Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Physics of Semiconductor Devices Part IV - Carrier Transport Phenomena
Luca Varani University Montpellier 2
November 8, 2011
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
The net flow of electrons and holes in a semiconductor will generate currents. The process by which these charged particles move is called transport. In this chapter we will consider the two basic transport mechanisms in a semiconductor crystal: drift the movement of charge due to electric fields, and diffusion the flow of charge due to density gradients.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
The carrier transport phenomena are the foundation for finally determining the current-voltage characteristics of semiconductor devices. We will implicitly assume that, though there will be a net flow of electrons and holes due to the transport processes, thermal equilibrium will not be substantially disturbed.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
Outline 1
Carrier Drift Drift Current Density Mobility Effects Conductivity Velocity Saturation
2
Carrier Diffusion
3
Graded Impurity Distribution
4
The Hall Effect L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
Carrier Drift
An electric field applied to a semiconductor will produce a force on electrons and holes so that they will experience a net acceleration and net movement, provided there are available energy states in the conduction and valence bands. This net movement of charge due to an electric field is called drift. The net drift of charge give, rise to a drift current.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
Drift Current Density If we have a positive volume charge density ρ moving at an average drift velocity vd , the drift current density is given by Jdrf = ρvd
(1)
where J is in units of A/cm2 . If the volume charge density is due to positively charged holes, then Jp|drf = epvdp
(2)
where Jp|drf is the drift current density due to holes and vdp is the average drift velocity of the holes. L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
The equation of motion of a positively charged hole in the presence of an electric field is F = mp∗ a = eE (3) where e is the magnitude of the electronic charge, a is the acceleration, E is the electric field, and mp∗ is the effective mass of the hole. If the electric field is constant, then we expect the velocity to increase linearly with time.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
However, charged particles in a semiconductor are involved in collisions with ionized impurity atoms and with thermally vibrating lattice atoms. These collisions, or scattering events, alter the velocity characteristics of the particle.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
As the hole accelerates in a crystal due to the electric field, the velocity increases. When the charged particle collides with an atom in the crystal, for example, the particle loses most or all of its energy. The particle will again begin to accelerate and gain energy until it is again involved in a scattering process. This continues over and over again.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
Throughout this process the particle will gain an average drift velocity which, for low electric fields, is directly proportional to the electric field. We may then write vdp = µp E
(4)
where µp is the proportionality factor and is called the hole (ohmic) mobility. The mobility is an important parameter of the semiconductor since it describes how well a particle will move due to an electric field. The unit of mobility is usually expressed in terms of cm2 /(Vs).
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
By combining Equations (2) and (4), we may write the drift current density due to holes as Jp|drf = (ep)vdp = eµp pE
(5)
The drift current due to hole, is in the same direction as the applied electric field. The same discussion of drift applies to electrons: Jn|drf = ρvdn = (−en)vdn
(6)
where Jn|drf is the drift current density due to electrons and vdn is the average drift velocily of electrons. The net charge density of electrons is negative. L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
The average drift velocity of an electron is also proportional to the electric field for small fields. However, since the electron is negatively charged, the net motion of the electron is opposite to the electric field direction. We can then write vdn = −µn E (7) where µn is the electron mobility and is a positive quantity.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
Equation (6) may be written as Jn|drf = (−en)(−µn E ) = enµn E
(8)
The conventional drift current due to electrons is also in the same direction as the applied electric field even though the electrons movement is in the opposite direction. Electron and hole mobilities are functions of temperature and doping concentration. Table 1 shows some typical mobility values at T = 300 K for low doping concentrations.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Silicon Gallium arsenide Germanium
Drift Current Density Mobility Effects Conductivity Velocity Saturation
µn (cm2 /Vs) 1350 8500 3900
µp (cm2 /Vs) 480 400 1900
Table 1: Typical mobility values at T = 300 K and low doping concentrations.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
Since both electrons and holes contribute to the drift current, the total drift current density is the sum at the individual electron and hole drift current densities: Total Drift Current Density Jdrf = e(nµn + pµp )E
L. Varani
Physics of Semiconductors Devices
(9)
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
Mobility Effects
Equation 3 related the acceleration of a hole to a force such as an electric field. We may write this equation as F = mp∗
dv = eE dt
(10)
where v is the velocity of the particle due to the electric field and does not include the random thermal velocity.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
If we assume that the effective mass and electric field are constants, then we may integrate Equation (10) and obtain v=
eEt mp∗
where we have assumed the initial drift velocity to be zero.
L. Varani
Physics of Semiconductors Devices
(11)
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
Figure 1 shows a schematic model of the random thermal velocity and motion of a hole in a semiconductor with zero electric field. There is a mean time between collisions which may be denoted by τcp . If a small electric field (E-field) is applied as indicated in Figure 1b, there will be a net drift of the hole in the direction of the E-field, and the net drift velocity will be a small perturbation on the random thermal velocity, so the time between collisions will not be altered appreciably.
L. Varani
Physics of Semiconductors Devices
C HAP T E R
5
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Levels Cruner Transport po Energy l€nomena
Drift Current Density Mobility Effects Conductivity Velocity Saturation
.' . . .
. ... . ;, .., ,..,
3
-
E fleld (h}
(a }
Figure Typical r.mdon\ h
, .J,
applied as indicated in Figure 5.1b. there will be a net drift of the hole in the directioo of the E··field. and the net drift velocity will be a small pel1urbation on the random L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
If we use the mean time between collisions τcp in place of the time t in Equation (11) then the mean peak velocity just prior to a collision or scattering event is: eτcp E (12) vd|peak = mp∗
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
The average drift velocity is one half the peak value so that we can write 1 eτcp hvd i = E (13) 2 mp∗ However, the collision process is not as simple as this model, but is statistical in nature. In a more accurate model including the effect of a statistical distribution, the factor 1/2 in Equation (13) does not appear.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
The hole mobility is then given by Hole mobility µp =
vdp eτcp = ∗ E mp
(14)
The same analysis applies to electrons: Electron mobility µn =
vdn eτcn = ∗ E mn
where τcn is the mean time between collisions for an electron. L. Varani
Physics of Semiconductors Devices
(15)
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
There are two collision or scattering mechanisms that dominate in a semiconductor and affect the carrier mobility: phonon or lattice scattering, and ionized impurity scattering. The atoms in a semiconductor crystal have a certain amount of thermal energy at temperatures above absolute zero that causes the atoms to randomly vibrate about their lattice position within the crystal. The lattice vibrations cause a disruption in the perfect periodic potential function.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
A perfect periodic potential in a solid allows electrons to move unimpeded or with no scattering through the crystal. But the thermal vibrations cause a disruption of the potential function, resulting in an interaction between the electrons or holes and the vibrating lattice atoms. This lattice scattering is also referred to as phonon scattering.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
Since lattice scattering is related to the thermal motion of atoms, the rate at which the scattering occurs is a function of temperature. If we denote µL as the thermal mobility that would be observed if only lattice scattering existed, then the scattering theory states that to first order µL ∝ T −3/2 (16)
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
Mobility that is due to lattice scattering increases as the temperature decreases. Intuitively we expect the lattice vibrations to decrease as the temperature decreases, which implies that the probability of a scattering event also decreases, thus increasing mobility.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Drift Current Density Mobility Effects Conductivity Velocity Saturation
Figure 2 shows the temperature dependence of electron and hole mobilities in silicon. In lightly doped semiconductors, lattice scattering dominates and the carrier mobility decreases with temperature as we have discussed. The temperature dependence of mobility is proportional to T −n . The inserts in the figure show that the parameter n is not equal to 3/2 as the first-order scattering theory predicted. However, mobility does increase as the temperature decreases.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
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Drift Current Density Mobility Effects Conductivity Velocity Saturation
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Figure 5.3 1Electron amI hule mobiJi(ics versus im»urity concentr.ui(ms for germanium. silicon. and gallium arsenide 31 T = 300 K .
Figure 3: Electron and hole mobilities versus impurity concentrations for germanium, silicon, and gallium arsenide at T = 300 K. fFromSu (121.)
TEST YOUR UNDERSTANDING
L. the Varani Physics ES.J (a} Using FigufC 5.2. find cl F,':,(x
+ tlx),
Continuity Equation for example. there will be a net increase in th Time-Dependent Diffusion Equations
of holes in the differentiaJ yolume! elemcm with (jme. If we generJlJize to
n ni , as we would expect for the p-type material. These results are for thermal equilibrium.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
If excess carriers are created in a semiconductor, we are no longer in thermal equilibrium and the Fermi energy is strictly no longer defined. However, we may define a quasi-Fermi level for electrons and a quasi-Fermi level for holes that apply for nonequilibrium.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
If δn and δp are the excess electron and hole concentrations, respectively, we may write: EFn − EFi (123) n0 + δn = ni exp kT and
p0 + δp = ni exp
EFi − EFp kT
(124)
where EFn and EFp are the quasi-Fermi energy levels for electrons and holes, respectively. The total electron concentration and the total hole concentration are functions of the quasi-Fermi levels. L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
Figure 21a shows the energy-band diagram with the Fermi energy level corresponding to thermal equilibrium. Figure 21b shows the energy-band diagram under the nonequilibrium condition. Since the majority carrier electron concentration does not change significantly for this low-injection condition, the quasi-Fermi level for electrons is not much different from the thermal-equilibrium Fermi level.
L. Varani
Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
CHAPT.R 8
N'onequilibfium Excess Carners in SemiConductors
O.2984cV
tI--------T--------.------'------'"
tl
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.... ... . . (a)
±
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t ,,,,-:1,.,,,.,.. , . ;; ----l---------------___ -___ t _________ ===",..!'f!
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(b)
fIgure 6.15 I (a) 'n1ermal-equilibtium energy-band diagram for N. = 10" cm- 1 and
no· = cm-Thermal-equilibrium .1. (b) Quasi-Fermi levels for electrons and holes jf lOllfor cm -:'I excess Figure 21:10 10(a) energy-band diagram −3 carriers Nd = 1015 arc cmpresent. and n0 = 1010 cm−3 . (b) Quasi-Fermi levels for electrons and holes if 1013 cm−3 excess carriers are present.
increased significantly so that the quasi-Fermi level for hole.' has concenlration moved much closer to Ihe valence band. We will consider the quasi-Fenni energy
le\'els agalo when we discuss forv.;ard-blased pn junctions, L. Varani Physics of Semiconductors Devices
Carrier Drift Carrier Diffusion Graded Impurity Distribution The Hall Effect Carrier Generation and Recombination Characteristics of Excess Carriers Ambipolar Transport Quasi-Fermi Energy Levels
The quasi-Fermi energy level for the minority carrier holes is significantly different from the Fermi level and illustrates the fact that we have deviated from thermal equilibrium significantly. Since the electron concentration has increased, the quasi-Fermi level for electrons has moved slightly closer to the conduction band. The hole concentration has increased significantly so that the quasi-Fermi level for holes has moved much closer to the valence band.
L. Varani
Physics of Semiconductors Devices
