1. ECSE-6230 Semiconductor Devices and Models I Lecture 4
Prof. Shayla Sawyer
Bldg. CII, Rooms 8225
Rensselaer Polytechnic Institute
Troy, NY
Tel. (518)
Fax. (518)
March 22, 2017
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2. Outline Carrier Concentration at Thermal Equilibrium
Introduction
Fermi Dirac Statistics
Donors and Acceptors
Determination of Fermi Level
Dopant Compensation
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3. Carrier Concentration Introduction
One of most important properties of a semiconductor is that it can be doped with different types and concentrations of impurities
Intrinsic material-no impurities or lattice defects
Extrinsic-doping, purposely adding impurities
N-type mostly electrons
P-type mostly holes
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4. Carrier Concentration Introduction
To calculate semiconductor electrical properties, you must know the number of charge carriers per cm3 of the material
Must investigate distribution of carriers over the available energy states
Statistics are needed to do so
Fermi-Dirac statistics
Distribution of electrons over a range of allowed energy levels at thermal equilibrium
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5. Fermi-Dirac Distribution
Probability that an available energy
state at E will be occupied by an
electron at absolute temperature T
Mathematically,
EF (Fermi Energy) is the energy at which f(E) = 1/2
The transition region in (E - EF)
from f(E) =1 to f(E) = 0 is within
3 k T.
When T  0,
E is discontinous at E = EF.
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6. Fermi-Dirac Distribution
To apply the Fermi-Dirac distribution, we must recall that f(E) is the probability of occupancy of an available state at E.
Where can we find available states?
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7. Carrier Concentration At Thermal Equilbrium
Number of electrons (occupied conduction band levels) given by:
Density of states g(E) can be approximated by the density near the bottom of the conduction band
MC is the number of equiv. minima
where
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8. Carrier Concentration At Thermal Equlibrium
The integral can be evaluated as
For the valence band, consider light and heavy holes for the density of states effective mass for holes (mdh)
and use similar equation
Where NC is the effective density of states in the conduction band given by:
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9. Carrier Concentration At Thermal Equlibrium: Intrinsic
For intrinsic material lies at some intrinsic level Ei near the middle of the band gap, electron and hole concentrations are
Law of mass action: product of maj. and min. carriers is fixed
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Donors and Acceptors
Doping by substituting Si atoms with Column III or V of the Periodic Table.
Very dilute doping level, typical 1014 to 1018 cm-3, results in discrete energy levels.
Donor level is neutral if filled with e-,
positively charged if empty.
e.g., P, As, and Sb in Si.
Acceptor level is neutral if empty,
negatively charged if filled with e-.
e.g., B and Al in Si.

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Donors and Acceptors
“Hydrogen-like” Model to describe dopant atom ionization.
Hydrogen Atom
Ground state (n=1) ionization energy of hydrogen is eV.
To estimate ionization energy of donors, replace m0 with m* and
0 and S (e.g., 11.70 for Si).
ED = (0 /S )2 ( m*/ m0 ) EH ~ eV for Ge,
0.025 eV for Si,
0.007 eV for GaAs
EA ~ eV for Ge,
0.05 eV for Si,
0.05 eV for GaAs
kT~0.026eV
Comparable to thermal energies so ionization
is complete at room temperature

12. Donors and Acceptor Levels

13. Determination of Fermi Level

14. Determination of Fermi Level
Intrinsic Semiconductor - EF ~ Eg / 2
Extrinsic Semiconductor - EF adjusted to preserve
space charge neutrality
Space Charge Neutrality
n0 + NA- = ND+ + p0
Total Neg. Charges = Total Positive Charges
electrons and ionized acceptors=holes and ionized donors
100% ionization assumed.
Ionized Concentration of Donors
When impurities are introduced:
where gD is the ground state degeneracy of donor impurity
gD = 2 (i) electrons with either spin
(ii) no electrons at all

15. Determination of Fermi Level
Ionized Acceptors
where gA is the ground state degeneracy of acceptor impurity
gA = 4 for Ge, Si, and GaAs because
(i) Acceptor levels can receive electrons with either spin and
(ii) Valence band double degeneracy.
Space Charge Neutrality
N-type Semiconductor is assumed.
n=ND++p ~ ND+ therefore

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Charge Neutrality
Since the material must balance electrostatically, the Fermi level must adjust such that charge neutrality remains.
The Fermi level therefore can be calculated for a set given ND, ED, NC, and T
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17. Graphical Determination of Fermi Level
Graphical solution of the space charge neutrality equation.
No need to assume 100% ionization.
Need to know the donor (or acceptor) level.
Degenerate Doping
Impurity levels are broadened into
impurity bands, thus reducing the
ionization energy of the dopants.
Ex. Phosphorus in Silicon.
EC - ED(ND) = x10-8 (ND)1/3
[eV] for ND > 1018 cm-3

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Dopant Compensation
When both n- and p-type (donor and acceptor) impurities are present,
the space charge neutrality condition n0 + NA- = ND+ + p0
holds, even when the impurities are deep levels.
In an n-type semiconductor where ND>>>NA
Fermi level can be obtained from

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Dopant Compensation
In an p-type semiconductor where NA>>>ND
Fermi level can be obtained from

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Example Problem
A hypothetical semiconductor has an intrinsic carrier concentration of 1.0 x 1010 cm-3 at 300 K, it has a conduction and valence band effective density of states NC and NV both equal to 1019 cm-3.
What is the band gap Eg?
If the semiconductor is doped with Nd = 1x1016 donors/cm3 , what are the equilibrium electron and hole concentrations at 300K?

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Example Problem
A hypothetical semiconductor has an intrinsic carrier concentration of 1.0 x 1010 cm-3 at 300 K, it has a conduction and valence band effective density of states NC and NV both equal to 1019 cm-3.
c) If the same piece of semiconductor, already having Nd = 1x1016 donors/cm3, is also doped with Na= 2x1016 acceptors/cm3 , what are the new equiliblrium electron and hole concentrations at 300 K?
Consistent with your answer to part (c), what is the Fermi level position with respect to the intrinsic Fermi level, EF – Ei?