Lecture Notes 7 PDF

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Agenda (Streetman 3.3)  Review: Intrinsic carrier vs. temperature Carrier Conc. Temperature Dependence

 Counting Electrons and Hole density Density of States (DOS) Fermi-Dirac Integral

 Using Maxwell Boltzmann Statistic to compute electron and hole density

ECE 440 Instructor: Mohamed Mohamed

n

i

as a function of Temperature

Intrinsic carrier concentration varies with temperature. (See Table Note: The larger bandgap, the smaller the intrinsic carrier density.

SEE FIGURE 3.17 STREETMAN ECE 440 Instructor: Mohamed Mohamed

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Electron Density as a function of Temperature

Si sample with a donor concentration of ND=1015 cm-3.

Fully ionized! ND =ND+

Full ionization at room temperature

ND=ND

+

ECE 440 Instructor: Mohamed Mohamed

Counting Electrons To compute current we need electron and hole densities:

no : po :

 qp ov hole,velocity J   qn ov electron,velocity

Mobile electron in the conduction band at Equilibrium Mobile hole in the valence band at Equilibrium

Equilibrium: No applied Bias or external perturbation The number of free electron and hole are given by n0  

Eto p EC

g C ( E) f ( E) d E

p0  

EV EBo tto m

g V ( E) 1  f ( E )  dE

ECE 440 Instructor: Mohamed Mohamed

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Density of States (DOS) Density of States (DOS) Describes the number of available States per unit energy per unit volume

DOS: Material Property! : Obtained from bandstructure detail ECE 440 Instructor: Mohamed Mohamed

Density of State and Band Diagram Energy Band diagram:

Conduction Band

E

Ec

Note DOS is NOT defined inside the bandgap!

ENERGY GAP

Ev

Valence Band

Notice in 3D:

ECE 440 Instructor: Mohamed Mohamed

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Fermi Dirac Distribution The function f(E) gives the probability that an available energy state at E will be occupied by an electron at temperature T

1

f E  

1  e

E  EF kBT

EF: Energy for which the probability of finding electron is 50%

What is the probability of finding hole in the valence band? ECE 440 Instructor: Mohamed Mohamed

Electron vs. Hole Probability

f E  

1 1  e

E EF k BT

1

1-f  E  

E F E k T 1  e B

ECE 440 Instructor: Mohamed Mohamed

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Fermi-Dirac: Energy Band diagram (1) 1

f E  

1  e

E EF k BT

ECE 440 Instructor: Mohamed Mohamed

Example 0

What is the probability that the conduction edge is filled given: a) EF is positioned Ec. b) EF is positioned 3kT below the conduction band Edge . c) EF is positioned 100kT below the conduction band Edge

ECE 440 Instructor: Mohamed Mohamed

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ECE 440 Instructor: Mohamed Mohamed

Fermi-Dirac: Energy Band diagram (2) Fermi functions moves Up or down the band diagram as we vary the doping

Go to the next slide to see a blow up version

ECE 440 Instructor: Mohamed Mohamed

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Blow up of the Fermi Dirac function inside the Conduction Band Which of these lines in the plot Represent higher carrier density?

Ec The zero energy in this plot is the Conduction Band Edge ECE 440 Instructor: Mohamed Mohamed

Fermi-Dirac & DOS OVERLAP is what matters!

f E  

1 1  e

E EF kB T

g 3D C (E )  E ECE 440 Instructor: Mohamed Mohamed

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Electron Density

1

f  E 

1  e

n0  

E top EC

E  EF kB T

g 3D C (E )  E

gC ( E) f ( E) d E ECE 440 Instructor: Mohamed Mohamed

Big Picture: Counting Electrons & Holes (intrinsic semiconductor example) n0  

E top EC

Energy Band

p0  

gC( E) f ( E) d E Density of states

EV EBottom

Fermi-dirac

g V ( E) 1 f ( E) d E Carrier concentration

ECE 440 Instructor: Mohamed Mohamed

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Counting Electrons and Holes Etop

EC

n0 

p0  

EV EBottom

g C ( E) f ( E) d E

g V ( E) 1 f ( E) d E

ECE 440 Instructor: Mohamed Mohamed

Integral of Fermi-dirac & DOS is hard to compute Analytically! n0 

Etop

 EC

gC ( E) f ( E) d E

p0  

EV EBottom

g V ( E) 1 f ( E) d E

ECE 440 Instructor: Mohamed Mohamed

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Maxwell Boltzmann Statistic f ( E) 

Fermi-Dirac Statistic: (Degenerate Stat)

1 ( E EF ) / kT

1e

( E EF ) / kT   f ( E)  e If E  E F  3 kT,  (E F E ) / kT  1  f ( E)  e

Fermi-Dirac is approximated as Maxwell-Boltzmann (non-degenerate) Statistics Where is that valid in the Band diagram? What doping Densities does it correspond to? ECE 440 Instructor: Mohamed Mohamed

Effective Density of States Replace DOS with Effective Density of States Assume all particles are effectively in the conduction band and valence band edges :

 2 mn k BT  N C  12   h2  

 2 mp kB T NV  12 h2 

32

for Si

32

  

for Si

at room-temperature (300 K)

NC  2.86 1019 cm 3 3

NV  2.66 10 cm 18

for Si

for Si

ECE 440 Instructor: Mohamed Mohamed

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Fermi-Dirac (FD)vs. Maxwell Boltzmann (MB) How does MB compare to FD in terms of accuracy?

ECE 440 Instructor: Mohamed Mohamed

Simplified Equations for calculating Carrier Density n0  

Etop EC

gC ( E) f ( E ) dE

p0  

EV EBototm

g V ( E) 1 f ( E) d E

ECE 440 Instructor: Mohamed Mohamed

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Summary: Intrinsic, n-type, p-type

ECE 440 Instructor: Mohamed Mohamed

Example 1

What is the probability that a state 3kT above the conduction edge is filled using both degenerate and non-degenerate statistics given: a) EF is positioned at Ec. b) An intrinsic semiconductor. c) EF is positioned 0.1kT below the conduction band Edge

ECE 440 Instructor: Mohamed Mohamed

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Example 2:Solution

ni  pi

For intrinsic semiconductor we know:

  E C  E i      E i  EV   N C exp   N V exp  kT kT    

N E i  E C  E V  E i  kT ln  V  NC

   

Ei 

Ec  EV   kT ln NV 2

2

  N   C

ECE 440 Instructor: Mohamed Mohamed

Example 2: Solution •

Let us find the location of the intrinsic Fermi-level for GaAs, Si and Ge

Ei 

E c  EV  2



kT  NV   ln 2  NC 

  

 35 meV GaAs : EG  1.42 eV kT  N V      - 7 meV Ge : E G  0.67 eV ln  2  N C    - 13 meV Si : E G  1.12 eV

kT  N V  3kT  mh , DOS  ln ln  4  me ,DOS 2  N C 

  

   

ECE 440 Instructor: Mohamed Mohamed

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Example 3 Silicon sample is uniformly doped with is given by 1x10 15/cm3 Boron atoms The sample is kept at T=300 K. Is this an n-type or p-type semiconductor? Compute the electron and hole densities? All dopants are FULLY ionized At room temperature So we expect that the number of  Holes to be: p  ni  N A

p-type (majority=holes): po  ni  N

electron (minority): no po  n2i

 A

no 

po  1 .5 1010  1 .0 1015  1.0 1015 cm 3



10 n 2i 1.5  10  15 po 1. 10



2

 1. 0  10 5cm  3

ECE 440 Instructor: Mohamed Mohamed

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