Abstract
A junction between two dissimilar materials is shown schematically in Fig. 2.1 below. Such a junction can take many forms; however, its defining characteristic is the asymmetry that exists upon crossing the junction from one side to the other.
“… the totality is not, as it were, a mere heap, but the whole is something besides the parts …”
Aristotle, Metaphysics
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Notes
- 1.
Often referred to simply as a crystal diode (detector).
- 2.
Mercury-arc valves were another early type of diode based on a liquid mercury cathode and carbon anode. In this case, electrons are preferentially emitted from the cathode upon formation of a flame or arc discharge (through the contained mercury vapor).
- 3.
The discovery of an unintentional junction in a rod of crystalline silicon by Ohl led to the terms “n-type” and “p-type” being coined to describe the different doping on either side of the junction. Ohl’s experiments also resulted in the demonstration of the first pn junction solar cell.
- 4.
W. Shockley, Bell Syst. Tech. J. 28, 435 (1949). This was a seminal paper in the history of semiconductor electronics and forms much of the foundation on which Chapters 2 and 3 are based.
- 5.
To equalize the Fermi levels there will be a net transfer of electrons toward the p-type region and recombination with holes results in the space-charge layer at equilibrium.
- 6.
The interface or plane where the doping type changes is known as the metallurgical junction. The same terminology is used to describe the physical or material transition interface in any type of solid-state junction.
- 7.
Throughout this book the device analysis and descriptions are mainly constrained to one spatial dimension (e.g., x) in order to emphasize and develop an understanding of the basic principles of solid-state electronic devices. The 1D picture will be extended to include additional dimensions/effects when necessary.
- 8.
Electron energies (and hence the energy band edges) are given by E = − qV. See Fig. A.13a in Appendix A for the general relation between the band edges and electric field.
- 9.
See Appendix A, Sect. A.2.
- 10.
This is also equivalent to the difference in work functions of the two separated semiconductors (cf. Fig. 2.2a).
- 11.
Using N a x p = N d x n.
- 12.
See Appendix A, Eq. (A.25).
- 13.
p + n and pn + junctions are often termed one-sided pn junctions.
- 14.
Note that, more rigorously, the Fermi levels under applied bias should be referred to as quasi-Fermi levels since the system is no longer in thermal equilibrium. In this textbook we will not make use of this distinction.
- 15.
Recall form Appendix A, Eq. (A.47), the excess carrier concentration is defined as the difference between the total concentration and the thermal equilibrium concentration.
- 16.
Recall that most of the voltage drop will occur inside the space charge region.
- 17.
This type of partial differential equation appears in many other fields including heat flow, mass transport in gases and fluids, etc.
- 18.
The diffusion length represents the average distance a minority carrier travels before recombining.
- 19.
The reason for the subscripts “B” and “E” in the pn junction symbols is for historical reasons relating to the bipolar transistor (Base, Emitter) discussed in Chap. 3.
- 20.
For simplicity p n0(x n) is written as p n0. Note that for a step junction p n0 will be constant throughout the neutral n-type region.
- 21.
Note that the x-coordinate is negative in this expression.
- 22.
The applied bias can be thought of as a small perturbation to the equilibrium properties of a material; hence, even when current flows through electronic devices they are still close to thermal equilibrium under most conditions.
- 23.
If one/both of the diode regions have intermediate lengths (i.e., x B or x E is comparable to L p or L n , respectively), in other words they are neither short nor long, then the diffusion equations lead to solutions for the excess carrier concentrations containing hyperbolic functions [1]. The net result once more is a modification of the saturation current in the ideal diode equation.
- 24.
It can be seen that E max ∝ V 1/2a for large reverse biases.
- 25.
One must take care however to avoid mechanical and/or thermal breakdown (in particular thermal runaway processes that cause uncontrollable positive current feedback loops), which can lead to device failure.
- 26.
The breakdown phenomenon of punchthrough may also occur for short-base diodes under reverse bias. This effect is discussed in Sect. 3.4.
- 27.
Note that this energy must be at least on the order of the band gap in order to create a new electron–hole pair.
- 28.
This process is more generally referred to as impact ionization.
- 29.
At very large doping levels the mean free path will also decrease significantly due to impurity scattering; however, in this regime avalanche breakdown is less likely to occur.
- 30.
Such field-induced interband tunneling can occur more generally in solids (e.g., insulators) and is often referred to simply as Zener breakdown; C. M. Zener, Proc. Roy. Soc. (London) A145, 523 (1934).
- 31.
See Appendix A, Example A.4.
- 32.
See Appendix B for the band gap energy as a function of temperature. Increased thermal energy results in larger crystal lattice spacing, which usually leads to the band structure resembling more of a free particle (i.e., a smaller band gap).
- 33.
Or, more generally, generation/recombination can occur in the space-charge region through a variety of mechanisms.
- 34.
The recombination current density in the space-charge region can be found by evaluating the expression
$$ {J}_{\mathrm{R}}=q{\displaystyle {\int}_{-{x}_{\mathrm{p}}}^{x_{\mathrm{n}}} Udx}, $$where U is the recombination rate (in units of s−1 cm−3). For SHR theory U can be approximated by Eq. (A.49c) in Appendix A, and by using Eq. (2.44) the integral above can be evaluated giving the result of Eq. (2.37). Note that in more detailed recombination models (and in practice) the factor multiplying k B T in the exponential can differ from 2.
- 35.
I GR0 is found from Eq. (2.37).
- 36.
Contact resistances may also be lumped into this parameter.
- 37.
The effects of series resistance may also be explicitly included here as in Eq. (2.41).
- 38.
In other words, we are assuming that quasi-equilibrium holds. It can be shown more generally that Eq. (2.44), often referred to as the “law of the junction”, is also valid inside the space-charge region and that \( pn\le {n}_{\mathrm{i}}^2{e}^{q{V}_{\mathrm{a}}/{k}_{\mathrm{B}}T} \) throughout the pn junction (H. K. Gummel, Solid-State Electron. 10, 209 (1967)).
- 39.
This can sometimes be used to approximately model pn junctions formed via gas phase diffusion processes. (An exponential distribution may also be used.)
- 40.
The results of this section are based on an ideal step pn junction diode unless otherwise noted. Non-idealities and/or other junction doping profiles can be included in a straightforward manner.
- 41.
This includes a device that is unbiased (i.e., zero bias).
- 42.
Not to be confused with the same symbol also used for carrier generation rate.
- 43.
This is simply the area of the triangle representing the excess minority carrier distribution vs. distance.
- 44.
- 45.
This equation has the same form as the capacitance of a parallel plate capacitor with the plates separated by the depletion width. It can be shown that this correspondence holds for arbitrary dopant profiles.
- 46.
It is possible to define a diode cutoff frequency as f T = (2π RC)−1, where R and C denote the overall (including parasitics, except for inductance) resistance and capacitance components that are dominant at a particular bias, respectively. f T can be considered an upper limit to how quickly the diode can respond to small-signal excitations.
- 47.
Ultimately, this can lead to so-called first-principles calculations that consider the individual atoms making up the device. Such calculations typically require very large computational times, but fortunately this kind of precision is not usually required for most devices at present.
- 48.
In other words, we are considering a p + n diode. Non-idealities (recombination in the space-charge region, etc.) are also ignored in this treatment.
- 49.
This result can be obtained by noting that both the stored charge and the current flowing through the junction can be expressed in terms of the bias appearing across the junction (via exponential factors) using the results of the ideal diode analysis carried out earlier.
- 50.
The recovery time is usually taken as the point at which the reverse current has dropped to 10 % of its initial value.
- 51.
Although an analytical expression for this parameter based on the diode material properties is possible, empirically a value of δ ~ 0.2 agrees quite well with data over a fairly broad range of currents.
- 52.
This equation overestimates the storage time by a progressively larger amount as I R/I F increases.
- 53.
- 54.
These critical IC interconnection techniques were first developed by Noyce and Lehovec working independently in 1959.
- 55.
Epitaxy is a type of thin film crystal growth. See Appendix A, Sect. A.3, for further details.
- 56.
Many of the important features of metal–semiconductor junctions (and pn junctions) were also developed theoretically by Davydov between 1938 and 1939.
- 57.
Known as the Schottky barrier.
- 58.
The analogous barrier height for a p-type semiconductor is given by qϕ B = E g − q(Φ M − Χ).
- 59.
Recall from electrostatics that free extra charge cannot exist in the interior of a (metallic) conductor.
- 60.
For a system obeying Maxwell–Boltzmann statistics, it can be shown that the electron (or hole) current across a plane due to carriers in thermal motion can be expressed as \( J= qn{\overline{v}}_{\mathrm{th}}/4 \), where \( {\overline{v}}_{\mathrm{th}} \) is the mean or average thermal velocity, \( {\overline{v}}_{\mathrm{th}}=\sqrt{\frac{8{k}_{\mathrm{B}}T}{\pi \kern0.1em {m}^{*}}} \).
- 61.
This is known as the thermionic emission model.
- 62.
For example, in a Mott barrier a thin, lightly doped semiconductor region contacts the metal, which transitions to a highly doped bulk region a short distance from the junction.
- 63.
At higher doping levels (~1018 cm−3) and/or lower temperature η begins to deviate from unity [2] as tunneling becomes more important.
- 64.
This may also be viewed as a measure of the screening length or the distance over which the free charges in the semiconductor rearrange themselves in response to the additional electrons in order to cancel out the electric field inside the bulk of the semiconductor. The Debye length is proportional to the density of free charges in a material; for most metals L D is typically well below 1 nm, while it can range from 10 to 100 nm or more in a semiconductor depending on the doping level, with a maximum value for the intrinsic case.
- 65.
Note that the equations given for a metal contact to n-type material apply equally for contacts to p-type material with appropriate substitutions for the doping type (N a for N d), effective mass (m p for m n ), etc.
- 66.
The importance of surface states on metal–semiconductor interfaces was pointed out by Bardeen in 1947. The effect of Fermi-level pinning is somewhat analogous to adding a very thin heavily doped layer between the metal and semiconductor.
- 67.
- 68.
Electrons that are emitted from the metal into the semiconductor under reverse bias will induce images charges of the opposite sign in the planar metal surface near the interface, which causes the barrier height to be lowered within a few nanometers from the metallurgical junction.
- 69.
Equation (2.79) is also valid for a one-sided abrupt pn junction (p + n or pn +).
- 70.
As for the pn diode, we can define a Schottky diode cutoff frequency as f T = (2π RC)−1. Typically, the fastest Schottky diodes are made from semiconductors with the highest carrier mobility in order to reduce the effect of series resistance.
- 71.
The external circuit parameters will largely determine how quickly a Schottky diode can be switched as opposed to intrinsic delays in the device itself.
- 72.
This edge or surface breakdown mechanism can also be important in pn diodes but is generally not as severe unless high power devices are required.
- 73.
J. B. Gunn, J. Electron. Control 4, 17 (1958) is an early study on isotype junctions.
- 74.
Shockley proposed using heterojunctions for devices in 1951 while in the same year Gubanov presented studies on their theoretical properties. In 1957, Kroemer provided important extensions to the earlier heterojunction studies and device proposals, followed by pioneering experimental studies by Anderson around 1960. Subsequently, the ability to grow different semiconductor layers with atomic precision and minimal lattice mismatch allowed high-quality heterojunction interfaces to be developed.
- 75.
R. L. Anderson, IBM J. Res. Dev. 4, 283 (1960). The Anderson model is similar to the idealized Schottky barrier model and can be modified to also include the effects of non-idealities such as interface states, etc.
- 76.
As temperature increases the forward voltage drop becomes progressively smaller and this ultimately limits sensitivity at high temperatures. The increased intrinsic carrier concentration may also affect the majority carrier levels in the semiconductor at elevated temperatures (see Appendix A) and alter the expected junction behavior.
- 77.
The Earth Observing System (EOS) Microwave Limb Sounder (MLS) is one prominent example, which is part of NASA’s Aura satellite and uses GaAs-based Schottky diode mixers to detect radiation via heterodyning. Some images of satellite data collection results are shown at the end of this chapter.
- 78.
This is however attenuated by the atmosphere before reaching the earth and the amount of “air mass” (AM) through which the light passes is used to describe the incident solar energy. Terrestrial solar cell performance is typically specified with respect to the AM1.5 spectrum (48° from vertical) or about 1,000 W/m2.
- 79.
Note the excess carrier concentration at the edge of the depletion region will be determined (for the case of low-level injection, as before) by the voltage appearing across the junction, V a (cf., Eq. (2.23)). (In the case of an illuminated junction V a is not strictly an applied bias, but we keep the same notation for simplicity.)
- 80.
Not to be confused with the diode ideality factor.
- 81.
Sources of solar cell losses include the inability to absorb photons with energy less than the band gap, heat generated by large energy photon absorption, reflection losses, carrier recombination, and parasitic resistances.
- 82.
Previously this was only possible in other areas of technology, for example, using the potential energy of a spring or the pressure/temperature differential of gases or of a chemical reaction to perform a useful task. Solid-state electronics has far surpassed the complexity of any other type of man-made “machine” in terms of the number of working parts operating together, as exemplified by the integrated circuit.
- 83.
This problem may be somewhat difficult and/or lengthy.
References
Shockley, W.: The Theory of p−n Junctions in Semiconductors and p−n Junction Transistors. Bell Syst. Tech. J. 28, 435 (1949)
Sze, S.M., Ng, K.K.: Physics of Semiconductor Devices, 3rd edn. Wiley Interscience, Hoboken (2007)
Kingston, R.H.: Switching Time in Juction Diodes and Junction Transistors. Proc. IRE 42, 829 (1954)
Muller, R.S., Kamins, T.I.: Device Electronics for Integrated Circuits, 3rd edn. Wiley, New York (2003)
Grove, A.S.: Physics and Technology of Semiconductor Devices. Wiley, New York (1967)
Shur, M.S.: Introduction to Electronic Devices. Wiley, New York (1995)
Neudeck, G.W.: The PN Junction Diode, 2nd edn. Prentice-Hall, Boston (1989)
Smith, R.A.: Semiconductors, 2nd edn. Cambridge University Press, Cambridge (1978)
Neamen, D.A.: Semiconductor Physics and Devices, 3rd edn. McGraw-Hill, New York (2003)
Green, M.A.: Solar Cells. Prentice-Hall, Upper Saddle River (1982)
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Appendices
Problems
-
1.
Long- vs. short-base diodes. An ideal silicon pn diode is formed by diffusing a high concentration of phosphorus into a 75-μm-thick boron-doped wafer having a resistivity of 5 Ω-cm and minority carrier lifetime of 5 μs. The junction is formed 10 μm below the surface with area 10−4 cm2. (1) Find the built-in potential. (2) Calculate the current flowing through the diode under an applied forward bias of 0.5 V. (3) Is I 0 for an ideal diode always constant?
-
2.
Ohmic voltage drops. Consider a silicon short-base diode with the following parameters:
$$ \begin{array}{l}\kern-0.18em {N}_{\mathrm{d}}=2\times {10}^{17}{\mathrm{cm}}^{-3}\ \mathrm{and}\ {N}_{\mathrm{a}}=5\times {10}^{18}{\mathrm{cm}}^{-3}\hfill \\ {}{x}_{\mathrm{B}}={x}_{\mathrm{E}}=5\kern0.5em \upmu \mathrm{m}\ \mathrm{and}\ A={10}^{-4}{\mathrm{cm}}^2\hfill \end{array} $$Find the voltage dropped in the neutral regions of the diode for an applied forward bias of 0.7 V.
-
3.
Footnote 82 Diode temperature dependence. An ideal long-base Si pn diode has N a = 1017cm− 3, N d = 7 × 1016cm− 3, and a cross-sectional area of 10− 3cm2. (1) If τ n = τ p = 1 μs, calculate the current flowing through the junction under an applied bias of 0.5 V. Repeat your calculation for a temperature of 500 K. (2) Sketch the thermal equilibrium band edge diagram of the pn junction for both temperatures. (3) If the junction in this question were required to absorb light, at what wavelength would it begin to absorb strongly?
-
4.
pn diode storage time. Compare the accuracy of Eqs. (2.61) and (2.62b) to the full solution of the continuity equation given by Eq. (2.62a), for I R/I F ranging from 0.01 to 100.
-
5.
Metal–semiconductor band edge diagrams. An ideal metal–semiconductor junction is formed between platinum (work function 5.3 eV) and p-type silicon. What type of contact results?
Microwave Limb Sounder onboard the Aura Satellite
The MLS (located near the foreground in the image below) observes microwave emission from gas molecules (e.g., O3, H2O, CO, SO2) from 118 GHz to 2.5 THz using GaAs-based Schottky diodes (solar panels based on silicon pn junctions power the satellite as well). The data shown is based on annual MLS Earth ozone concentration measurements taken over the south pole.
Source: JPL/NASA
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Papadopoulos, C. (2014). Junctions and Diodes. In: Solid-State Electronic Devices. Undergraduate Lecture Notes in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8836-1_2
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