3-July-7Pm-Resistivity-Mobility-Conductivity-Current-Density-ESE-Prelims-Paper-I.Pdf
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Carrier drift Current carriers (i.e. electrons and holes) will move under the influence of an electric field because the field will exert a force on the carriers according to F = qE In this equation you must be very careful to keep the signs of E and q correct. Since the carriers are subjected to a force in an electric field they will tend to move, and this is the basis of the drift current. Figure illustrates the drift of electrons and holes in a semiconductor that has an electric field applied. In this case the source of the electric field is a battery. Note the direction of the current flow. The direction of current flow is defined for positive charge carriers because long ago, when they were setting up the signs (and directions) of the current carriers they had a 50:50 chance of getting it right. They got it wrong, and we still use the old convention! Current, as you know, is the rate of flow of charge and the equation you will have used should look like
Id = nAVdq where
Id = (drift) current (amperes), n = carrier density (per unit volume) A = conductor (or semiconductor) area,
vd = (drift) velocity of carriers, q = carrier charge (coulombs) which is perfectly valid. However, since we are considering charge movement in an electric field it would be useful to somehow introduce the electric field into the equations. We can do this via a quantity called the mobility, where
d = E In this equation: = the drift mobility (often just called the mobility) with units of cm2V-1s-1 E = the electric field (Vcm-1) We know that the conduction electrons are actually moving around randomly in the metal, but we
will assume that as a result of the application of the electric field Ex, they all acquire a net velocity in the x direction. Otherwise, there would be no net flow of charge through area A.
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The average velocity of the electrons in the x direction at time t is denoted Vdx(t). This is called the drift velocity, which is the instantaneous velocity Vx in the x direction averaged over many electrons (perhaps, 1028 m-3); that is ΔqenAVΔt J==enV dx xdx AΔtAΔt
This general equation relates Jx to the average velocity Vdx of the electrons. It must be appreciated that the average velocity at one time may not be the same as at another time, because the applied
field, for example, may be changing: Ex=Ex(t). We therefore allow for a time-dependent current by writing
Jx(t) = enVdx(t)
Figure: Drift of electrons in a conductor in the presence of an applied electric field. Electrons drift with an average
velocity Vdx in the x direction
To calculate the drift velocity Vdx of the electrons due to applied field Ex, we first consider the
velocity Vxi of the ith electron in the x direction at time t. Suppose Velocity gained along x
Present time vx2 – ux2
its last collision was at time ti; therefore, for time (t-ti), it accelerated free of collisions, as
indicated in Figure. Let uxi be the velocity of electron i in the x direction just after the collision.
We will call this the initial velocity. Since eEx/me is the acceleration of the electron, the velocity
Vxi in the x direction at time t will be
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eEx V=u+t-txixii me
However, this is only for the ith electron. We need the average velocity Vdx for all such electrons along x. We average the expression for i=1 to N electrons, as in Equation. We assumed that immediately after a collision with a vibrating ion, the electron may move in any random
direction; that is, it can just as likely move along the negative or positive x, so that uxi averaged over many electrons is zero. Thus,
1eE x V=V+V+...... +V=t-tdxx1x2xNi Nme
where t - ti is the average free time for N electrons between collisions. Suppose that is the mean free time, or the mean time between collisions (also known as the
mean scattering time). For some electrons, (t-ti) will be greater than, and for others, it will be
shorter, as shown in figure. Averaging (t-ti) for N electrons will be the same as . Thus, we can
substitute for (t – ti) in the previous expression to obtain eτ V=Edxx me Equation shows that the drift velocity increases linearly with the applied field. The constant of
proportionality e/me has been given a special name and symbol. It is called the drift mobility d, which is defined as
Vdx = dEx Where eτ μ=d me Equation relates the drift mobility of the electrons to their means scattering time . To reiterate, , which is also called the relaxation time, is directly related to the microscopic processes that causes the scattering of the electrons in the metal; that is, lattice vibrations, crystal imperfections, and impurities, to name a few.
From the expression for the drift velocity Vdx, the current density J, follows immediately by substituting Equation, that is,
Jx = endEx
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Therefore, the current density is proportional to the electric field and the conductivity is the
term multiplying Ex, that is,
= end It is gratifying that by treating the electron as a particle and applying classical mechanics (F=ma), we are able to derive Ohm's law. We should note, however, that we assumed to be independent of the field. Drift mobility is important because it is a widely used electronic parameter in semiconductor device physics. The drift mobility gauges how fast electrons will drift when driven by an applied field. If the electron is not highly scattered, then the mean free time between collisions will be long, will be large, and by Equation the drift mobility will also be large; the electrons will therefore be highly mobile and imply high conductivity, because also depends on the concentration of conduction electron n. The mean time between collisions has further significance. Its reciprocal 1/ represents the mean frequency of collisions or scattering events; that is, 1/ is the mean probability per unit time that the electron will be scattered (see example). Therefore, during a small time interval t/. The probability of scattering per unit time 1/ is time independent and depends only on the nature of the electron scattering mechanism. Mobility variation with temperature The mobility will be made up of at least two components associated with lattice (scattering) interactions and impurity (scattering) interactions. If the mobility is made up of several
components say 1, 2, 3, then the total mobility is given by
-1 111 μ=++ μμμ123
If we consider just two components due to lattice scattering, L, and ionized impurity scattering,
I, then the mobility is given by
-1 11 μ= + μμIL As the lattice warms up, the lattice scattering component will increase, and the mobility will decrease. Experimentally, it is found that the mobility decrease often follows a T-1.5 power law:
-1 L = C1 T ======YourPedia For any query or detail Contact us at: 9855273076, or visit at www.yourpedia.in 4 ______
where C1 is a constant. As we cool a piece of semiconductor down, the lattice scattering becomes less important, but now the carriers are more likely to be scattering by ionized
impurities, since the carriers are moving more slowly. It is found that the ionized impurity scattering often follows a T1.5 power law giving:
3/2 I = C2 T Figure shows the variation in carrier mobility as a function of temperature due to the presence of both ionized impurity and lattice scattering effects. MATTHIESSEN'S RULE AND THE TEMPERATURE COEFFICIENT OF RESISTIVITY ()
The theory of conduction that considers scattering from lattice vibrations only works well with pure metals; unfortunately, it fails for metallic alloys. Their resistivities are only weakly temperature dependent. We must therefore search for a different type of scattering mechanism.
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Consider a metal alloy that has randomly distributed impurity atoms. An electron can now be scattered by the impurity atoms because they are not identical to the host atoms, as illustrated in figure. The impurity atom need not be large than the host atom; it can be smaller. As long as the impurity atom results in a local distortion of the crystal lattice, it will be effective in scattering. One way of looking at the scattering process from an impurity is to consider the scattering cross- section.
Figure: Two different types of scattering processes involving scattering from impurities alone and from thermal vibrations alone.
We now effectively have two types of mean free times between collisions; one, T, for scattering
from thermal vibrations only, and the other, 1, for scattering from impurities only. We define T
as the mean time between scattering events arising from thermal vibrations alone and 1 as the mean time between scattering events arising from collisions with impurities alone. Both are illustrated in figure. In general, an electron may be scattered by both processes, so the effective mean free time
between any two scattering events will be less than the individual scattering times T and J. The electron will therefore be scattered when it collides with either an atomic vibration or an impurity
atom. Since in unit time 1/ is the net probability of scattering, 1/T is the probability of scattering
from lattice vibrations along, and 1/1 is the probability of scattering from impurities alone, then within the realm of elementary probability theory for independent events, we have 111 =+ τττ T1 In written equation for the various probabilities, we make the reasonable assumption that, to a greater extent, the two scattering mechanisms are essentially independent. Here, the effective
mean scattering time is clearly smaller than both T and 1. We can also interpret equation as
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follows: In unit time, the overall number of collisions (1/) is the sum of the number of collisions
with impurities alone (1/1).
The drift mobility d depends on the effective scattering time via d = e/me, so equation can also be written in terms of the drift mobilities determined by the various scattering mechanisms. In other words, 1 1 1 =+ μ μdL1 μ
where L is the lattice-scattering-limited drift mobility, and 1 is the impurity-scattering-
limited drift mobility. By definition, L = eT/me and 1 = e1/me. The effective (or overall)
resistivity of the material is simply 1/end, or 111 ρ=== enμenμenμdL1 which can be written
= T + 1
where 1/enL is defined as the resistivity due to scattering from thermal vibrations, and 1/en1 is the resistivity due to scattering from impurities, or 11 ρ=T1 and ρ enμenμL1 The final result in equation simply states that the effective resistivity is the sum of two
contributions. First, T = 1/enL is the resistivity due to scattering by thermal vibrations of the host atoms. For those near-perfect pure metal crystals, this is the dominating contribution. As
soon as we add impurities, however, there is an additional resistivity, 1=1/en1, which arises from the scattering of the electrons from the impurities. The first terms is temperature dependent
-1 because T T but the second term is not.
The mean time 1 between scattering events involving electron collisions with impurity atoms depends on the separation between the impurity atoms and therefore on the concentration of those
atoms (see figure). If 1 is mean separation between the impurities, then the mean free time
between collisions with impurities alone will be 1/u , which is temperature independent because
-1/3 1 is determined by the impurity concentration N1(i.e., 11=N ), and the mean speed u of the
electrons is nearly constant in a metal. In the absence of impurities, 1 is infinitely long, and thus ======YourPedia For any query or detail Contact us at: 9855273076, or visit at www.yourpedia.in 7 ______
1=0. The summation rule of resistivities from different scattering mechanism, as shown by equation is called Matthiessen's rule. There may also be electrons scattering from dislocation and other crystal defects, as well as from grain boundaries. All of these scattering processes add to the resistivity of a metal, just as the scattering process from impurities. We can therefore write the effective resistivity of a metal as
=T+R
Figure: The formation of a 2s energy band from the 2s orbital's when N Li atoms come together to form the Li solid. There are N 2s electrons, but 2N states in the band. The 2s band is therefore only half full. The atomic 1s orbital is close to the Li nucleus and remains undisturbed in the solid. Thus, each Li atom has a closed K shell (full 1s orbital) 23 The single 2s energy level E2x therefore splits into N( 10 ) finely separated energy levels, forming an energy band, as illustrated in Figure. Consequently, there are N separate energy levels, each of which can take two electrons with opposite spins. The N electrons fill all the levels up to and including the level at N/2. Therefore, the band is half full. We do not mean literally that
the band is full to the half-energy point. The levels are not spreads equally over the band from EB
to ET, which means that the band cannot be full to the half-energy point. Half filled simply means half the states in the band are filled from the bottom up.
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Figure: Typical electron energy band diagram for a metal. All the valence electrons are in an energy band, which they only partially fill. The top of the band is the vacuum level, where the electron is free from the solid (PE=0)
Fermi energy and work function of selected metals (polycrystalline) Metal Ag Al Au Cs Cu Li Mg Na (eV) 4.26 4.28 5.1 2.14 4.65 2.9 3.66 2.75
EEO (eV) 5.5 11.7 5.5 1.58 7.0 4.7 7.1 3.2 At absolute zero, all the energy levels up to the Fermi level are full. The energy required to excite an electron from the Fermi level to the vacuum level, that is, to liberate the electron from the metal, is called the work function of the metal. As the temperature increases, some of the electrons get excited to higher energy levels. To determine the probability of finding an electron at an energy level E, we must consider what is called "particle statistics", a topic that is key to understanding the behavior of electronic devices. Clearly, the probability of finding an electron at
0K at some energy E < EFO is unity, and at E > EEO, the probability is zero. Table summarizes the Fermi energy and work function of a few selected metals.
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Figure: (a) Energy band diagram of a metal. (b) in the absence of a field, there are as many electrons moving right as there are moving left. The motions of two electrons at each energy cancel each other as for a and b. (c) In the presence of a field in the –x direction, the electron a
accelerates and gains energy to a' where it is scattered to an empty state near EFO but moving in the –x direction. The average of all moment's values is along the +x direction and results in a net electric current.
Figure: Conduction in a metal is due to the drift of electrons around the Fermi level. When a voltage is applied, the energy band is bent to be lower at the positive terminal so that the electron's potential energy decreases as it moves toward the positive terminal.
Figure: A simplified energy band diagram of a semiconductor. CB is the conduction band and VB is full of electrons and the CB
is empty. If a photon of energy hf > Eg is incident on the semiconductor, it can be absorbed by an electron in the VB, which becomes photo excited into the CB. Some electrons in the VB can be excited into the CB by thermal excitation, that is, occasional rupturing of Si-Si bonds by energetic lattice vibrations. Thermal generation creates electron and hole pairs.
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FERMI ENERGY SIGNIFICANCE METAL-METAL CONTACTS: CONTACT POTENTIAL Suppose that two metals, platinum (Pt) with a work function 5.36eV and molybdenum (Mo) with a work function 4.20eV, are brought together, as shown in figure. We know that in metals, all the energy levels up to the Fermi level are full. Since
Figure: There is no current when a closed circuit is formed by two different metals, even though there is a contact potential at each contact. The contact potentials oppose each other.
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Figure: The Seebeck effect.
A temperature gradient along a conductor gives rise to a potential difference. (Note that the EF in the hot region is not exactly the same as that in the cold region. Had the two Fermi levels not aligned in equilibrium, the difference could have been used to do external work. What is important is that, in equilibrium, the Fermi level is uniform throughout the combined system as in figure b.
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Figure: Thermal vibrations of atoms can break bonds and thereby create electron-hole pairs.
` Figure: (a) Energy band diagram. (b) Density of states (number of states per unit energy per unit volume). (c) Fermi-Dirac probability function (probability of occupancy of a state). (d) The product of g(E) and f(E) is the energy density of electrons in the CB (number of electrons per unit
energy per unit volume). The area under nE(E) versus E is the electron concentration.
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Figure: Energy band diagram for an n-type Si doped with 1ppm As.
There are donor energy levels just below EC around As+ sites.
Figure: Energy band diagram for a p-type Si doped with 1ppm B. There are acceptor - energy levels Ea just above EV around B sites. These acceptor levels accept electrons from the VB and therefore create hole in the VB
A derivation of Ohm's law Drift current in any specimen is due to electric field and defined as
Id = nAqd first convert to current density (Am-2) to obtain
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where Jx is the current density in the x-direction. Current density is defined in term of effect mass and density as below:
nτq2 J =xx E* m where we now have a relationship between current density and electric field. This equation is in fact just Ohm's law in a slightly different from. In order to see that equation really is just Ohm's law we need to introduce the conductivity. The (electrical) conductivity of a material is simply the proportionality factor between the current density and the applied electric field, which in one dimension can be written as
Jx = xEx
-1 where x is the electrical conductivity with units of Sm . The reciprocal of conductivity is called the resistivity: 1 ρ=x σx
where x is the electrical resistivity with units of m. You can see immediately from equation that the conductivity, in terms of fundamental quantities, is given by
nq2τ σ=* m Finally, convince yourself by referring to figure that the resistance of a block of material of resistivity across the two end faces of area A is just l R= A where you can resort to analysis of the units for confirmation. We can now transform equation into the more familiar Ohm's law. Starting with, nq2 τ Jxx =* E =σE m
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write down the current density as current divided by area, and electric field as potential divided by length to obtain IV xx=σ ALx Rearrange to get LI V= xxρ x A Finally, use equation to obtain
Vx = IxR which is the form of Ohm's law with which you'll be most familiar. Drift current equations We have now covered enough material to look at the drift current equations with some numerical examples. Considering intrinsic semiconductors first, in n type semiconductor electron concentration is greater and dominating. Hence the entire conductivity is contributed by electron only
Jn = nqEn
where n = electron mobility For holes the current density will be
Jp = pqEp
where p = hole mobility
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Since we are dealing with intrinsic material ni = n = p and so the total current density will be given by
J = Jn + Jp = Eniq(n+p) giving an intrinsic conductivity of J σ==nqμ+μiinp E For extrinsic semiconductor the current density is again due to electron and hole flow:
J = nqEn + pqEp However, for n-type semiconductor material where n > p we obtain
n = qNdn
where Nd is the shallow donor concentration. Similarly, for a p-type semiconductor where p>n we get
p = qNap
with Na being the shallow acceptor concentration.
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