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**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 March 22, 2017 1

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**Outline Carrier Concentration at Thermal Equilibrium**

Introduction Fermi Dirac Statistics Donors and Acceptors Determination of Fermi Level Dopant Compensation March 22, 2017 March 22, 2017 2

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**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 March 22, 2017 March 22, 2017 3 3 3

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**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 March 22, 2017 March 22, 2017 4 4 4

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**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. March 22, 2017 5

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**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? March 22, 2017 6

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**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 March 22, 2017 7

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**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: March 22, 2017 8

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**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 March 22, 2017 9

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**sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html**

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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**sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html**

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., 11.70 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

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**Donors and Acceptor Levels**

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**Determination of Fermi Level**

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**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

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**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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**sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html**

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 March 22, 2017 16 16

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**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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**sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html**

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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**sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html**

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

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**sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html**

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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**sawyes@rpi.edu www.rpi.edu/~sawyes/courses.html**

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?

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