1. Nonhomogeneous Semiconductors: Constancy of the Fermi Level at Equilibrium
Chapter 4 Section 1
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2. Nonhomogeneous semiconductors
So far have examined homogeneous semiconductors
Doping constant everywhere in the crystal
Material is the same everywhere (e.g. silicon)
In this chapter will look at nonhomogeneous semiconductor
Doping may vary with location
Material may vary with location
We’ll see that these can create internal electric fields that are very useful

3. Constancy of Fermi level at equilibrium
In a system at equilibrium, the Fermi level is constant everywhere

4. Consider two materials in contact
Let them have different band gaps
Figure shows instant of contact – state is not stable
Each has its own Fermi level and associated carrier distributions
Only showing electrons here
Condition shown is electrical neutrality
Will see why shortly
Neutrality

5. Neutrality continued Consider energy range shaded
Recall electrons travel at constant energy between collisions
In this range, there are more electrons on the left than on the right
Result is net flux of electrons from A to B
Neutrality

6. Must eventually reach equilibrium
Result is net flux of electrons from A to B, but can’t go on forever
When equilibrium is reached, net current is zero
Number of electrons flowing from A to B must equal number flowing from B to A
This is what picture will look like- let’s see why
Equilibrium

7. Consider each material separately for a moment
Let the density of states functions in each material be SA(E) and SB(E)
Let probability of occupancy in each material be fA(E) and fB(E)
Next, let flux of electrons from A to B be and flux from B to A be
Neutrality

8. More electrons in dE in A than in B
Since conduction bands are mostly empty, lots of empty states for electrons to flow to
Concentration of empty states in B is
And
(C is some constant)
Neutrality

9. Flux from B to A
At equilibrium, both fluxes must be equal
And
Thus

10. Must include Evac to get the whole picture

11. Write out the Fermi functions
Thus
Equilibrium

12. Comments
On each side, the Fermi level is located on the energy-band diagram according to the doping in that material
Band gap is fixed for a given material
Thus, can draw energy band diagram on each side based on knowledge of material
To draw combined diagram, must line up Fermi levels

13. More comments
We have done the example for a heterojunction (junction between dissimilar materials)
Principle of constancy of Fermi level at equilibrium applies to all systems
We have idealized this somewhat- we’ll see details in a later chapter

14. Key points
In a system at equilibrium, the Fermi level is constant everywhere
Next section: what happens when the doping concentration varies with position?

15. Graded Doping Chapter 4 Section 2 Copyright © 2016
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16. Introduction
In last section we combined two materials with different band gaps
Here, will consider a single material but let the doping vary with position

17. Compensated materials
Often a semiconductor contains both donors and acceptors
Example: might start with a p—type substrate (containing acceptors) and add donors to some sections to compensate and make regions n-type
Define
n-type
p-type

18. Consider a sample with graded doping

19. Where there are more acceptors there are more holes
This is neutrality: at any location, the number of positive charges equals the number of negative charges
So you might expect this:

20. Assuming all acceptors are ionized
Then concentration of holes is
But NA’ is a function of x
Quantities NV, k, T are all constants
Thus Ef-EV must vary with position
But, Ef is constant at equilibrium
Then EV must vary with position
How to fix energy band diagram?

21. Let the diffusion begin
Holes diffuse to regions of lower concentration
The leave behind ionized acceptors
(negatively charged)

22. Ions do not move Remember acceptors are atoms (e.g. boron)
They are chained to the lattice by atomic bonds

23. Result is uncovered negative charges
More negative charges toward the left (uncovered acceptor ions)
More positive charges to the right (holes that have diffused)
Results in an electric field

24. But at equilibrium, current is zero
If there is a field, there must be drift current
If there is a concentration gradient, there must be diffusion current
At equilibrium, these two must be equal and opposite
Equilibrium

25. If field is present, must be a gradient in the potential energy
Recall that EV is the potential energy for holes in the valence band
Recall EC is the the potential energy for electrons in the conduction band
Thus
Since Eg is constant, EC is parallel to EV
Since χ is constant, Evac is parallel to EC
Thus
Equilibrium

26. Write out the currents
Drift plus diffusion must equal zero at equilibrium
Examine diffusion term

27. But Ef is constant We had
Thus the electric field is proportional to slope in EV

28. For a given material Band gap Eg is constant, so EC is parallel to EV
Electron affinity is constant, so Evac is parallel to EC
Now able to draw entire figure

29. Procedure for drawing energy band diagrams
Assume electrical neutrality in every macroscopic region
Using the vacuum level as a reference (i.e. draw Evac as constant for each region)
Assemble the energy band diagram for each region using knowledge of band gap and electron affinity
Use knowledge of doping to draw in the Fermi level
Adjust (tilt) the diagram to make Ef constant everywhere
Redraw.

30. Return to previous results
Combine to get
Einstein relation again!

31. Recall under graded doping:

32. This is a built-in field
It is not applied
It is the result of doping gradient
It does create a drift current even at equilibrium
There is also a diffusion current at equilibrium
These two cancel at equilibrium

33. Currents

34. Comments One side is at a higher potential than the other
There is a built-in voltage Vbi from one end to the other
Can it be a battery? No.

35. Suppose there is a wire connecting one end to the other
There is metal from one end to the other
There are additional built-in voltages at the junctions to the wire
Net voltage around the loop is zero
Cannot “access’ the built-in in voltage from outside

36. Key points
If the doping varies with position, there is a change in carrier concentration
Sets up a diffusion current
As carriers (charged) diffuse, they leave “uncovered” ionized donors or acceptors
Donors and acceptors cannot move!
The charges set up an electric field
Electric field causes a gradient in the potential energy (band edges)
The electric field causes drift, which exactly compensates the diffusion
Both drift and diffusion going on all the time, even at equilibrium- but they cancel

37. Key points, continued
Varying the doping concentration creates built-in internal electric fields
Fields create drift currents
Varying doping creates diffusion currents
At equilibrium, these cancel
For both electrons and holes
There are additional built-in fields at contacts to sample, so is not a battery

38. Nonuniform Composition
Chapter 4 Section 3
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39. Introduction
We saw that nonuniform doping in a material created built-in internal electric fields
Band gaps, electron affinities were constant but doping caused band edges to slope
Now we will examine what happens if the material is not uniform
Band gap not uniform- what happens?

40. Consider a graded alloy
Let the sample be pure silicon at one end
Let germanium be introduced in increasing amounts along sample
Results in alloy of SiyGe1-y
Variable y denotes fraction of Si
Both Si and Ge are in column four
Will bond covalently
Have same crystal structure
But they have different band gaps

41. Let the material be doped
Let’s say acceptors
Say doped such that Ef is same distance from EV throughout
Doping is not quite uniform because Si and SiGe alloys will have different slightly NV (because of different effective masses)

42. Start by drawing neutrality case
Draw Evac first
Then construct rest of diagram
Si and Ge have nearly the same electron affinity χ so EC relatively constant
Band gap shrinks as Ge fraction increases

43. Think about carrier concentrations
Hole concentration about the same for both because Nv’s are close and Ef-EV constant
Electron concentrations not the same
Band gaps different, so Ef get closer to EC

44. Expect diffusion
Holes do not particularly diffuse, since there is no concentration variation
Electrons diffuse from right to left- leaving behind??
Holes
No donors in this example

45. Equilibrium case Redraw figure with Ef constant
There is a gradient in EC and EC is the potential energy for electrons
Thus there is an electric field for electrons
Electric field (for electrons) accelerates them to the right
Electrons diffuse to left
Electron drift and diffusion cancel at equilibrium

46. Equilibrium case for holes
There is no diffusion of holes (no concentration gradient)
There is no electric field for holes (no slope in potential energy for holes, which is EV)

47. Define effective electric fields
Recall force is minus the gradient of the potential
For holes, we get
(Why no minus sign? Hole energy increases downward, so slope as drawn is actually negative that for holes)

48. What is the “true” electric field?
By definition, electric field is force on a unit positive test charge
Drill an imaginary hole in the sample and put a test charge on a stick down into the hole
Potential energy on the test charge is Evac

49. Example
Consider a transistor with a Si:Ge base in which the band gap varies linearly by 0.1 eV across the 0.05 µm base width. Assume that the net acceptor concentration is constant. Find the effective electric field and force for holes and electrons, and the true electric field.

50. Solution
Since ΔEg varies (by about 7.5 meV per atomic percent Ge), grading is linearly graded in Ge content from 0 to ≈13%
From previous slides, if NA≈constant, EV is constant for SiGe

51. Since band gap decreases Ec decreases
Base with is 0.05 µm
Effective field for electrons is

52. True field

53. Key points
A gradient in composition can produce an internal built-in field
Field for electrons not necessarily the same as for holes
True field is indicated by slope in Evac

54. Graded Doping and Nonuniform Composition Combined
Chapter 4 Section 4
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55. Introduction
In an actual SiGe transistor base may have both composition gradients and doping gradient
Effect is to speed up transistor even more
Higher operating frequencies

56. Example
A Si:Ge transistor has a base that is compositionally graded as in earlier example, but in addition, the net acceptor density decreases exponentially from 1018 cm-3 to 3x1016 cm-3 over this region. Find the effective electric field for electrons.

57. Solution
Effective field due to composition was 0.1 eV over 0.05 µm, or -20kV/cm as found before
Doping:
Recall
Thus
NA(0)=1018 cm-3
NA(0.05 µm)=3x1016 cm-3

58. Additional tilt of EV due to doping
Let WB=base width, or 0.05 µm
Minus sign means EV moves down (away from Ef)

59. Energy band diagram under neutrality

60. Energy band diagram at equilibrium

61. Key points
Both doping gradients and compositional variations can be used to create internal, built-in electric fields
These can be combined to increase the effect
Effective fields for electrons may be different from those for holes

62. Summary Chapter 4 Section 5 Copyright © 2016
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63. Key points Found we can create internal electric fields by
Varying the doping concentration
Varying the composition
Both
The field is proportional to the slope of the potential energy
Electrons:
Holes:

64. Key points continued The field cannot be accessed from outside
Not a battery
Reason is that there are additional fields where sample meets the real world
Wire, or vacuum, or anything
These fields sum to zero around a closed loop

65. Road map
Real semiconductor devices are usually made of combinations of n-type and p-type materials, joined
Example: a diode can be made from a junction between an n-type region and a p-type region
The internal fields will have some very interesting effects
Next up: Part 2 Diodes