Carriers Concentration and Current in Semiconductors

Carriers Concentration and Current in Semiconductors

Carrier Transport Two driving forces for carrier transport: electric field and spatial variation of the carrier concentration. Both driving forces lead to a directional motion of carriers superimposed on the random thermal motion. To calculate the directional carrier motion and the currents in a semiconductor, classical & nonclassical models can be used. The classical models assume that variation of E-field in time is sufficiently slow so that the transport properties of carriers (mobility or diffusivity) can follow the changes of the field immediately. If carriers are exposed to a fast-varying field, they may not be able to adjust their transport properties instantaneously to variations of the field, and carrier mobility and diffusivity may be different from their steady-state values nonstationary Nonstationary carrier transport can occur in electron devices under both dc and ac bias conditions. 2

Classical Description of Carrier Transport Assume thermal equilibrium for a semiconductor having a spatially homogeneous carrier concentration with no applied E-field. No driving force for directional carrier motion. The carriers not in standstill condition but in continuous motion due to kinetic energy. For electron in the conduction band, * 3 mn 2 Ekin kbt vth where v th is the thermal velocity, m n * is the conductivity 2 2 effective electron mass. The average time between two scattering events is the mean free time and the average distance a carrier travels between collisions is the mean free path. Fig. 2.5 (a) Applying V, the E-fields adds a directional component to the random motion of the electron. Fig. 2.5 (b) 3

The mean electron velocity: v n = -μ n E The directed unilateral motion of carriers caused by E-field is drift velocity. Similarly, v p = μ p E A change in E-field instantaneously results in a change of the drift velocity. 4

Fick s First Law: relating diffusion current to carrier concentration gradient. e D e dn dx e = electron flux, D e = diffusion coefficient of electrons, dn/dx = electron concentration gradient Electron Diffusion Current Density dn J D,e e e ed e dx J D, e = electric current density due to electron diffusion, e = electron flux, e = electronic charge, Where: D e = diffusion coefficient of electrons, dn/dx = electron concentration gradient

Hole Diffusion Current Density dp J D,h e h ed h dx J D, h = electric current density due to hole diffusion, e = electronic charge, h = hole flux, D h = diffusion coefficient of holes, dp/dx = hole concentration gradient Total Electron Current Due to Drift and Diffusion J e en e E x ed e dn dx J e = electron current due to drift and diffusion, n = electron concentration, e = electron drift mobility, E x = electric field in the x direction, D e = diffusion coefficient of electrons, dn/dx = electron concentration gradient

Total Currents Due to Drift and Diffusion J h = hole current due to drift and diffusion, p = hole concentration, h = hole drift mobility, E x = electric field in the x direction, D h = diffusion coefficient of holes, dp/dx = hole concentration gradient J h ep h E x ed h dp dx J e en e E x ed e dn dx J e = electron current due to drift and diffusion, n = electron concentration e = electron drift mobility, E x = electric field in the x direction, D e = diffusion coefficient of electrons, dn/dx = electron concentration gradient J total = J h +J e

Einstein Relation: diffusion coefficient and mobility are related! D e e kt e and D h h kt e D e = diffusion coefficient of electrons, e = electron drift mobility, D h = diffusion coefficient of the holes, h = hole drift mobility

Carrier diffusion due to doping level gradient. This is a common device fabrication step. Exposed As + Donor n 2 n 1 V o E x represents electrons (majority carriers in this case) Note: the As + are fixed, non-mobile charges! Diffusion Flux Drift Net current = 0 Diffusion occurs until an electric field builds up! We call this the built-in potential. Fig. 5.32: Non-uniform doping profile results in electron diffusion towards the less concentrated regions. This exposes positively charged donors and sets up a built-in field Ex. In the steady state, the diffusion of electrons towards the right is balanced by their drift towards the left. From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca

Built-In Potential and Concentration V 2 = potential at point 2, V 1 = potential at point 1, n 2 = electron concentration at point 2, n 1 = electron concentration at point 1 V 2 V 1 kt e ln n 2 Built-In Field in Nonuniform Doping E x = electric field in the x direction, b = characteristic of the exponential doping profile, e = electronic charge. E x kt be n 1 Exposed As + Donor n 2 n 1 V o E x Diffusion Flux Drift Net current = 0 Fig. 5.32: Non-uniform doping profile results in electron diffusio towards the less concentrated regions. This exposes positively ch donors and sets up a built-in field Ex. In the steady state, the dif electrons towards the right is balanced by their drift towards the From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002) http://materials.usask.ca

Carrier creation: Photoinjected charge carriers If we shine light on a semiconductor, we will generate new charge carriers (in addition to those thermally generated) if E photon >E gap. If the light is always on and of constant intensity, some steady state concentration of electrons and holes will result.

Carrier creation: Photoinjected charge carriers Let s consider the case of n-type material Consider an n-type semiconductor with a doping concentration of 5 x 10 16 cm -3. What are the carrier concentrations? Let s define some terms; n no majority carrier concentration in the n-type (only thermally ionized carriers) (i.e. the electron concentration in n-type) p no minority carrier concentration in the n-type (only thermally ionized carriers) (i.e. the hole concentration in n-type) semiconductor in the dark semiconductor in the dark Note: the no subscript implies that mass action law is valid!

When we have light: With light of E photon >E gap hitting the semiconductor, we get photogeneration of excess charge carriers. n n excess electron concentration such that:: n n = n n -n n0 p n excess hole concentration such that:: p n = p n -p n0 & Note that photogenerated carriers excited across the gap can only be created in pairs i.e. p n = n n and now (in light) n n p n n i2 i.e. mass action not valid!

Carrier density change under illumination If the temperature is constant, n n0 and p n0 are not time dependent, so dn n dt d n n dt and dp n dt d p n dt Consider the case of weak illumination, which creates a 10% change in n n0 i.e. n n = 0.1n n0 Or if the doping level is n no =5 x 10 16 cm -3, then n n = 0.1n n0 = 0.5 x 10 16 cm -3 And p n = n n = 0.5 x 10 16 cm -3 Which change is more important? Majority or minority?

Recall the intrinsic carrier concentration For Si n i is roughly 1.5x10 10 cm -3 At room temperature Since p no =n i2 /n n0 = (1.5x10 10 ) 2 /5x10 16 p no =4500 cm -3

p n = n n = 0.5 x 10 16 cm -3 An extremely important concept! Minority carrier concentration can be controlled over many orders of magnitude with only a small change in majority concentration.

Carrier creation followed by recombination

Carrier creation followed by recombination Mostly majority carriers in the dark Almost equal Carrier concentration In light The extra minority carriers recombine once the generation source is removed. How quickly do the carriers recombine?

Minority carrier lifetime h for n-type h = average time a hole exists in the valance band from its generation until its recombination And so 1/ h is the average probability (per unit time) that a hole will recombine with an electron. h depends on impurities, defects and temperature. The recombination process in a real semiconductor usually involves a carrier being localized at a recombination center. can be short (nanoseconds) allowing fast response (e.g. switch) or slow (seconds) for a photoconductor or solar cell

Excess Minority Carrier Concentration d p n dt G ph p n h p n = excess hole (minority carrier) concentration, t = time, G ph = rate of photogeneration, h = minority carrier lifetime (mean recombination time) h = average time a hole exists in the valance band from its generation until its recombination

Carrier concentration versus time with pulsed illumination t is time after illumination is removed

Continuous illumination provides increased conductivity Often used as a switch or motion detector

Carrier diffusion away from high concentration holes in this p-type example

Carrier motion: via diffusion (due to concentration gradient) and drift (due to electric field) Both diffusion and drift occur in semiconductors. Note here that holes (minority carriers) drift and diffuse in the same direction; but electrons (majority carriers) do not! With light we alter minority carrier concentration