents of about 100 pA/cm2 have been achieved, which is about 3 times the theoretical value (Holland 2004).

APPENDIX E THE DIODE EQUATION

Fig. E.4 Reverse current of Si and Ge diodes at room temperature.