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Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Chapter 6 Carrier Transport Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport DRIFT Definition-Visualization Drift is charge-particle motion in response to an applied electric field. The relaxation time (ππ ) can be interpreted as mean free time (π‘) between collisions, if a particle reaches equilibrium by the collision once. The probability that a particle experience collision during time dt: = If there are n(t) particles, Collision becomes less with time. ππ(π‘) = βπ(π‘) Number of particles experience a collision during dt. Average time, π‘ = β 0 π‘π(π‘)ππ‘ β 0 π(π‘)ππ‘ Mean free time ππ‘ ππ ππ‘ ππ = ππ ππ(π‘) π(π‘) =β ππ‘ ππ π = π‘ β π£π‘β Mean free path π(π‘) = π 0 π βπ‘/ππ Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport In steady-state, carriers drifts at a constant drift velocity by balancing between acceleration by electric field and deceleration by collision. πππ₯ πππ₯ |πβπππππ + | =0 ππ‘ ππ‘ ππππππ πππ πππ₯ acceleration, | = βππππ₯ ππ‘ πβπππππ deceleration, In 3-D, < ππ₯ > ππ‘ = β π ππβ ππβ π₯ π£π = β π£π = β ππ,π πππ₯ = βππ₯ ππ‘ β π = ππ π ππ,π ππ‘ β π = ππ π ππ,π 1 1 1 =3 + + ππβ ππ‘β ππ‘β ππ‘ π‘ Momentum change due to collision during dt πππ₯ ππ₯ |ππππππ πππ = β ππ‘ π‘ Average momentum per electron, < π£π₯ >= Total momentum at t < ππ₯ >= ππ₯ πππ₯ =β π‘ = βπ π‘ππ₯ π πππ‘ 1-D expression for electron β β , ππ,π Where ππ,π are conductivity effective masses. for hole β1 β ππ,π = β ππβ 3/2 β + πββ 3/2 β 1/2 β 1/2 ππβ + πββ Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Two effective masses of carrier 1) Density of state effective mass, ππβ , in the density of state function 2) Conductivity (or mobility) effective mass, ππβ , in the expression for mobility The density of states effective mass for electrons and holes is given by, β ππβ = πππ = ππ₯π₯ ππ₯π₯ ππ₯π₯ 1/3 = ππ₯π₯ for the π€ -valley (= ππ₯π₯ = ππ¦π¦ = ππ§π§ ) = ππ ππ‘2 ππβ = β 3/2 πββ + β 3/2 ππβ 2/3 1/3 for the X or L -valley β ππβ 3/2 β = πββ 3/2 β + ππβ 3/2 The conductivity (or mobility) effective mass for electrons and holes is given by, 1 1 1 1 1 1 = + + = β ππ,π 3 ππ₯π₯ ππ¦π¦ ππ§π§ ππ₯π₯ = 1/2 1/2 β β 1 ππβ + πββ β = β 3/2 β 3/2 ππ,π ππβ + πββ for the π€ -valley (= ππ₯π₯ = ππ¦π¦ = ππ§π§ ) 1 1 2 + 3 ππ ππ‘ for the X or L -valley β π½π = ππππππ π = πββ ππββ + ππβ πππβ π β πββ β = πββ 1/2 β + ππβ 1/2 β = ππβ 3/2 β 1 β = ππ,π 3 2 β 1 β β + ππβ πββ β πββ 3/2 β + ππβ 1 β ππβ 1 3/2 β ππ,π 3/2 β Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Drift Current The formal definition of current, πΌπ|πππππ‘ = πππ£π π΄ In vector notation, π½π|πππππ‘ = πππ£π Excluding situations involving large β° fields, π£π = ππ β° where ππ ,the hole mobility, is the constant of proportionality constant π½π|πππππ‘ = πππ πβ° similarly, π½π|πππππ‘ = πππ πβ° Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Mobility The carrier mobility varies inversely with the amount of scattering taking place within the semiconductor. To theoretically characterize mobility it is therefore necessary to consider the different types of scattering events that can take place inside a semiconductor. (i) (ii) (iii) (iv) (v) Phonon (lattice) scattering Ionized impurity scattering Scattering by neutral impurity atoms and defects Carrier-carrier scattering Piezoelectric scattering For the typically dominant phonon and ionized impurity scattering, single-component theories yield, respectively, to first order ππΏ β π β3/2 ππΌ β π β3/2 /ππΌ where ππΌ = ππ·+ + ππ΄β Matthiessenβs Rule Noting that each scattering mechanism gives rise to a βresistance-to-motionβ which is inversely proportional to the component mobility, and taking the βresistanceβ to be simply additive (analogous to a series combination of resistors in an electrical circuit), one obtains 1 1 1 = + +β― ππ ππΏπ ππΌπ 1 1 1 = + +β― ππ ππΏπ ππΌπ Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Doping/Temperature Dependence The Si carrier mobility versus doping and temperature plots presented respectively in Figs 6.5 and 6.6 were constructed employing the empirical-fit relationship π = ππππ + π0 1 + (π/ππππ )πΌ where π: carrier mobility N: doping density(either NA or ND) All other quantities are fit parameters that exhibit a temperature dependence of the form π Ξ· πΌ = π΄0 ( ) 300 where π΄0 : temperature-independent constant T: temperature in Kelvin Ξ· βΆ temperature exponent for the given fit parameter Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport ππΏ β π β3/2 from first order theory Experimental values for lightly doped Si, π β ππΏ β π β2.3±0.1 β π β2.2±0.1 for electron for hole Advanced Semiconductor Fundamentals For GaAs, Chapter 6. Carrier Transport Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport High-Field/Narrow-Dimensional Effects Under low electric field i) ii) Carrier gains energy from the electric field and loses the energy through collisions with low energy acoustic phonons or impurities. 3 The averages energy of the electrons β ππat thermal equilibrium (π»π β π»πππππππ ) 2 iii) Drift velocity π£π β π and current density π½ β π. Velocity Saturation under high electric field i) ii) Electrons gain energy from the field faster than they can lose it to the lattice. The electron distribution can be characterized by effective temperature, ππ . (π»π > π»πππππππ : hot electron effect) iii) Drift velocity π£π and current density π½ are no longer linear with π. (nonohmic) iv) Electrons can transfer energy to the lattice by the generation of high energy optical phonons. This causes saturated drift velocity (ππ πππ ). In Si at 300 K, π£ππ ππ‘ β 107 cm/sec for both electrons and holes at π β 107 V/cm. Temperature dependence of π£ππ ππ‘ for electrons in Si can be modeled by the empirical-fit expression. 0 π£ππ ππ‘ π£ππ ππ‘ = 1 + π΄π π/ππ 0 π£ππ ππ‘ = 2.4 × 107 ππ/π ππ A = 0.8 Td = 600 K Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Intervalley Carrier Transfer For GaAs ellipsoidal constant energy surface βΞπΏ = 0.29 spherical constant energy surface eV π£π β Electrons in Ξ; ππΞ,π = ππβ = 0.063 π0 Electrons in L; ππβ = 1.9 π0 , ππ‘β = 0.075 π0 β πππΏ,π = 0.55 π0 π 3 ππ = 3.3 × 10 π/ππ β β πππΏ,π β 10ππΞ,π Under normal circumstances, the Ξ βvalley is the only one occupied, but for an applied field of ~ 3.5 KV electrons begin to be transferred to the L-valley. The resulting negative differential conductance occurs when the carriers are transferred from low mass, high velocity states to high mass, low velocity states is referred to as the βGunn Effectβ. Advanced Semiconductor Fundamentals 1 2πΞβ πΞ = 2 2π β2 1 2πΞβ πππ = 4 πβ 2 3/2 1 πΈ 2 ππ₯π 0 3/2 ππ₯π 1 2ππΏβ ππΏ = 4 β 2 2π β2 nπΏ ππΏβ πππΏ =4 nΞ πΞβ ππ β 3/2 Chapter 6. Carrier Transport β πΈ β πΈπΉ πππ πΈπΉ πππ β βΞπΏ 3/2 ππ₯π β 1 πΈ 2 ππ₯π ππΈ for nondegenerated semiconductor where Te is an electron temperature β πΈ β πΈπΉ ππππΏ ππΈ βΞπΏ πΈπΉ 1 1 ππ₯π β ππππΏ π πππΏ ππ = 2πΞβ πππ πβ 2 3/2 ππ₯π β βΞπΏ πΈπΉ ππ₯π ππππΏ ππππΏ If TeL = Te is an electron temperature, ππΏ β πΞ at Te = 950 K. For temperature higher than this, the upper valley has a higher density of states occupied. Thus when an electron initially in the Ξ-valley at energy of E = βπ€πΏ is scattered, it is more likely to undergo an intervalley scattering to L-valley. The total conductivity for carriers in the two set of valleys, π = nΞ ππΞ + nπΏ πππΏ where n = nΞ +nπΏ The change in the conductivity with electric field, assuming π is only a very weak function. ππ πnΞ πnπΏ πnΞ β ππΞ + πππΏ = πΞ β ππΏ ππ ππ ππ ππ πnΞ nπΏ =β ππ ππ Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport From the current density equation, π½ = ππ The differential conductivity, ππ½ ππ ππ½ ππ πnΞ =π+π = nΞ ππΞ + nπΏ πππΏ + ππ πΞ β ππΏ ππ ππ ππ if ππ½ ππ < 0, (-)function nΞ ππΞ + nπΏ πππΏ < βππ πΞ β ππΏ πnΞ ππ β πΞ β ππΏ π πnΞ nπΏ n ππ > 1 πΞ + ππΏ Ξ nΞ ππͺ > ππ³ π β 7000 ππ2 /π β π ππ Ξ πL β 100 ππ2 /π β π ππ also, β for GaAs. πnΞ nΞ > ππ π ππΏβ where πnΞ = β nπΏ = β4 ππ ππ πΞβ 3/2 ππ₯π β βΞπΏ πnΞ π βΞπΏ + nΞ ππ₯π β πππ ππ ππ πππ nπΏ ππΏβ πππΏ =4 nΞ πΞβ ππ 3/2 ππ₯π β βΞπΏ πΈπΉ 1 1 ππ₯π β ππππΏ π πππΏ ππ Advanced Semiconductor Fundamentals πnΞ ππΏβ =β4 ππ πΞβ =β 3/2 ππ₯π β nπΏ πnΞ βΞπΏ πππ + nΞ 2 nΞ ππ πππ ππ Chapter 6. Carrier Transport βΞπΏ πππ πnΞ βΞπΏ πππ + nΞ 2 ππ πππ ππ =β nπΏ πnΞ βΞπΏ πππ β nπΏ 2 nΞ ππ πππ ππ nΞ + nπΏ πnΞ βΞπΏ πππ = βnπΏ 2 nΞ ππ πππ ππ β πnΞ βΞπΏ nΞ nπΏ πππ nΞ = > ππ πππ ππ nΞ + nπΏ ππ π β πnΞ nΞ > ππ π nπΏ βΞπΏ π πππ >1 nΞ + nπΏ πππ ππ ππ Assuming that the electron temperature increases linearly with electric field, π πππ β1 ππ ππ βΞπΏ nΞ 1 ππΏβ >1+ =1+ πππ ππΏ 4 πΞβ β3/2 βΞπΏ ππ₯π πππ simple transcendental equation nπΏ ππΏβ =4 nΞ πΞβ 3/2 ππ₯π β βΞπΏ πππ Advanced Semiconductor Fundamentals βΞπΏ nΞ 1 ππΏβ >1+ =1+ πππ ππΏ 4 πΞβ There are two regions of βΞπΏ πππ Chapter 6. Carrier Transport β3/2 ππ₯π βΞπΏ πππ where this inequality is not satisfied: i) At very high electron temperature( not of interest) ii) At low electron temperature of interest Upper limit of βΞπΏ πππ β 5.8. Lower limit of electron temperature, ππ β 600 πΎ ππΏ β 0.15 nΞ So that negative differential conductivity sets when as little as 15 % of the electrons transferred to the upper valleys. π£ increasing βΞπΏ π Read βBallistic transport/velocity overshootβ. Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Related Topics Resistivity/Conductivity πΊ = Οπ± or 1 π π± = ππΊ = πΊ In a homogeneous material, π± = π±πππππ‘ = π±π|πππππ‘ + π±π|πππππ‘ = q(ππ n +ππ p)πΊ β΄ resistivity , π = 1 q(ππ n + ππ p) [Ξ©β ππ] conductivity, π = q(ππ n + ππ p) 1 qππ ππ· 1 π= qππ ππ΄ π= for n-type semiconductor for p-type semiconductor Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Sheet Resistance π π = π π‘ [Ξ©/β] π =π πΏ πΏ πΏ =π = π π A Wβt W Four-point probe technique 1) For thick sample (s << t) At probe 1, π π = π π½ π = βπ π π =β = ππ(π) ππ 1 D πΌ where π½ π = π 2ππ 2 ππΌ ππ + πΆ1 2ππ 2 πΌπ + πΆ1 2ππ π21 π31 = πΌπ + πΆ1 2π(2π ) t I In spherical coordinate system π πΌπ = + πΆ1 2ππ 2 3 4 π π Due to symmetry of current path, V is not function of π and β Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport At probe 4, I ππΌ ππ + πΆ2 2ππ 2 πΌπ =β + πΆ2 2ππ π π = π24 = β π π π πΌπ πΌπ + πΆ2 π34 = β + πΆ2 2π(2π ) 2ππ πΌπ πΌπ πΌπ + πΆ1 β + πΆ2 = + πΆ1 + πΆ2 2ππ 2π 2π 2π 2π πΌπ πΌπ πΌπ = + πΆ1 β + πΆ2 = β + πΆ1 + πΆ2 2π 2π 2ππ 2π 2π π2 = π21 + π24 = π3 = π31 + π34 π = π2 β π3 = πΌπ 2ππ β΄ π = 2ππ If t >> s, πΉ1 β 1, π‘/π If t << s, πΉ1 β , 2ππ2 π πΉ πΌ 1 where πΉ1 : thickness correction factor π‘/π πΉ1 = π‘ sinh π π 2ln π = 2ππ π‘ πΌ sinh 2π ππ‘ π π= ππ2 πΌ Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport 2) For thin sample (s >> t) At probe 1, π π = ππ½ π = βπ π π =β π21 = β ππ(π) πΌ where π½ π = π ππ 2πππ‘ ππΌ πΌπ ππ + πΆ1 = β πππ + πΆ1 2πππ‘ 2ππ‘ πΌπ πππ + πΆ1 2ππ‘ π31 = β 1 2 3 4 D t πΌπ ππ2π + πΆ1 2ππ‘ I π At probe 4, π π = πΌπ πππ + πΆ2 2ππ‘ In cylinderical coordinate system πΌπ πΌπ ππ2π + πΆ2 π34 = πππ + πΆ2 2ππ‘ 2ππ‘ πΌπ πΌπ πΌπ π2 = π21 + π24 = β πππ + πΆ1 + ππ2π + πΆ2 = ππ2 + πΆ1 + πΆ2 2ππ‘ 2ππ‘ 2ππ‘ πΌπ πΌπ πΌπ π3 = π31 + π34 = β ππ2π + πΆ1 + πππ + πΆ2 = β ππ2 + πΆ1 + πΆ2 2ππ‘ 2ππ‘ 2ππ‘ πΌπ π = π2 β π3 = ππ2 ππ‘ π24 = Advanced Semiconductor Fundamentals β΄π= Chapter 6. Carrier Transport ππ‘ π πΉ ππ2 πΌ 2 where πΉ2 : size correction factor ππ2 π· ( )3 +3 π ππ2 + ππ π· ( )3 β3 π ππ‘ π If D >> s, πΉ2 β 1, π = ππ2 πΌ πΉ2 = : same as before summary i) If t >> s and D >> s, π πΌ ii) If the condition for t >> s and D >> s is not satisfied, π = 2ππ π = 2ππ π πΉπΉ πΌ 1 2 iii) In most cases, t << s and D >> s, π= ππ‘ π ππ2 πΌ π π = πΉ1 β π‘/π , πΉ2 β 1, 2ππ2 π π π = 4.53 ππ2 πΌ πΌ Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Hall Effect Lorenz force in the sample assuming p-type, πΉ = π π£π × π΅ + π π π₯πΉπ₯ π¦πΉπ¦ π§πΉπ§ π₯ = π π£ππ₯ 0 π¦ π£ππ¦ 0 π₯ππ₯ π§ π£ππ§ + π π¦ππ¦ π΅π§ π§ππ§ πΉπ₯ = ππ£ππ¦ π΅π§ + πππ₯ πΉπ¦ = βππ£ππ₯ π΅π§ + πππ¦ Lorenz force in y-direction must be balanced under steady state. πΉπ¦ = βππ£ππ₯ π΅π§ + πππ¦ = 0 Moreover, π½π₯ = πππ£ππ₯ β π£ππ₯ = Hall coefficient, π π» = π½π₯ ππ ππ¦ 1 = π½π₯ π΅π§ ππ π π» = β 1 ππ β΄β for p-type for n-type π½π₯ π΅π§ + ππ¦ = 0 ππ Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport With measurable quantity, ππ¦ 1 ππ» /π ππ» π€ 108 ππ» π€ π π» = = = = = πΌ ππ π½π₯ π΅π§ π΅πΌ π΅πΌ π΅ π€π If VH is given in volts, w in cm, B in gauss, I in amps, and RH in cm3/coul. The resistance of the bar is just VA/I. π = ππ΄ π =π πΌ π€π β π= 1 ππ΄ π€π = ππππ» πΌ π The Hall mobility, ππ» = 1 1 π π» β = ππ π π More exacting analysis gives, ππ» ππ ππ» π π» = β ππ π π» = for p-type Hall factor ππ» β 1 for n-type The relationship between hall mobility and drift mobility ππ» = ππ» ππππππ‘ For GaAs, ππ» > 1 β ππππππ‘ < ππ» Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Anisotropic Conductivity The equations, so far, for the mobility, conductivity, and Hall constant are applicable for electrons in spherical band minima. The situation is somewhat more complicated, when the carrier transport in an ellipsoidal minima. For nonspherical energy surface with one ellipsoidal conduction band minimum at π€ (π = 0) ππ¦ transverse ππ₯ ππ¦2 β2 π 2 β2 ππ₯2 ππ§2 πΈ(π) β πΈπΆ β = + + 2ππβ 2 ππ₯π₯ ππ¦π¦ ππ§π§ ππ§ longitudinal π 2 ππ‘ π½π₯ = π = ππππ₯ ππ₯ ππ₯π₯ π₯ π 2 ππ‘ π½π¦ = π = ππππ¦ ππ¦ ππ¦π¦ π¦ π 2 ππ‘ π½π§ = π = ππππ§ ππ§ ππ§π§ π§ The total current density is therefore, π½ = π 2 ππ‘ 1 βπ πβ where 1 πβ is the effective mass tensor. Advanced Semiconductor Fundamentals 1 = πβ 1 ππ₯π₯ 0 0 0 1 ππ¦π¦ 0 0 1 ππ§π§ 0 π½π₯ π½π¦ π½π§ ππππ₯ 0 = 0 0 ππππ¦ 0 Chapter 6. Carrier Transport This can also be put in the form, π½ = π βπ where π is the conductivity tensor. ππππ₯ 0 π = 0 0 ππ₯ 0 β ππ¦ ππ§ ππππ§ 0 ππππ¦ 0 0 0 ππππ§ If π is off axis and three diagonal terms are not equal, the current is not in same direction as π. βanisotropic conductivityβ y π is off axis with ellipsoidal conduction band minima. (ππ₯π₯ β ππ§π§ β ππ¦π¦ , ππππ₯ β ππππ¦ β ππππ§ ) π π is on axis. π½ π is off axis but spherical conduction band minima. (ππ₯π₯ = ππ¦π¦ = ππ§π§ , ππππ₯ = ππππ¦ = ππππ§ ) π½ π π is on axis. x Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport For multiple equivalent ellipsoidal conduction band minimum at X or L (π β 0) ky 2 2 2 2 2 2 πΈ(π) = kx For example, Si: 6 equivalent conduction band minima in the directions of π kz ππ¦ β π β ππ₯ ππ§ = + + 2ππβ 2 ππ₯π₯ ππ¦π¦ ππ§π§ (π β π β π ) π₯π₯ π¦π¦ π¦π¦ π = (1, 0, 0) π ππ₯ Concentration of electron in each minima is n/6. When the electric field in x-direction, the total current in the x-direction π 2 ππ‘ 2 2 2 ππππ₯ π½π₯ = + + ππ₯ = ππ₯ + ππ¦ + ππ§ = πππ₯ 6 ππ₯π₯ ππ¦π¦ ππ§π§ 3 Similar expressions can be obtained for y- and z-directions and for any electric field. π 2 ππ‘ Compare with the expression for the conductivity, π½ = β π = ππππππ π = ππ ππππ ππ₯ + ππ¦ + ππ§ : conductivity mobility ππππ = 3 1 1 1 1 1 Isotropic conductivity = + + : conductivity β ππππ 3 ππ₯π₯ ππ¦π¦ ππ§π§ The current and the electric field effective mass are always in the same direction. scalar quantity Advanced Semiconductor Fundamentals DIFFUSION Definition-Visualization Diffusion Current SIMPLIFYING ASSUMPTIONS: 1) Carrier motion and concentration gradients are restricted to1-D. 2) All carriers move with the same velocity, π£ = π£π‘β. 3) The distance traveled by carriers between collisions is a fixed length, π which corresponds to the mean distance traveled by carriers between scattering events Chapter 6. Carrier Transport Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Consider the p-type semiconductor bar of cross-sectional area A and the steady-state hole concentration gradient shown in the figure. If t is arbitrarily set equal to zero at the instant all of the carriers scatter, if follows that half of the holes in a volume ππ΄ on either side of x = 0 will be moving in the proper direction so as to cross the x = 0 plane prior to the next scattering event at π/π£. π = π= π»ππππ πππ£πππ ππ π‘βπ + π₯ π΄ ππππππ‘πππ π€βππβ ππππ π π‘βπ = 2 π₯ = 0 πππππ ππ π π‘πππ π/π£ π»ππππ πππ£πππ ππ π‘βπ β π₯ π΄ ππππππ‘πππ π€βππβ ππππ π π‘βπ = 2 π₯ = 0 πππππ ππ π π‘πππ π/π£ 0 π π₯ ππ₯ βπ π π π₯ ππ₯ 0 Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Since π is typically quite small, the first two terms in a Taylor series expansion of p(x) about x = 0 will closely approximate p(x) for x values between β π and + π . π π₯ βπ 0 β ππ | π₯ ππ₯ 0 1 1 ππ π2 π = π΄ππ 0 β π΄ |0 , 2 2 ππ₯ 2 . . . . . βπ β€ π₯ β€ π 1 1 ππ π2 π = π΄ππ 0 + π΄ |0 2 2 ππ₯ 2 The net number of + x directed holes that cross the x = 0 plane in a time π/π£. ππ π2 π β π = βπ΄ |0 ππ₯ 2 The net number crossing the x = 0 plane per unit time due to diffusion, dropping the β|0 β. πΌπ|ππππ = π(π β π) 1 ππ = β ππ΄π£π π/π£ 2 ππ₯ π£π ππ π½π|ππππ = βπ( ) 2 ππ₯ π£π π£π (exact three dimensional analysis leads to π·π β‘ ) 2 3 ππ = βππ·π ππ₯ Finally, introducing π·π β‘ π½π|ππππ In three dimension, π±π|ππππ = βππ·π π»π π±π|ππππ = βππ·π π»π π·π , π·π : The hole and electron diffusion coefficient with standard unit of cm2/sec. Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Einstein Relationship Consider a nununiformly doped semiconductor under equilibrium conditions, as shown below. ππΈπΉ =0 ππ₯ π»πΈπΉ = 0 under equilibrium condition Nonzero electric field is established inside nonuniformly doped semiconductors under 1 1 1 equilibrium conditions. π = π»πΈ = π»πΈ (π₯) = π»πΈ (π₯) π π πΆ π π π±π|πππππ‘ + π±π|ππππ = πππ ππ + ππ·π π»π = 0 β΄ πππ ππ + ππ·π π»π = π π ππ π β π = ππΆ β±1/2 Ξ·πΆ π»π = where Ξ·πΆ = (πΈπΉ β πΈπ )/ππ π ππ π· ππ πΞ·πΆ π ππ 1 ππ π ππ π»Ξ·πΆ = β π»πΈπΆ (π₯) = β π πΞ·πΆ ππ πΞ·πΆ ππ πΞ·πΆ β΄ 0 = πππ ππ + ππ·π π»π = π π ππ π β π ππ π· ππ πΞ·πΆ π π·π ππ π = ππ π ππ πΞ·πΆ generalized form of Einstein relationship Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport In the nondegenerate semiconductor, π π β ππΆ ππ₯π Ξ·πΆ , β1 ππ πΞ·πΆ π·π ππ = ππ π Einstein relationship for electrons π·π ππ = ππ π Einstein relationship for holes Similar argument for holes, (Home work) Prove that Einstein relationship is also valid even under nonequilibrium conditions (π»πΈπΉ β 0). Use the relation for β±π Ξ·πΆ , πβ±π Ξ·πΆ = β±πβ1 Ξ·πΆ πΞ·πΆ Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport EQUATIONA OF STATE Current Equations Carrier Currents π½π = πππ πβ° β ππ·π π»π π½ = π½π + π½π π½π = πππ πβ° + ππ·π π»π : total particle current at steady state Dielectric Displacement Currents The change in polarization may be viewed as given rise to a nonparticle current, the dielectric displacement current. Ex) Current flow through a capacitor under a.c. and transient conditions π½π· = ππ· ππ‘ For a linear dielectric, π· = πΎπ π0 β° β΄π = π½π + π½π + ππ· ππ‘ : total current under a.c. and transient conditions Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Quasi-Equilibrium and Quasi-Femi Energies For nondegenerate semiconductors, π = ππ π (πΈπΉ βπΈπ ) π = ππ π (πΉπ βπΈπ ) due to perturbation π = ππ π (πΈπ βπΈπΉ ) equilibrium π = ππ π (πΈπΉ βπΈπ ) π = π π (πΈπ βπΈπΉ ) π π = ππ π (πΈπ βπΉπ ) nonequilibrium with small perturbation Carrier distributions remain almost unchanged from the equilibrium distributions even when small perturbation applied to the semiconductor. This is referred to the quasi-equilibrium and the Fermistatistics developed at equilibrium still can be used with quasi-Fermi energies, FN and FP, at quasi-equilibrium. π = ππ π (πΉπ βπΈπ ) π = π π (πΈπ βπΉπ ) π πΉπ = πΈπ + ππ ln(π/ππ ) πΉπ = πΈπ β ππ ln(π/ππ ) Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Differentiating the nonequilibrium carrier concentration, π»π = ππ /ππ β π πΈπ βπΉπ (π»πΈ β π»πΉ ) π π where β° = 1 π»πΈ π π π½π = π(ππ βππ·π /ππ)π β° + (ππ·π /ππ)ππ»πΉπ where ππ·π /ππ = π π π½π = ππ ππ»πΉπ = πππ π π½π = ππ ππ»πΉπ = πππ π π»πΉπ = ππ ππππ π π»πΉπ = ππ ππππ π Ohmβs law Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Continuity Equations ππ ππ ππ ππ ππ = |πππππ‘ + |ππππ + |π βπΊ + | ππ‘βππ ππ‘ ππ‘ ππ‘ ππ‘ ππ‘ ππππππ π ππ ππ ππ ππ ππ ππ = |πππππ‘ + |ππππ + |π βπΊ + | ππ‘βππ ππ‘ ππ‘ ππ‘ ππ‘ ππ‘ ππππππ π ππ By introducing, ππ ππ β‘ | ππ‘βππ ππ‘ ππππππ π ππ ππ β‘ ππ | ππ‘βππ ππ‘ ππππππ π ππ and noting the particle divergence, ππ ππ 1 |πππππ‘ + |ππππ = π» β π½π ππ‘ ππ‘ π ππ ππ 1 |πππππ‘ + |ππππ = β π» β π½π ππ‘ ππ‘ π ππ 1 = π» β π½π β ππ + ππ ππ‘ π ππ 1 = β π» β π½π β ππ + ππ ππ‘ π ππ β‘ β ππ | ππ‘ π βπΊ ππ β‘ β ππ | ππ‘ π βπΊ Advanced Semiconductor Fundamentals Chapter 6. Carrier Transport Minority Carrier Diffusion Equations Assumptions: 1) 1-D 2) minority carrier only 3) π β 0 4) uniformly doped 5) low-level injection 6) no other processes except possibly photo generation 1 1 ππ½π π» β π½π β π π ππ₯ π½π = πππ ππ + ππ·π ππ ππ β ππ·π ππ₯ ππ₯ ππ ππ0 πβπ πβπ = + = ππ₯ ππ₯ ππ₯ ππ₯ ππ = βπ ππ ππ = πΊπΏ ππ ππ0 πβπ πβπ = + = ππ‘ ππ‘ ππ‘ ππ‘ 1 π 2 βπ π» β π½π β π·π π ππ₯ 2 πβππ π 2 βπ βππ = π·π β + πΊπΏ ππ‘ ππ₯ 2 ππ πβππ π 2 βπ βππ = π·π β + πΊπΏ ππ‘ ππ₯ 2 ππ Minority carrier diffusion equations