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Transcript
CHAPTER 3: CARRIER CONCENTRATION PHENOMENA Part 2 CHAPTER 3: Part 2 Continuity Equation Thermionic Emission Process Tunneling Process High-Field Effect CONTINUITY EQUATION Continuity Equation – to consider the overall effect when drift, diffusion, and recombination occur simultaneously in s/c material. Fig. 3.15 – infinitesimal slice with a thickness dx located at x – it may derive by one dimensional continuity equation. The number of electron in the slice – increase due to net current flow into the slice & the net carrier generation in the slice. The overall rate of electron increase is algebraic sum of 4 components: A–B+C–D where; A = number of electrons flowing into the slice at x B = number of electrons flowing out at x + dx C = rate which electrons are generated D = rate at which they are recombined with holes in the slice Figure 3.15. Current flow and generation-recombination processes in an infinitesimal slice of thickness dx. CONTINUITY EQUATION (cont…) The overall rate of change in the number of electrons in the slice: J n ( x) A J n ( x dx) A n Adx (Gn Rn ) Adx t q q (1) For 1-D, under low injection condition, the continuity eq. for minority carriers: yk yk 2 yk ( yk yko ) E zyk k k E Dk Gk 2 t x x x k (2) where, y = n, k = p, and y = p, k = n, (i.e np – in p-type s/c, and pn – n-type s/c. CONTINUITY EQUATION (cont…) • In addition to continuity equations, Poisson’s equation: dE s dx s (3) must be satisfied, and s – s/c dielectric permittivity, s – space charge density, where s = q(p – n + ND+ - NA-) Continuity Equation (cont…) Solve the Continuity Equation Steady-state injection from one side Minority carriers at the surface The Haynes-Shockley Experiment Steady-State Injection From One Side n-type semiconductor • Assume that light is negligibly small, and assumption of zero field & zero generation at x > 0. • At steady state there is a concentration gradient near surface. From (2) the diff. equation for minority carriers inside s/c is pn 2 pn ( pn pno ) 0 Dp 2 t x p Pn: Holes in n-type s/c (4) Figure 3.16. Steady-state carrier injection from one side. (a) Semi-infinite sample. (b) Sample with thickness W. Steady-State Injection From One Side (cont..) The solution of pn(x) by considering the boundary conditions, pn(x = 0) = pn(0) = constant value, and pn (x ) = pno, thus pn ( x) pno pn (0) pno exp( x / L p ) L p D p p (5) 1/ 2 Where, is called diffusion length. • With thickness x = W, thus pn ( x) pno pn (0) pno Where, sinh (W x) / L p (6) sinh( W / L p ) Current density at x = W is given by J p q pn (0) pno Dp L p sinh( W / L p ) (7) Minority Carriers at the Surface When surface recombination is introduced (Fig. 3.17), the hole current density flowing into the surface from the bulk of the s/c. It’s given by qUs. Assume that the sample is uniformly illuminated with uniform generation of carriers. Surface recombination leads to a lower carrier concentration at the surface. The solution of the continuity equation based on boundary condition (x = 0, and x = ), is S exp( x / L (8) ) pn ( x) pno p GL 1 p lr ( L p p Slr ) Graph pn(x) versus x in Fig. 3.17 for a finite Slr. When Slr , thus, pn ( x) pno p GL 1 exp( x / L p ) Slr: low injection surface recombination velocity p (9) Minority Carriers at the Surface (cont.) Figure 3.17. Surface recombination at x = 0. The minority carrier distribution near the surface is affected by the surface recombination velocity. The Haynes-Shockley Experiment -One of the classic experiments in semiconductor physics to demonstrate drift and diffusion of minority carriers. Phys. Rev. Vol 81 pg. 835 (1951) • The voltage source V1 establishes an electric field in the +x direction in the n-type semiconductor bar. Excess carriers are produced and effectively injected into the semiconductor bar at contact (1) by a pulse. Without applied field • Contact (2) may collect a fraction of the excess carriers as they drift through the s/c. Carrier distributions With applied field Figure 3.18. The Hayes-Shockley experiment. (a) Experimental setup. (b) Carrier distributions without an applied field. (c) Carrier distributions with an applied field. The Haynes-Shockley Experiment (cont…) After a pulse, by setting Gp = 0, and E/x = 0 (applied electric is constant across the conduction bar), thus transport equation is given by: p n p n 2 p n ( p n p no ) (10) And the solution may be written as p E Dp 2 t x p x p n ( , t ) 2 t exp p no 4D p t 4 D p t p N (11) For no electric field applied along the sample, = x, and with electric field, = x - pEt . N = number of electrons or holes generated per unit area. Illustrated by Fig. 3.18(b). For E = 0, carriers diffuse away from the point of injection and recombine. For E 0, all excess carriers move with drift velocity pE, and diffuse outward and recombine as in the field-free case. Thermionic Emission Process At the s/c surface, carriers may recombine with recombination centers due to the dangling bonds of the surface region. Thermionic Emission Process – condition where the carriers have sufficient energy to ‘thermionically’ emitted into the vacuum. Fig. 3.19(a) – band diagram of an isolated n-type s/c. q is the energy difference between the condition band edge & the vacuum level in the s/c. qs – work function (energy between Fermi level & vacuum level in the s/c). If energy > q - electron can be thermionically emitted into the vacuum. • Electron density with energies > q may be written as (a) q ( Vn ) nth n( E )dE N C exp (12) kT q NC – effective density of states in cond. band. Vn – is the difference between bottom of cond. band & Fermi level. Figure 3.19. (b) (a) The band diagram of an isolated ntype semi-conductor. (b) The thermionic emission process. TUNNELING PROCESS • Fig. 3.20a – the energy diagram when two isolated s/c samples are brought close together. •qV qVo =two q. s/c sample and Distance between o respectively. potential barrier height represents by d and • If d<<<, electron at left-side s/c may transport across the barrier & and move to the other side (even if electron is << barrier height.) – called Quantum Tunneling Phenomena. Figure 3.20. (a) The band diagram of two isolated semiconductors with a distance d. (b) One-dimensional potential barrier. (c) Schematic representation of the wave function across the potential barrier. TUNNELING PROCESS (cont…) Classic case: particle is always reflected (if E < qVo). Quantum case: particle has finite probability to transmit or ‘tunnel’ through the potential barrier. As usual, the behavior of particle (conduction electron) in the region with qV(x) = 0 can be described by Schrödinger equation: 2 d 2 E 2 2mn dx or 2m n d 2 2 E 2 dx (13) mn – effective mass, ħ - reduced Planck constant, E – kinetic energy, and - wave function of the particle. The solution of (13) are ( x) A exp( jkx) B exp( jkx) ( x) C exp( jkx) Where k = (2mnE/ ħ2)1/2. for x 0 for x d (14) Schrodinger Equation Additional Fact!! The Schrodinger equation plays the role of Newton’s Law and conservation of energy in classical mechanics - i.e., it predicts the future behavior of a dynamic system. It is a wave equation in terms of the wave function which predicts analytically and precisely the probability of events or outcome. The detailed outcome is not strictly determined, but give a large number of events, the Schrodinger equation will predict the distribution of results. The kinetic and potential energies are transformed into the Hamiltonian which acts upon the wave function to generate the evolution of the wave function in time and space. The Schrodinger equation gives the quantized energies of the system and gives the form of the wave function so that other properties may be calculated TUNNELING PROCESS (cont…) For x 0 : incident-particle wave function with amplitude A. : reflected wave function with amplitude B. For x d : transmitted wave function with amplitude C. Inside the potential barrier, wave function may be written as d 2 qV0 E 2 2mn dx or d 2 2mn (qV0 E ) 2 2 dx (15) The solution of E < qVo: ( x) F exp( x) G exp( x) Where = {2mn(qVo – E)/ħ2}1/2. (x) illustrated at Fig. 3.20(c). (16) TUNNELING PROCESS (cont…) Transmission coefficient may be written as (qV0 sinh d ) C 1 4 E ( qV E ) A 0 2 2 1 (17) Transmission coefficient decreases as E decreases. When d >> 1, (C/A) <<< and varies as C ~ exp( 2d ) exp 2d A 2 2mn (qV0 E ) * Used for tunneling diodes in Chapter 8 (18) HIGH-FIELD EFFECT At low electric field, drift velocity is proportional to the applied field, and assume that time interval between collision c is independent to applied field. Fig. 3.21 shows the measured drift velocity of electrons and holes in Si as a function of the electric field. Drift velocity increases less rapidly when electric field increased. At large field, the drift velocity approaches a saturation velocity. Thus, from experimental investigation, it may be expressed by Drift velocity: vn , v p vs 1 E / E 1/ (19) 0 Where vs – saturation velocity (107cm/s for Si at 300K). E0 – constant field which is 7 x 103 V/cm for electrons and E0 = 2 x 104V/cm for holes in high-purity Si materials. - 2 for electrons and 1 for holes. vs at high field is particularly likely for FET – discuss more in Chapter 6. HIGH-FIELD EFFECT (cont…) Figure 3.21. Drift velocity versus electric field in Si. HIGH-FIELD EFFECT (cont…) Fig 3.22 shows the differentiation between high-field transport in n-type GaAs and Si. For n-type GaAs – vs reached maximum level, then decreases when the field increases. This phenomenon is due to energy bands structure of GaAs that allows the transfer of conduction electrons from high mobility energy minimum (called valley) to low mobility. Means that, electron transfer from the central valley to the satellite valleys along [111] direction (discussed in Chapter 2). HIGH-FIELD EFFECT (cont…) Figure 3.22. Drift velocity versus electric field in Si and GaAs. Note that for n-type GaAs, there is a region of negative differential mobility. HIGH-FIELD EFFECT (cont…) Fig. 3.23 gives a clear view of the phenomena in Fig. 3.22, where it considers the simple two-valley model of n-type GaAs at various conditions of electric fields. Energy separated between two-valleys is E = 0.31eV. The lower valley’s electron effective mass, electron mobility, and electron density are represented by m1, 1, and n1 respectively. The upper level represents by the same expression with subscript 2. HIGH-FIELD EFFECT (cont…) Figure 3.23. Electron distributions under various conditions of electric fields for a twovalley semiconductor. HIGH-FIELD EFFECT (cont…) Total electron concentration is given by n = n1 + n2. The steadystate conductivity of n-type GaAs may be written as q(1n1 2 n2 ) qn The average mobility is Drift velocity may be written as v s E 1 n1 2 n2 n1 n2 (20) (21) (22) At Fig. 2.32(a), E << and all electrons remain in the lower valley. Fig. 2.32(b), E is higher and some electrons gain sufficient energies from the field to move to the higher valley. Fig. 2.32(c), E >>, it may transfer all electrons to the higher valley. HIGH-FIELD EFFECT (cont…) In mathematical view; For 0 < E < Ea n1 n Ea < E < E b n n n 1 2 E > Eb n1 0 The drift velocity: For 0 < E < Ea For E > Eb v n 1 E vn 2 E and and and n2 0 n2 n n2 n (23) (24) If 1E > 2E – there is a region which the vs decreases with an increasing field at Ea < E < Eb shown in Fig. 3.24. With this characteristic of n-type GaAs drift velocity – this materials is used in microwave transferred-electron devices (discuss in Chapter 8). HIGH-FIELD EFFECT (cont…) Figure 3.24. One possible velocity-field characteristic of a two-valley semiconductor. HIGH-FIELD EFFECT (cont…) When E in s/c is increased above a certain value – the carriers gain enough K.E to generate electron-hole pairs by an avalanche process shown in Fig. 3.25 (electron in cond. band represented by 1). If E >>> - electron can gain K.E before it collides with the lattice. On impact with the lattice – electrons imparts most of its K.E to break a bond – to ionize a valence electron from the valence band to the cond. band & generate an electron-hole pair (represented by 2 and 2’). This process continued to generate another electron-hole pairs (e.g 3 and 3’, 4 and 4’) and so on. This process called avalanche process. This process will results in breakdown in p-n junction (discussed in Chapter 4). Figure 3.25. Energy band diagram for the avalanche process. HIGH-FIELD EFFECT (cont…) Consider the process of 2 – 2’: Just prior to the collision, fast moving electron (no. 1) has a K.E = ½ m1vs2, and momentum, p = m1vs, (m1 – effective mass). After collision, there are 3 carriers: the original electron + electron hole pair (no.2 and 2’). If we assume the 3 carriers have same effective mass, same K.E, and same p, thus the total K.E = 3/2 (m1vf2), and total p = 3m1vf. vf - velocity after collision. To conserve both energy and momentum before and after the collision, thus and 1 3 m1v s2 E g m1v 2f 2 2 m1v s 3m1v f respectively. (25) (26) HIGH-FIELD EFFECT (cont…) Eg – band gap corresponding to the minimum energy required to generate an electron-hole pair. By substitute (26) into (25), thus the required K.E for the ionization process may be written as E0 1 m1v s2 1.5 E g 2 (27) E0 > Eg for the ionization process to occur. It depend on the band structure, where for Si, electron and hole are E0 = 3.6eV (3.2Eg) and E0 = 5.0eV(4.4Eg) respectively. The number of electron-hole pairs generated by an electron per unit distance traveled – ionization rate, where for the electron and hole is represented by n and p respectively. HIGH-FIELD EFFECT (cont…) Measurement of ionization rates for Si and GaAs are shown in Fig. 2.36. (n and p are strongly dependent on the electric field. For large ionization rate (say 104cm-1), the corresponding electric field is 3 x 105 V/cm for Si and 4 x 105 V/cm for GaAs. Electron-hole pair generation rate GA from the avalanche process is given by GA n | J n | p | J p | (28) q Where Jn and Jp are the electron and hole current densities, respectively. This expression may be used in the continuity equation for devices operated under the avalanche condition. HIGH-FIELD EFFECT (cont…) Figure 3.26. Measured ionization rates versus reciprocal field for Si and GaAs. CONCLUSION Excess carriers in s/c cause non-equilibrium condition, where most of s/c devices operate under this circumstances. Carriers may be generated by: forward-bias of p-n junction, incident light, and impact ionization. Continuity equation – the governing equation for the rate of charge carriers. Thermionic emission occurs when carriers in the surface region gains enough energy to be emitted into vacuum level. Tunneling process – based on the quantum tunneling phenomena that results in the transport of electrons across a potential barrier even if the electron energy is less than the barrier height. When the electric field become higher, drift velocity departs from its linear relationship with the applied field & approaches a saturation velocity. This phenomena is important in the study of short-channel field-effect transistor (Chapter 6). CONCLUSION (cont…) When the electric field exceeds a certain value, the carriers gain enough K.E to generate electron hole-pair by colliding with the lattice & breaking a bond. This effect particularly important in the study of p-n junctions. Impact ionization or avalanche process – high field accelerates a new electron-hole pairs, which collide with the lattice to create more electron-hole pairs. From avalanche process, the p-n junction breaks down and conducts a large current (you may learn more about this topic in Chapter 4). "Twenty years from now you will be more disappointed by the things that you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover." ~ Mark Twain ~ American Writer “The longest journey begins with a single step” ~Confucius~