1. UCONN_ECE 4211/LW2 Week 2-1, 2-2 January 26, and 28, 2016 (FCJ) Chapter 1 Quantization of levels in a quantum well Energy Band E-K Diagram: Direct and Indirect Electron and hole concentrations Electrical conductivity-resistivity 1 Chapter 2 Work function and electron affinity Work function difference for n- & p- Si; Method I for built-in voltage V bi Junction formation and Methods II and III for Vbi Forward biased p-n junction (carrier distribution and I-V equaiton Reverse biased p-n junciton Avalanche and Zener breakdown Heterojunctions

2. Schrodinger Equation and infinite well representing single atom (p.20) Quantization of energy levels and momentum In the region x = 0 to x = L, V(x) = 0. Eq. 1 simplifies to: (1) Boundary conditions x = 0 and x = L, (x) = 0., for  (L) = 0 2

3. Nearly free electrons: energy bands 3 Fig.21. Electrons kinetic energy (KE) as a function of momentum.

4. 1.6.3 Schrodinger equation in one-dimensional chain: Kronig-Penny Model: Energy Bands Theory II 4 Fig.22. Potential in Kronig-Penney model representing perturbing potential due to ionic cores shown in Fig. 19 (right). The solution is modulated [u k (x)] electron plane waves [e ikx ].

5. Energy-Momentum diagram 5 Fig. 24(a) Allowed electron energy E as a function of ka. The dashed curve is the plot due to nearly free electron theory and the segments corresponds to AB, CD, EF and other allowed electron energies. Fig. 25(a). Energy vs wave vector, E-k, band structure for germanium (left panel), silicon (middle), gallium arsenide. E g is the energy gap. (J.R. Chelikowsky and M.L. Cohen, Phys. Rev. B14, 556 (1976))

6. Direct and Indirect Energy Gaps 6 Fig. 25(b). Energy-wavevector (E-k) diagrams for indirect and direct semiconductors.

7. Summary: Carrier Concentrations 7 In n-type case, n n = N D, and p-Si: Acceptor atoms in Si lattice If N A is the concentration of boron atoms (or other acceptor atoms) in Si, the acceptor concentration is assumed to be all ionized and greater than intrinsic carrier concentration n i (N A >> n i ). In this case, p p = N A, and. Exact and accurate determination of carrier concentration using Fermi level E f Actual calculation of carrier concentration involves the knowledge of density of states N (E) and statistical distribution function (like Fermi-Dirac distribution f (E)=[1/(1+exp (E-E f /kT)]). Here, E f is the Fermi level. The electron concentration is expressed by Eq. 84 (84A) (85A) n=Nc, (84-AA),. (75) (75) (84A), here E c =0 and E v =-E g. (85A)

8. Electron concentration Fig 28(a) 8

9. Charge neutrality condition to find carrier concentration 9 N D + + p no = n n (92) (93) (94)

10. 1.10.4 Electrical Resistivity Carrier Transport and Current Conduction 10 Semiconductors with majority carriers of electrons (n >> p) are referred to as n-type; when the majority carriers are holes (p>>n), they are called p-type. The conductivity is expressed as Eq. 116 as a function of electron and hole concentrations. N-type:, Eq. 116A, (116B) Here, q is the magnitude of electron charge = 1.6  10 -19 coulomb, n the electron concentration, p the hole concentration, and  n and  p are the electron and hole mobility, respectively. (122) P-type:

11. Resistivity and Mobility 11 Figure 35. Resistivity versus impurity concentration for silicon at 300 K. (Beadle, Plummer, and Tsai, Quick Reference Manual for Semiconductor Engineer, Wiley, New York, 1985). Fig. 37. Mobility as impurity concentration (N a + N d ) for Ge, Si and GaAs at 300  K.

12. 2.3 Work function q  and electron affinity q  in p- and n-semiconductors 12 Chapter 2: Fig. 1. (a) Energy levels in p-type semiconductor, (b) in n-type semiconductor. ECE 4211 LW2 Lecture Week 2- lecture 2 F. Jain February 3, 2015 See also ECE course pages L2 http://www.ee.uconn.edu/undergraduate-program/course-pag

13. Work function difference and built-in voltage V bi 13 The work function for p-Si is obtained from Fig. 1(a), (here, all three terms are added), The difference in the work function for an n-p Si junction is We get V n -V p = V bi Thus, the voltage difference from n-Si to p-Si is obtained by dividing the work function difference (energy) by –q. The work function for n-Si is obtained from Fig. 1(b), (two arrows are added and one subtracted,

14. 14 Here, the reverse saturation current I s = A J s, and reverse saturation current density J s is Here, D p is the diffusion coefficient of holes, p no is the hole concentration under equilibrium on n-Si side, L p is the diffusion length of holes (L 2 p = D p *  p ).  p is the average lifetime of injected holes. Similarly, L n is the diffusion length of electrons injected from n-side into p-side and n po is the minority electrons on p-side at equilibrium. Fig. 7 P-n Junction under bias: Shockley's equation

15. Space charge region or junction formation under equilibrium 15 When p-Si and n-Si are joined, there is a transfer of electrons from n-side to p-side and holes from p-side to n-side and an equilibrium state is reached. When holes leave the p-side they leave behind negatively charged acceptors and similarly, when electrons leave n-side they leave behind positively charged donors. This establishes a space charge density is shown in Fig. 4b and 4c. This in turn establishes an electric field E. Fig. 3 (above) shows the profile of carrier concentrations in the junction region. The concentration gradient dn/dx and dp/dx and electric field oppose each other. This results in an equilibrium junction width W o.

16. 16 2.5.1.1 Evaluation of built-in voltage, V bi, in the junction region: Method II One method to calculated built-in voltage using the work functions has been described. This is the second approach and it depends on the fact that there is no net electron current or hole current flow under equilibrium after junction is formed (10) Using, (13) Using, (16A)

17. 17 2.5.1.2 Evaluation of built-in voltage using Poisson’s Equation Method III: Gauss’s law  ·D =  ρ = q[p(x) + N D + - n(x) – N A - ](25) Poisson’s Equation In abrupt p-n homojunctions, assuming all donors and acceptors are ionized (qN A - ~ qN A, and qN D + ~ qN D ) and neglecting n(x) and p(x) in the junction, we simplify the derivation for electric field E(x) in the junction and the derivation of built-in voltage V bi. Using Eqs. (22), (26) and (27) Here, D is the displacement,  is the charge density and E is the electric field and psi is the voltage. (26) (27) Integrating Eq. (28) from –x po to 0, and keeping E(-x po ) = 0 and E(0) = E m (28) (29) Integrating Eq. (29) from 0 to x no, and keeping E(x no ) = 0 and E(0) = E m (30) (31)

18. Built-in Voltage 18 Voltage across the junction is obtained by integrating the electric field E from –x po to x no. Alternatively, it is obtained by taking the area of electric field plot. W is the junction width under equilibrium. It is W o to distinguish from forward bias width W f. (23) Fig. 5(a) Voltage  as a function of x.

19. 19 Fig. 5. Schematic representation of energy band diagram for a p-n junction showing electron energy.

20. 20 P-n Junction at Equilibrium Use (V bi – V f ) for forward-biased junctions, and (V bi + V r )for reverse-biased junctions. Fig. 2 (40B) Built-in voltage: Method I

21. 21 Equilibrium (VA)(VA) -xp-xp xnxn 0 lnln lplp pn N A = 10 19 cm -3 VAVA I Forward bias x nene pepe -x po x no 0 lnln lplp pn oo N A = 10 19 cm -3 N D = 10 16 cm -3 n(x)on(x)o p(x)op(x)o E(x)E(x) NANA NDND Carrier distribution in a p-n junction under equilibrium Space charge Electric field Carrier distribution under forward biasing

22. 2.5.2. Forward-Biased p-n Junction 2.5.2.1 Distribution of Carrier Concentrations using Continuity Equation: 22 Figure 6A. Carrier concentrations in a forward- biased p-n junction. When a forward bias is applied, it alters the balance between concentration gradient and electric field that was established under equilibrium. As a result more holes flow from p-side to the n-side and electrons from n-side to the p-side. New electron and hole carrier concentration profiles are established in the junction region as well as in the neutral n and p-regions. The junction width W f is = x n + x p. This results in net current flow I in the external circuit. Higher the V f, greater is the current flow.

23. Continuity Equation and finding carrier concentration of holes p n (x) in n-region and n p (x) in the p-region 23 The continuity equation gives the hole concentration distribution in the n- region [p n (x)]. It can also be written for electrons in the p-region. It expresses rate of increase of hole concentration. (47) Eq. 47 expresses the rate of increase of hole concentration p n (x) on n-side (x > x n ) outside the junction, is due to the divergence term and rate of recombination term that depends on the excess concentration (p n –p no ) and the lifetime of holes  p. The divergence of (J p /q) expresses the net outflow of carriers per second per unit volume. Putting a ‘-‘ sign makes it as net inflow of carriers per unit volume per unit time. That is, rate of increase or rate of inward flow due to non-uniform current density. (48) (49) Under steady state (with no time variation dp n /dt =0), Eq. 49 simplifies to = 0 (50A) and d  p(x)/dx = dp n (x)/dx, and dp no /dx = 0 as p no is constant, Eq. 50A becomes (51A) The solution of partial differential Eq. (51) is: (53A)

24. Carrier concentration of holes p n (x) and electrons n p (x) 24 A and B in Eq. 53A are obtained using suitable boundary conditions. These are (53A) (57) (58) Substituting boundary condition Eq. (58) in Eq. (53A) Substituting boundary condition Eq. (57) in Eq. (53A) p n (x) = p no + (60A) (60B) The excess hole concentration decays exponentially as shown by Eq. (60A). The hole concentration is The excess electron concentration in n-region decays exponentially as shown by Eq. (61A). (61A) n p (x) = n po +

25. 2.5.2.2 Diffusion currents in the n- and p-neutral regions of diode: Shockley Equation 25 The current due to injected minority holes on n-side is obtained using: (64), Multiply by cross-section area A of diode (64B) Electron diffusion on the p-side is expressed by (66B)

26. I-V Equation (cont.) 26 Adding Eqs. (64) and (66) gives total current density throughout the diode. It assumes that there is no electron-hole recombination in the junction region, that is, J p and J n are constant in the junction. (67) Using Eq. (68), Eq. 67 simplifies to (68) (69) (70)

27. Current density and carrier concentrations 27 Figure 8A. Current density variation along p-n diode in bulk (neutral) regions and junction. Forward biased p-n junctions: Minority holes injection and recombination with electrons Minority carriers are injected as we forward bias a p-n junction. For example, holes are injected from the p-side across the junction into the n-region. The level of injection, i.e. hole concentration p e at the junction/n-region boundary is p e = p no [exp(qV f /kT)]. Similarly, electrons are injected from the n-region into the p-region, and n e = n po [exp(qV f /kT)]. Here, p no is the equilibrium minority hole concentration on the n-side and n po (n i 2 /p po =n i 2 /N A, N A is acceptor concentration) is the minority electron concentration on the p-side. If p-region is more heavily doped that the n-region, p e >>n e, and hole injection dominates and determines the device I-V and other characteristics. Injected minority holes recombine in the n-region where electrons are in majority. As the holes recombine, their concentrations p n (x) is reduced as they travel towards the Ohmic contact. Injected holes have an average lifetime  p which is related as L p 2 = D p  p, here L p is the diffusion length and D p is the diffusion coefficient. D p is related to mobility  p as D p = (kT/q)  p.

28. 28 Carrier recombination and photon emission: (see also LEDs and Lasers) Radiative (photon emission) and non-radiative transitions: Since hole belongs to the valence band and the electrons to the conduction band, the recombination of an electron-hole pair results in energy release. When the energy is released as a photon, the transition is called radiative. On the other hand, when the energy released as phonons (or lattice vibrations) it is nonradiative, and causes heating of the material. Number of photons produced per second (in a region defined by x) depends on the current I p (x). the total number of photons produced per second in n-region is I p (x n )*  q /q. Quantum efficiency h q defines the ratio of probability of a radiative transition. It is expressed as  q = (1/  r )/[(1/  r ) + (1/  nr )], here,  r and  nr are the radiative and non-radiative lifetimes, respectively. The quantum efficiency is higher in the direct energy gap semiconductors than in indirect energy gap semiconductors. It takes longer for an electron-hole pair to recombine in indirect energy gap semiconductors as they need the assistance of a phonon to conserve the momentum. Both energy and momentum are conserved in a downward transition, such as that occurs when an electron-hole pair recombines. In direct energy gap semiconductors, the recombination via photon emission is more likely than in indirect semiconductors, such as Si and Germanium. In indirect energy gap semiconductors, the electrons and holes have different momentum or wavevectors (k). [Momentum is related to k via the Planck’s constant; p=h/ =(h/2p)k]. Photons have comparable energy (E=h ) ) to electrons and holes. However, they have very small momentum when compared to them. As a result, photon absorption and emission, involving electrons and holes, are shown by vertical lines on the E-k diagram (or energy band diagram). By contrast, lattice vibrations or phonons travel with the speed of sound, and have very small energy (~20 to 80 milli electron volt). But their momentum is comparable to that of electrons and holes. As a result phonons play a crucial role in transitions involving photons in indirect transitions. Extraction of generated photons: once the photons are generated they travel in different directions. Depending on the application, sometime we want to stay inside a cavity (lasers) or move along waveguides (photonic integrated circuits) and other times we want to extract them out (light-emitting diodes, LEDs). See the section on waveguiding layers to configure LEDs as lasers.

29. 29 2.5.7 Energy band diagrams under forward and reverse biasing: Fig. 10 shows energy band diagrams for equilibrium, forward bias and reverse biasing conditions. We outline steps to obtain these diagrams: Step 1. Draw a straight line and call it Fermi energy under equilibrium (dashed line). Step 2. Draw two vertical lines representing junction. Note that the equilibrium width W o (left figure) is larger than forward bias diode (center W f ) and smaller than the reverse biased diode (W r right). Step 3. Draw new Fermi level in the neutral p-semiconductor with respect to the n-semiconductor. There is no change in equilibrium or left figure. Under forward bias, p-region is qV f lower than the equilibrium value as shown in the center figure. In forward bias of 0.2V, p-side is positive, but the electron energy band diagram is 0.2eV below the equilibrium Fermi level. The case is opposite in the reverse bias of -0.2V. Here, p-side is V r = -0.2V negative with respect to n-side, but the new Fermi level is 0.2eV higher than the equilibrium level. Step 4. Draw valence band and conduction band on the p-side using new Fermi level. Step 5. Join conduction bands and valence bands between the two vertical lines.

30. 30 Energy band diagram: Heterojunction