# Carriers Concentration, Current & Hall Effect in Semiconductors. Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

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1 Carriers Concentration, Current & Hall Effect in Semiconductors 1 Prof.P. Ravindran, Department of Physics, Central University of Tamil Nadu, India

2 Conductivity Charge carriers follow a random path unless an external field is applied. Then, they acquire a drift velocity that is dependent upon their mobility, µn and the strength of the field, E V d = -µn E The average drift velocity, v av is dependent Upon the mean time between collisions, 2τ

3 Charge Flow and Current Density Current density, J, is the rate at which charges, cross any plane perpendicular to the flow direction. J = -nqv d = nqµ n E = σe n is the number of charges, and q is the charge (1.6 x C) The total current density depends upon the total charge carriers, which can be ions, electrons, or holes J = q(nµ n + pµ p ) x OHM s Law: V = IR Resistance, R(W) is an extrinsic quantity. Resistivity, r(w m), is the corresponding intrinsic property. r = R*A/L Conductivity, σ, is the reciprocal of resistivity: σ(w m) -1 = 1/r

4 Electron and Hole Mobilities m p v v = = qet q Et m p mp mp v = p E v = - n E = p qt m mp p = n qt m mn n p is the hole mobility and n is the electron mobility

5 Electron and Hole Mobilities v = E ; has the dimensions of v/e cm/s V/cm = 2 cm. V s Electron and hole mobilities of selected semiconductors Si Ge GaAs InAs n (cm 2 /V s) p (cm 2 /V s) Based on the above table alone, which semiconductor and which carriers (electrons or holes) are attractive for applications in high-speed devices?

6 Drift velocity [cm/s] Carrier mobilities Si electrons holes n= 1500 cm2/vs p= 350 cm2/vs Electric field [V/cm] 6

7 Carrier mobilities Mobility decreases with increasing doping concentration 300 K At around room temperature mobilities decrease as temperature increases ~ T -3/2 Si, holes 7

8 Conductivity of a Semiconductor = en e + ep h = conductivity, e = electronic charge, n = electron concentration in the CB, e = electron drift mobility, p = hole concentration in the VB, h = hole drift mobility Note: this expression is valid for both intrinsic and extrinsic semiconductors! We now see that n p for extrinsic semiconductors.

9 9 Drift of carriers in electric field The carriers are in constant random motion due to heat and there is no net motion. Under an electric field there is net motion. x p n x x n x n x x x x E p n q J E qn J qn E J qn J ) ( v = = = = = -

10 Both electrons and holes can drift in an electric field

11 . V(x) Electron Energy x Electrostatic PE(x) = -ev E x E c E d E F E Fi For an n-type semiconductor (e.g. Arsenic doped) only the electrons are mobile. The ionized dopant atoms do not move. A E v n-type Semiconductor V B Thus we have added both mobile and fixed or localized charge! Fig. 5.13: Energy band diagram of an n-type semiconductor connected to a voltage supply of V volts. The whole energy diagram tilts because the electron now has an electrostatic potential energy as well From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002)

12 Diffusion Diffusion results in a net flux of particles from the region of higher concentration to the region of lower concentration This flux leads to current (movement of charged particles) Magnitude of current depends on the gradient of concentration J n, diffusion ( x) = qd n dn( x) dx D n is the diffusivity coefficient Diffusivity is related to mobility by Einstein s relationship D D n p kt = = q Typical values for Si at room temp D n = 34 cm 2 /s and D p = 13 cm 2 /s n p

13 Variations In The Properties Of Silicon The conductivity of a semiconductor depends on both p and n and μ p andμ n. Because semiconductor devices are subject to a wide range of operating temperatures, the variations of these parameters with temperature are important. Mobility Impurity scattering T 3/2 (affects highly doped samples more than lowly doped samples) Lattice scattering T -3/2 (affects can be seen in lowly doped samples) Depends on effective mass as well At high electric field the mobility can saturate or even become smaller.

14 Mobility with electric field T -m Mobility μ decreases with temperature Mobilities are also functions of the electric field intensity and doping levels. In n-type silicon, μ is constant at a given temperature only if ξ < 10 3 V/cm. For ξ > 10 4 V/cm, μ n is inversely proportional to ξ and drift velocities approach 10 7 cm/s Between 10 3 and 10 4 V/cm, μ n varies approximately as ξ -1/2.

15 Intrinsic Concentration With increasing temperature, the density of holeelectron pairs increases in an intrinsic semiconductor. n 2 = i A T EG KT where E G0 is the energy gap (the energy required to break a covalent bond) at 0 K in electron volts, k is the Boltzmann constant in electron volts per degree kelvin (ev/k) A o is a constant independent of T. 0 3 e 0

16 Conductivity The conductivity of an intrinsic semiconductor increases with increasing temperature because the increase in hole-electron pairs is greater than the decrease in their mobilities. For extrinsic semiconductors, in the temperature range 100 to 600 K, the number of majority carriers is nearly constant but diminished mobility causes the conductivity to decrease with temperature.

17 DIFFUSION It is possible to have a nonuniform concentration of particles in a semiconductor. concentration gradient dn/dx in the density of carriers In a given time interval, more electrons will cross the surface from the side of greater concentration to the side of smaller concentration than in the reverse direction. This constitutes a current in the positive x direction.

18 Carrier motion in a concentration gradient How can we describe how charge carriers move in response to a concentration gradient? It is just simple diffusion; i.e Ficks first law; but of course it is with charged particles Diffusion current density = (charge) x flux Where Flux = N/A t or #of charge carriers passing unit area per unit time Ficks first law relates flux to concentration gradient Flux = e= -De dn/dx for electrons Where De = diffusion coefficient for electrons

19 Carrier motion in a concentration gradient So the current density (due to diffusion) becomes: Diffusion current density = (charge) x flux note: diffusion is away from high concentration JD e = -e e for charge carriers due to diffusion, or JD e = ed e dn/dx for electrons and JD h = -ed e dp/dx for holes

20 Carrier motion in a concentration gradient So, now for the total current we have both drift and diffusion currents for both types of charge carriers! J total = J drift + J diffusion J e = en e E x + ed e dn/dx for electrons, and J h = ep h E x - ed h dp/dx for holes. We relate the diffusion coefficient to temperature and mobility by the Einstein relationship: D e = (kt/e) e or D e / e = (kt/e) and D h / h = (kt/e) We also find that D=L2/ t where L is the mean free path, and t is the mean time between scattering events.

21 Definition of Particle Flux = N A t = particle flux, N = number of particles crossing A in t, A = area, t = time interval Definition of Current Density J = electric current density, J = Q Q = charge of the particle, = particle flux

22

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