5.4. Recombination and Minority Carrier Injection 5.4.1 Direct and Indirect Recombination A free electron in CB "meets" a hole in VB: the excess energy > a photon energy. Energy CB ψ cb (k cb ) ψ vb (k vb ) E c E v hυ = E g VB Distance Fig.5.22: Direct recombination in GaAs. k cb = k vb so that momentum conservation is satisfied
5.4.2 Minority Carrier Lifetime What happens when an ntype semiconductor, doped with donors, is uniformly illuminated with appropriate wavelength light to photogenerate electronhole pairs(ehps)? CB E c E d E v VB Fig. 5.24: Low level photoinjection into an ntype semiconductor in which n n < n no From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGrawHill, 2002) http://materials.usask.ca In an ntype semiconductor, electrons>the majority carriers; holes> the minority carriers.
Define = the majority carrier (electron) concentration in thermal equilibrium in the dark. = the minority carrier (hole) concentration in thermal equilibrium in the dark. In thermal equilibrium, the mass action law Under illumination, excess EHPs are created by photogeneration. : Let = excess electron concentration; the instantaneous = excess hole concentration, the instantaneous Log(carrier concentration) (cm 3 ) 5 10 16 1.5 10 10 4.5 106 5.5 10 16 nno n i p no n n = n no + n n 0.5 10 16 p n = p no + p n p n = 0.5 10 16 n i p no (a) In the dark: np = n i 2 (b) In light: np n i 2 Fig. 5.25: Low level injection in an ntype semiconductor does not affect n n but drastically affects the minority carrier concentration p n. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGrawHill, 2002) http://materials.usask.ca
Photoexcitation creates an equal number of electrons and holes: The mass action is not obeyed: and After illumination, recombination of excess electrons and holes: minority carrier lifetime Illumination A + + ntype semiconductor in the dark. p n = p no <<n no B + + + + + + + + + + + + + + + + + + Illumination with hυ >E g creates excess holes: p n = p no + p n n n C + + + + + + + + + + In dark after illumination. Excess holes are disappearing by recombination. Fig. 5.26: Illumination of an n type sem iconductor results in excess electron and hole concentrations. After the illumination, the recombination process restores equilibrium; the excess electrons and holes simply recombine. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGrawHill, 2002) http://m aterials.u sask.ca
Rate of increase in excess hole concentration = Rate of photogeneration Rate of recombination of excess holes = the rate of photogeneration (The time evolution of the excess minority carrier concentration) 5.5. Diffusion and Conduction Equations, Random Motion The particle flux ; the current density with the charge Q n ( x,t) Net electron diffusion flux Electric current n(x,t) n 1 n 2 x o x o (a) x o + x o x o x o + Fig. 5.29: (a) A rbitrary electron concentration n ( x,t) profile in a semiconductor. There is a net diffusion (flux) of electrons from higher to lower concentrations. (b) Expanded view of two adjacent sections at x o. There are m ore electrons crossing x o com ing from left (x o ) than com ing from right (x o + ). x (b) x From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGrawHill, 2002) http://m aterials.u sask.c a
Let n 1 (or n 2 )= the electron concentrations at. The number of electrons moving toward the right to cross The net number of electrons crossing per unit time per unit area in the +x direction I is the electron flux For small mean free path l, Then, where the diffusion constant of electrons (Fick's first law) In case of the hole concentration gradient, The "diffusion" current density: Total ("drift"+"diffusion") current density: and
Einstein relation: Light Semitransparent electrode ntype Semiconductor Electron Diffusion Electron Drift Hole Diffusion Hole Drift x E x Fig. 5.31: When there is an electric field and also a concentration gradient, charge carriers move both by diffusion and drift. (E x is the electric field.) From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGrawHill, 2002) http://materials.usask.ca
5.6. Continuity Equation 5.6.1 TimeDependent Continuity Equation Consider an infinitesimally thin elemental volume in which the hole concentration is in an ntype semiconductor slab. p(x,t) z J h J h + δj h A Semiconductor y x x+δx x Fig. 5.33: Consider an elemental volume Aδx in which the hole concentration is p(x,t) From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGrawHill, 2002) http://materials.usask.ca
The current density at x due to holes flowing into the volume: The current density due to holes flowing out at is NOT uniform along x. If, the current leaving the vol. is less than that entering the vol.: an increase in the hole concentration in. Rate of increase in the hole concentration the change in. Recombination taking place in removes holes from this volume (hole recombination time ) There exists photogeneration at x at time t. ( = the photogeneration rate) continuity equation for holes : with for drift and diffusion For constant current density and photogeneration, : the net rate of change of
5.6.2 SteadyState Continuity Equation Assume that there is no bulk photogeneration,, under the continuous illumination of one end of an ntype semiconductor slab by light (absorption in a very small thickness ) x o ntype semiconductor Light Excess concentration p n (0) p n (x) n n (0) n n (x) (a) Currents (ma) 4 0 I D,h Diffusion I drift,e x Diffusion Drift I D,e 4 x (µm) 0 20 40 60 80 (b) Fig. 5.34: (a) Steady state excess carrier concentration profiles in an n type sem iconductor that is continuously illum inated at one end. (b) M ajority and m inority carrier current com ponents in open circuit. T otal current is zero. From Principles of Electronic M aterials and Devices, Second Edition, S.O. Kasap ( McGrawHill, 2002) http ://M a te rials.u sa sk.c a
The steadystate: For very small electric field, with the diffusion length of Assume weak injection: : Illumination causes the excess hole concentration at x = 0: boundary condition Then, the hole diffusion current Find : under steadystate, the hole generated per unit time in must be removed by the whole current at at the same rate., so that
Similarly, for the electron case, with and : This is greater than 1 for Si. 5.9. Schottky Junction 5.9.1 Schottky Diode What happens when a metal and an ntype semiconductor are in contact? Work functions for the metal and the semiconductor:: Fermi levels for the metal and semiconductor: The electron affinity (the minimum energy required to remove an electron from the semiconductor): Assume that
V o E o Vacuum level E Fm Metal Φm ntype Semiconductor Φn χ CB E c E Fn E v VB E Fm Metal Depletion region W Φ B Neutral semiconductor region Φm Φn Φm Φn=eV 0 CB E c Fn Before Contact E v VB After Contact Fig. 5.39: Formation of a Schottky junction between a metal and an ntype semiconductor when Φm > Φn. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGrawHill, 2002) http://materials.usask.ca The more energetic electrons in CB of the semiconductor tunnel into the metal in search of lower empty levels (just above ) and accumulate near the surface of the metal.
Electrons tunneling from the semiconductor leave behind an electrondepleted region of width W where there are net positive space charge: The contact potential. the builtin potential develops between the metal and the semiconductor; The depletion region constitutes a space charge layer (SCL) in which there is a nonuniform internal field directed from the semiconductor to the metal surface. The Fermi level throughout the whole solid, the metal and the semiconductor in contact, must be uniform in equilibrium: and line up In the W region, depleted of electrons, must increase so that n decreases. The bands must bend to increase towards the junction. The Schottky barrier height = the potential energy for electrons moving from the metal to the semiconductor: The current due to electrons being thermally emitted from the metal to the CB of the semiconductor = where is a constant. The current due to electrons being thermally emitted from the CB of the semiconductor to the metal: where is a constant. In equilibrium under open circuit conditions,
Under forward bias conditions, the semiconductor side is connected to the negative terminal. The potential barrier for thermal emission of electrons from the semiconductor to the metal is now V V r Metal ntype Semiconductor e(v 0 V) CB E c Φ B e(v 0 +V) CB E c E v VB E v VB (a) Forward biased Schottky junction. Electrons in the CB of the semiconductor can eadily overcome the small PE barrier to enter the metal. I (b) Reverse biased Schottky junction. Electrons in the metal can not easily overcome the PE barrier Φ B to enter the semiconductor. 1 µa 1 ma 10 µa 0.2 V V (c) IV Characteristics of a Schottky junction exhibits rectifying properties (negative current axis is in microamps) Fig. 5.40: The Schottky junction. From Principles of Electronic Materials and Devices, Second Edition, S.O. K asa p ( M cg raw H ill, 2002) http://materials.usask.c a
The net current : where is a constant depending on the material and the surface properties of the two solids. Under reverse bias conditions, : The reverse bias current is essentially limited by only and is very small. For pn junctions, under a forward bias, For the Schottky junctions between a metal and a ptype semiconductor, 5.10 Ohmic Contacts A junction between a metal and a semiconductor that does NOT limit the current flow. The formation of an ohmic contact between a metal and an ntype semiconductor:
E Fm Φ m Φn CB E c E Fn E v VB Metal ntype Semiconductor Before Contact Accumulation Region Ohmic Contact Bulk Semiconductor CB E Fm E E c Fn E v VB Metal ntype Semiconductor After Contact Fig. 5.43: W hen a metal w ith a smaller workfunction than an n type sem iconductor are put into contact, the resulting junction is an ohm ic contact in the sense that it does not lim it the current flow. From Principles of Electronic Materials and Devices, Second Edition, S.O. K asap ( M c G raw H ill, 2 002) http://materials.usask.ca Accumulation region near the junction in the semiconductor region: The semiconductor energy bands are bending downward to decrease so as to increase n.