Semiconductors- Part 2 Carrier transport phenomena: PDF

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Summary

mobility, drift current, diffusion current and the Einstein relation...

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4/30/2021

Semiconductors

1

OUTLINE ฀ ฀ ฀ ฀ ฀ ฀ ฀ ฀ ฀ ฀

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Carrier Drift Band Bending Carrier Diffusion Total Current Density Einstein Relations Excess Carriers in Semiconductors Carrier injection Generation and Recombination Continuity Equation Poisson and Laplace equations Semiconductors

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Carrier Drift In this section we would now like to consider the nature of carrier DYNAMICS under a variety of conditions Thermal velocity, Drift velocity and Mobility

=0

• At room temperature and in the absence of an applied electric field ( = 0) electrons and holes in a semiconductor undergo a rapid but random thermal motion (thermal velocity, vth) that the charge carriers execute due to their thermal energy From Thermodynamics

1 2 3 mvth  kT 2 2

&

vth 

l tr

Schematic path of an electron in a semiconductor

where l  mean free distance and 𝑡𝑟  mean free time or relaxation time

In spite of this rapid motion the net current flowing through the crystal is zero x

• If an electric field is applied to semiconductor, it superimposes a slow net drift velocity (vd) on top of the thermal velocity. [Note vd   ] Combined motion due to random thermal motion and an applied electric field.

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Carrier Drift • At steady state, the mean momentum gain should equal the impulse (force x time) applied to the electron during a mean free time period: mn*vdn  q x t rn The electron drift velocity is For holes:

 q x trn    n x mn* q x t rp    p x m *p

v dn 

(1)



n 

v dp

(3)



p 

qtrn mn* qtrp

m *p

(2)

(4)

 Eqs.(1) &(3) predict a linear dependence of the drift velocity on electric field which is indeed found at low electric fields (see measured data at next figure)  At higher fields however the electron and hole drift velocities saturate at comparable values to the thermal velocity of the carriers  in this regime added energy imparted by the field is transferred to the lattice rather than increasing the carrier velocity 4/30/2021

Semiconductors

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Carrier Drift Example For simplicity consider the case of FREE electrons at room temperature (300 K). If the relaxation time of the electrons at this temperature is 10-12 s, estimate their DRIFT and THERMAL velocities in the presence of an applied electric field of 10 Vcm-1. Since the electric field is low, we are in the LINEAR regime and we may estimate the drift velocity as

qt rn 1.6  1019  10 12 3 10  176 ms 1   vd  31 9.1 10 m

To estimate the thermal velocity we recall that in a CLASSICAL gas the AVERAGE thermal energy associated with each particle is 3kT/2

 vth 

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3kT  m

3  1.38  1023  300  117 kms 1 31 9.1 10 Semiconductors

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Carrier Drift • Eq.(1) states that the electron drift velocity is proportional to the applied electric field. The proportionality factor is called the electron mobility μn with units of m2/V-s, Mobility is an important parameter for carrier transport because it describes how strongly the motion of an electron or a holes is influenced by an applied electric field. • Mobility depends strongly on Temperature and Doping Concertation.

silicon at 300 K

silicon

The mobility decreases with increasing temperature because of increasing lattice vibration with increasing temperature 4/30/2021

For a given temperature, the mobility decreases with increasing impurity concentration because of enhanced impurity scatterings. Semiconductors

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Mobility and Scattering (Mathiessen's rule) • The mobility parameter is a measure of the ease of carrier motion in a crystal.  The carrier mobility varies inversely with the amount of scattering taking place within the semiconductor.  The dominant scattering mechanisms are typically (i) lattice scattering involving collision with thermally agitated lattice atoms, and (ii) ionized impurity • The probability that a carrier is scattered in a time dt only by lattice scattering is 𝑑𝑡/𝑡 where 𝑡 is the average time between two lattice scattering events.

e

• Similarly, the probability that a carrier is scattered only by ionized impurity is 𝑑𝑡/𝑡 where 𝑡 is the average time between two ionized impurity scattering events.

e

• The total probability 𝑑𝑡/𝑡 that a carrier is scattered in the time interval dt is then the sum of the probabilities of being scattered by each mechanism.

e

dt dt dt   tr tL tI



1 1 1   tr tL tI 

Because the mobility is related to relaxation time by 𝜇 = ∗ , we can write last Eq. as 

1





1

L



1

I

In general, each scattering mechanism is associated with a specific mobility. The net mobility is determined by Mathiessen's rule : 1/ = 1/1 + 1/2 + 1/3 +………….. The lowest mobility is the dominating one.

Problemset2,Solve:T&F1→ 5

MCQ 1→ 8

Problems

1→ 6

Carrier Drift Drift Current • The current (I) across an area in the wire is defined quantitatively as the net charge flowing across the area per unit time {I = dq/dt } . I

The current in the wire can be expressed in terms of the electrons drift velocity (vd), number of free electrons per unit volume (n) and cross-section area A as follows:

A

     V        d

 Suppose all electrons moving with an average drift velocity vd in the opposite direction of the field 

vd dt

 In a time dt each electron advances a distance vddt. In this time, the number of electrons crossing any cross-section of the wire will be contained in a volume Avdt. The number of electrons in this volume is nAvdt.  If each electron has a charge e, then the charge flowing across the cross-section during time dt is

dq  enAv d dt

Hence

then 4/30/2021

I 

dq  enAv d dt

I  enA  Semiconductors

 vd   Drift Current 9

Carrier Drift • The Current Density (J) is defined as the current per cross-sectional unit area I J   env d  en  A J  

Ohm’s law

where  = en is the conductivity of the material note that the resistivity  = 1/ =1/ en

• When the electric field is applied to any semiconductor material  The hole (positive charge) moves in the OPPOSITE direction to the electron in the presence of an applied electric field  The direction of motion of the positive carriers has been conventionally assumed to be the direction of a current.  The currents carried by the electrons and holes both point in the same direction as the electric field however and so add rather than canceling

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Semiconductors

Jp h

h e h e

e e

h

Jn



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Carrier Drift Electron current density is:

J n   qnvn  qn n   n 

(5)

Hole current density is:

J p  qnvp  qp p   p 

(6)

where n and p are the electron and hole conductivity of the material respectively.

Jt  J n  J p

Total drift current density is:

t  n   p

Total conductivity is:

(amps/m2) (ohm-m)-1

 q n n  q  p p Total resistivity is:

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

1





1 q n n  q  p p

Semiconductors

(ohm-m)

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Band Bending •

•

•

In the absence of an applied voltage or electric field the potential energy of the charge carriers is independent of position and the energy bands are consequently flat

V

 

When a voltage is applied across a uniformly doped semiconductor, the carrier potential energy becomes a function of position and so causes bending of the energy bands. To determine this band bending we must remember that the voltage between two points measures the work done to move a unit positive charge between the points.



V

V 0 dV 0 dx

0 Ec

x 

-qV

Ei Ev

Ec(x) Ei (x) Å

Ev(x) Eref

Eref is any convenient reference energy

 By definition, a positive voltage raises the potential energy of a hole and lowers the energy of an electron. The band diagram is higher where the voltage is lower.

Ec(x) = constant – qV(x)

assume the voltage is dropped linearly along its length.

The “constant” takes care of the unspecified and inconsequential zero references for Ec and V. 4/30/2021

Semiconductors

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Band Bending The “constant” drops out when one considers the electric field. dV dx 1 dEc 1 dEv 1 dEi    q dx q dx q dx

  

Example Next figure presents an energy band diagram of nonuniform doped silicon semiconductor sample at 300 K with Ei -Ef = EG/4 at x =  L and Ef -Ei = EG/4 at x = 0. Use Ef as the energy reference level. Use the cited energy band diagram to answer the following questions •

(b) - x direction

(c) no current

The potential energy of electron 3 and hole1 are (a) 0.84 &0.84

•

0

The direction of drift electron current at x= -L/2 is (a) + x direction

•

Electronenergy

In other words, the slope of Ec, Ev and Ei indicates the electric field

(b) 0.84 & 0

(c) 0.84 & 0.28

(d) 0 & 0.28

The kinetic energy of electron 2 and potential energy of hole 2 are (a) 0.56 & 0 4/30/2021

(b) 0 & 0.65

(c) 0.56 & 0.28 Semiconductors

Electrons 1 and 2 have the same PE

(d) 0 & 0.28 13

Carrier Diffusion Diffusion is driven by the existence of carrier concentration gradients in the semiconductor and the desire to achieve a uniform distribution of carriers To drive the expression of diffusion current, let us define the flux (j) as amount of particles passing per unit time per unit area. Electron Density n(x)

Assumptions: Electron density n(x) varies along the x direction, no applied electric field and uniform temperature

At one mean free path away from x = 0 to the left, half the electrons n(-l) will cross the plane x = 0 during one mean free time period

n(l) n(0) n(-l)

The average flux across the plane x=0 from left is j1 

n  l  l 2 t rn



-l

n  l  vth _x 2

Similarly, the average flux across the plane x=0 from right is The net flux from left to right is j  j1  j 2  4/30/2021

v th _ x 2

j2 

nl  2

0

l

x

vth _ x

 n  l  n l 

Semiconductors

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Carrier Diffusion Approximate the densities at ± l by a Taylor series expansion, we get: j

vth _ x   dn   dn   dn   n  0   l    n  0   l     vth _ x l dx   dx   dx 2 

The electron diffusion current is J n diff   q j  qvth _ x l

dn dn  qDn dx dx

DIFFUSION CURRENT

where Dn = vth-x l is the diffusivity

DIFFUSION CURRENT

e ELECTRON DIFFUSION (flux)

e

e

e

e

e

e

e

e

e

e

e

e

e

e

h HOLE DIFFUSION (flux)

h

h

h

h

h

h

h

h

h

h

h

h

h

h

x

J n diff  qDn

dn dx

(7)

x

J p diff   qDp

dp dx

(8)

NOTE THE SIGNS in Eq.(7) & Eq.(8) 4/30/2021

Semiconductors

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Total Current Density In the presence of an applied electric field both carrier drift and diffusion are effective, we may therefore write expressions for the net electron and hole currents as:

dp dx dn  q  nn  qDn dx

J P  J p drift  J p diffusion  q  p p  qD p J n  J n drift  J n diffusion Total current density

J  JP  Jn

(9) (10)

(11)

ASSIGNEMENT Next figure is a part of the energy band diagram of a p-type semiconductor bar of length L. The valence band edge is sloped because doping is non-uniform along the bar. Assume that Ev rises with a slope of Δ ∕ L (a) Write an expression for the electric field inside this semiconductor bar. (b) what is the electron concentration n(x) along the bar? Assume that n(x = 0) is no. Express your answer in terms of no, Δ, and L. 4/30/2021

Semiconductors

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Einstein Relations Recall that

1 * 2 3 mn vth  kT 2 2

&

n 

qtrn m*n

The equipartition theorem in statistical mechanics states that the energy is equally divided in the degrees of freedom. Thus for the one-dimensional case being considered, we have: 1 * 2 1 mn vth _ x  kT 2 2 For electrons

2

Dn  vth _ xl  trn vth  Dn

n Similarly, for holes

vth2 _ x 

Dp

p

 

kT m*n

t rnkT q  kT   mn* q  q

kT q kT q

 n 

Einstein relations

The Einstein relations show that mobility and the diffusion constant are proportional to each other at fixed temperature 4/30/2021

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Example Consider a silicon sample at 300 K that is doped with acceptors at a concentration of 1015 cm-3. A stream of minority carriers is injected at x = 0 and the distribution of these carriers is assumed to be linear, decreasing from 1011 cm-3 at x = 0 to the equilibrium value at x = W = 10 mm. Determine the diffusion current density of electrons. ni  1010 cm3 , p  N A  1015 cm3 ni2 n   105 cm 3 p

n  1300 cm 2 / Vs from  vs. N A curve  J n diff  qDn

dn dn  kT n dx dx

Si at 300K

10  10   1.38  1023  300 1300  4   10 10    0.54 mA/ cm2 5

11

NA

The motion of electrons is in the positive x direction Causes a diffusion current in the -x direction! 4/30/2021

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Example A semiconductor maintained at 300 K is characterized by the energy band diagram shown below. Use the cited energy band diagram to select the correct answer. 1. The semiconductor is degenerate at (a) –W...