The PN Junction Diode
2.12 Summary
2.1. In a p-n junction an energy barrier is formed, which is characterized by a built-in potential V0. The Fermi energy is constant across the junction in equilibrium. An electric field is present in the transition region of the junction.
2.2. Four currents flow in a p-n junction. In equilibrium all these currents add to zero.
In forward bias the diffusion currents due to majority carriers dominate and high currents can flow. In reverse bias drift currents due to minority carriers flow but these reverse bias currents are limited in magnitude due to the limited number of minority carriers and constitute the reverse saturation current I0.
2.3. The contact potential V0 may be calculated from carrier concentrations and the resulting position of the Fermi energy relative to the conduction and valence band edges on either side of the junction. In equilibrium the Fermi energy is constant.
2.4. The transition region can be modelled using the depletion approximation in which a fully depleted space charge layer of width W is assumed. Values for equilibrium charge densityρ(x), electric fieldε(x) and potential V (x) result from the depletion approximation.
2.5. The diode equation derives the net diode current I obtained due to an applied potential V. It may be derived by considering the change in carrier concentrations at the edges of the depletion region implied by a change in net contact potential, and then using the diffusion equation to determine the resulting diffusion currents. Minority carrier concentrations decay exponentially with distance from the depletion region in both n-type and p-type material.
2.6. Reverse breakdown in a diode can arise from carrier avalanching. Field ionization and impact ionization in high electric fields can occur at or near the junction if the reverse bias is large enough. In addition highly doped p-n junctions may also exhibit electron tunnelling. Both mechanisms occur in the Zener diode.
2.7. If even higher doping levels are present the Fermi energy can enter the conduction and valence bands and the condition of degenerate doping is established. Tunnelling of electrons in both directions across the junction results, and a tunnelling junction results. A tunnelling junction allows efficient current flow across a p-n junction, which is important for multi-junction solar cells.
2.8. Carriers crossing over the depletion region may recombine due to deep traps. This will modify the diode equation resulting in a diode ideality factor n with values between 1 and 2. This trapping is most important at low diode currents.
2.9. Diodes require ohmic contacts to allow current to flow between metal contacts and the semiconductor. A metal-semiconductor junction may form a Schottky diode, which can be understood by thermionic emission that occurs at a metal surface. The mechanism for thermionic emission is derived to obtain the Richardson–Dushman equation. By using high doping levels tunnelling behaviour can be obtained at a metal- semiconductor junction to create a highly conductive or ohmic contact rather than a rectifying contact. A detailed analysis of metal- semiconductor junctions is influenced by surface defects, traps, dangling bonds and impurities at the metal-semiconductor interface.
2.10. Heterojunctions are formed between two semiconductors having different compo- sitions. This permits the bandgap to change through a device. Heterojunctions are
important for both solar cells and LEDs since they have desirable electronic and optical properties for these devices such as the localization of optical absorption and the localization of electron-hole pair recombination.
2.11. An understanding of AC and transient diode behaviour requires that two mechanisms of charge storage are discussed. In reverse bias, charge is stored in the depletion region in the form of ionized donors and acceptors, which leads to diode capacitance.
In forward bias, charges are stored as minority carriers on either side of the depletion region. This leads to switching delays, which can be characterized by a storage delay time. Modelling this is complex.
Suggestions for Further Reading
Neamen DA. Semiconductor Physics and Devices, 3rd edn. McGraw Hill, 2003.
Roulston DJ. An Introduction to the Physics of Semiconductor Devices. Oxford University Press, 1999.
Streetman BG and Banerjee SK. Solid State Electronic Devices, 6th edn. Prentice Hall, 2006.
Problems
2.1 Draw a diagram showing the excess minority carrier concentrations as a function of position in a p-n junction diode under the following conditions. Assume approximately equal doping levels on both sides of the junction:
(a) Reverse bias (b) No bias
(c) Weak forward bias (d) Strong forward bias
What assumption is no longer valid if forward bias current is increased beyond a certain level? Explain carefully.
2.2 An abrupt Si p-n junction has the following properties:
p-side: Na=1018cm−3 n-side: Nd =1015cm−3 junction area A=10−4cm2
(a) Sketch a band diagram of the diode under forward bias showing hole and electron quasi-Fermi levels.
(b) Calculate the depletion region width with a reverse voltage of 10 volts.
(c) Find the diode current with the reverse bias of 10 volts. What is this current called?
(d) Find the peak value of depletion region electric field at a reverse bias of 10 volts.
(e) If the silicon exhibits avalanche breakdown at an electric field of 1×105V cm−1 find the reverse bias voltage at which breakdown will occur at the junction.
2.3 An abrupt Si p-n junction has the following parameters:
n-side: Nd =5×1018cm−3 p-side:Na=1017cm−3 junction area A=10−2cm2 (a) Find the built-in potential V0. (b) Find the reverse saturation current I0.
(c) Find the ratio of hole injection current to electron injection current at a forward current of 1 mA.
(d) Find the total minority carrier charge on each side of the diode at a forward current of 1 mA.
(e) In the p-side of the diode, at a certain distance away from the depletion region, the hole and electron currents are equal in magnitude, but opposite in direction.
Find this distance.
(f) Find the quasi-Fermi level separations Fn−Fpat a distance of 0.1μm from the edges of the depletion region on either side of the junction (i.e. at a depth of 0.1μm into the neutral p-type and n-type regions) at a forward bias of 1 mA.
2.4 An abrupt GaAs p-n junction has the following parameters:
n-side: Nd =2×1018cm−3 p-side: Nd =2×1017cm−3 junction area A=10−4cm2 (a) Find the built-in potential V0.
(b) Find the ratio of hole current to electron current crossing the junction at a total forward current of 5 mA.
(c) Find the reverse saturation current I0.
(d) At a forward current of 5 mA, find the total minority carrier charge on each side of the diode.
(e) Find the distance into the p-side at which the minority electron concentration is half the maximum value. Sketch it as a function of distance into the p-side starting from the edge of the depletion region.
2.5 Calculate the electric field as a function of position in a p-n junction having a depletion region of width W and constant doping levels of Nd and Na on the n- and p-sides respectively. To do this, first find the width of the depletion region on either side of the junction as a function of W. Then use Gauss’s law to determine the electric field at any point in the depletion region by considering a Gaussian surface that covers only a fraction of the space charge on either side of the junction. Show that the electric field increases linearly and reaches a maximum value at the junction when the Gaussian surface encloses all the space charge on one side of the junction. Sketch the field as a function of position.
2.6 If the diode of Problem 2.4 is forward biased, sketch how the electric field would vary as a function of position throughout the depletion region. Repeat for a reverse bias.
Compare these sketches to the sketch without bias.
2.7 A GaAs diode is reverse biased and it exhibits an increasing reverse current as bias increases. Explain how this occurs based on generation or recombination in the depletion region. What changes would be made to the properties of the semiconductor to reduce this effect?
2.8 A tunnel diode is formed using highly doped silicon with a total depletion width under equilibrium conditions of 5 nm. According to the depletion approximation, what doping level would be needed to achieve this? Assume equal doping levels on both sides of the junction.
2.9 The forward current in a silicon diode increases with bias voltage with the following relationship:
I ∝exp q V
1.5kT
(a) Explain the physics underlying this dependence. Plot a representative graph of the current versus voltage dependence using linear x- and y-axes and compare its shape to the graph of a diode having a relationship that obeys the diode equation.
(b) At low forward bias voltages a diode is observed to behave according to I ∝ exp(1.5kTq V ) but at higher voltages its current–voltage dependence approaches the diode equation. Explain carefully.
(c) A silicon diode at a given forward bias voltage range behaves according to I ∝ exp(2kTq V) at a junction temperature of 100◦C but at−50◦C its current–voltage dependence for the same bias voltage range follows the diode equation. Explain carefully.
2.10 Figure 2.27 shows the band structure of an effective ohmic contact that works by tunnelling applied to n-type silicon.
(a) Sketch the analogous band structure for an ohmic contact applied to p-type silicon that relies on tunnelling.
(b) Propose a metal contact material that forms a tunnelling type ohmic contact to n-type GaAs. Look in the literature and see if your answer is a good prediction of materials used in practice.
(c) Repeat (b) for p-type GaAs.
2.11 A Schottky diode is composed of a junction between p-type silicon and aluminium.
The barrier height Ebis 0.4 eV and junction area is 1000μm2. (a) Calculate Ieat room temperature.
(b) Find and plot the diode current as a function of applied voltage at room temperature for both forward and reverse bias. You may neglect reverse breakdown.
2.12 (a) Suggest why the boundary conditions for travelling waves relevant to the Schottky diode applied to Equation 1.27 (kx =2naxπ etc.) are a factor of two different from the boundary condition for standing waves. Using the travelling wave boundary conditions, show that you obtain the number of available electron states per unit volume of crystal as dn=h22d pxd pyd pz.
(b) The Born–von Karman boundary condition is relevant here and is the basis for the travelling wave boundary condition. Look up this boundary condition and explain it in the context of the travelling wave boundary condition.
2.13 A silicon p-n junction diode has the following parameters: Nd=2×1018cm−3, Na= 2×1016cm−3,τn=τp =2×10−6s, Dn=25 cm2s−1and Dp=8 cm2s−1. A light source is incident only on the depletion region, producing a generation current density of Jgen=50 mA cm−2. The diode is open circuited. The generation current density forward biases the junction, inducing a forward- bias current in the opposite direction to the generation current. A steady-state condition is reached when the generation current density and forward-bias current density are equal in magnitude. What is the induced forward-bias voltage at this steady-state condition?
2.14 An abrupt Si p-n junction has the following parameters:
p-side: Na=1017cm−3 n-side: Nd=1014cm−3
junction area A=1×10−5cm2 Find:
(a) V0, the built-in potential.
(b) I0, the reverse saturation current.
(c) The depletion region width at 0 volts.
(d) The depletion region width at 10 volts reverse bias.
(e) The peak electric field at 10 volts reverse bias.
(f) The ratio of hole-to-electron current flow in forward bias.
(g) The diode capacitance at reverse bias of 5 volts, 10 volts and 15 volts.
2.15 An abrupt Si p-n junction has the following properties:
p-side: Na=1017cm−3, n-side: Nd=1015cm−3 junction area A=10−4cm2
(a) Sketch an equilibrium band diagram showing Efand V0. (b) Calculate V0.
(c) Calculate the space charge width with zero applied voltage.
(d) Find the maximum electric field at a reverse bias of 10 volts.
(e) Find Io, the diode reverse saturation current.
(f) Find the breakdown field for silicon if the diode has a reverse breakdown voltage of 100 volts. Hint: Use the highest field in the depletion region for this calculation.
(g) Find the depletion region width just before reverse breakdown.
(h) Find the diode capacitance at a reverse bias of 10 volts.
(i) Find the voltage across the diode at a forward current at 1 A.
2.16 An abrupt Si p-n junction has the following properties:
p-side: Na=1018cm−3 n-side: Nd=1016cm−3 junction area A=10−4cm2
(a) Sketch a band diagram of a diode under forward bias showing hole and electron quasi-Fermi levels.
(b) Calculate the space charge width with a reverse voltage of 10 volts.
(c) Find the diode capacitance at an applied voltage of 10 volts reverse bias.
(d) Find the diode current with a reverse bias of 10 volts.
(e) Find the ratio of hole to electron current that crosses over the depletion region.
(f) Find the peak value of depletion region electric field at a reverse bias of 10 volts.
2.17 In a p+-n junction at room temperature, the n-doping Nd is doubled. How do the following two parameters change if everything else is unchanged?
(a) breakdown voltage (b) built-in voltage
2.18 Ohmic contacts are needed on a p-n junction made with silicon. Metal contact pads will be deposited on the p and n regions, and then diffused in for a short time to form ohmic contacts. List some suitable materials for p and n ohmic contacts.
2.19 Show that you can obtain expression 2.36 from 2.37. Sketch the depletion region of a diode and show the locations of differential charge dQ obtained by a small change in applied voltage dV.
3
Photon Emission and Absorption
3.1 Introduction to Luminescence and Absorption 124
3.2 Physics of Light Emission 125
3.3 Simple Harmonic Radiator 128
3.4 Quantum Description 129
3.5 The Exciton 132
3.6 Two-Electron Atoms 135
3.7 Molecular Excitons 141
3.8 Band-to-Band Transitions 144
3.9 Photometric Units 148
3.10 Summary 152
Suggestions for Further Reading 153
Problems 155
Objectives
1. Introduce the basic forms of luminescence.
2. Present the dipole model of luminescence based on radiation from the acceleration of charges.
3. Introduce the quantum mechanical description of acceleration of charges that can be used to calculate radiation rate and radiative power.
4. Introduce the free exciton and the mechanism by which an exciton emits a photon through dipole radiation.
5. Describe the two-electron atom and the concept of indistinguishable particles.
6. Present the resulting molecular excitons and their classification as singlet excitons or triplet excitons.
Principles of Solar Cells, LEDs and Diodes: The role of the PN junction, First Edition. Adrian Kitai.
© 2011 John Wiley & Sons, Ltd. Published 2011 by John Wiley & Sons, Ltd.
7. Describe the luminescent properties of fluorescence and phosphorescence that are observed from singlet and triplet excitons respectively.
8. Describe the band-to-band emission and recombination model that determines absorption or radiation spectra based on band states and their probability of occupancy.
9. Introduce the human visual system and the units of luminescence and colour that allow light sources to be described with relevance to human perception.