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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering SIZE DEPENDENT TRANSPORT IN DOPED NANOWIRES Qin Zhang Anubhav Khandelwal Jeffrey Bean December 13, 2004

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering OUTLINE Introduction and Motivation Bandgap variation in 1D wires Impurity binding energy Carrier concentration in 1D wires Roughness scattering limited momentum relaxation time Mobility in nanowires Conclusion

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Introduction and Motivation In low dimensional structures, such as nanowires, quantum effects change electrical properties: electronic band gap impurity binding energy carrier concentration carrier mobility Doping in nanowires Mobility in nanowires

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Bandgap Variation For a quantum wire, confinement energy is given by: in nm

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Effective Mass Approximation Advantages Simplest Dimensional Effect Surface Effect Limits effective mass from bulk semiconductors is not good assumption when d is very small parabolic band structure is not a good approximation when E g is small

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Impurity Binding Energy The binding energy was calculated using the expression 1 : where: We have considered the cases when the impurity is located on the axis, at the midpoint between the axis and edge, and on the edge(t 0 =0, ½, 1 respectively) of the wire for different values of d 1. J. W. Brown and H. N. Spector, J. Appl. Phys 59, 1179 (1986)

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering We have determined the hydrogenic binding energies (in meV) as a function of wire radius (in nm) for CdSe, GaAs, and Si using parameters as listed below. 174.1310.2 0.13CdSe 5210.7812.9 0.063GaAs 9780.6311.7 0.98Si R 0 * (meV) a 0 * (nm) ε/ε 0 m e * /m 0 Binding Energy Binding energy vs. Wire radius for CdSe Binding energy vs. Wire radius for GaAs Binding energy vs. Wire radius for Si

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Carrier Concentration For the 1-D case, the total electron concentration in the conduction band is: Where: g c 1 D (E) is the density of states (DOS) for 1-D f(E) is the Fermi distribution function. Under non-degenerate conditions, where:

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Carrier Concentration Using the charge-neutral relationship for n-type material: Since E b depends on the position of impurities, n needs to be averaged: This equation is solved numerically. Here we assume the doping is uniform along the axis of the nanowires.

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering CdSe Carrier Concentrations… Electron concentration in the conduction band of CdSe: n vs. temperature for constant doping density and different wire radii n as T and d Electron concentration in the conduction band of CdSe: n vs. Temperature with constant radius and different doping densites n as T and N d Electron concentration in the conduction band of CdSe: n vs. d with T=300K and different doping densites n as d and N d

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Carrier Concentration Electron concentration for CdSe, GaAs, and Si vs. wire radius with doping density of 5*10 5 cm -1 at 300K

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Roughness Scattering Limited Momentum Relaxation Time For a quantum wire, confinement energy is given by: For the ground state wave function: Roughness potential V(z) is given by: Roughness S(z) is assumed to be Gaussian and is expressed as where Δ is the maximum height and Λ is the full width half max of the roughness.

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Roughness Scattering Limited Momentum Relaxation Time Momentum relaxation time is given by: where: total carrier density N 1 V = nL, n is carrier density (cm -1 ), L is the wire length, θ is the angle between the initial and final wavevectors k and k’ Calculating the matrix elements: where:

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Momentum relaxation time is calculated as: where k 10 d=2.405 (first root of J 1 (k 10 d)), which gives the final expression for m -1 The momentum relaxation time is given by: The mobility is then given by: Roughness Scattering Limited Momentum Relaxation Time where: k F is the Fermi wave vector given by n/2

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Roughness Scattering Limited Momentum Relaxation Time mobility µ as a function of n for different X(=Λ) mobility µ as a function of d for different X(=Λ) mobility µ as a function of X(=Λ) for two values of n equal to 104 cm -1 and 105 cm -1 1: d=20nm 2: d=10nm 3: d=5nm

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Advanced Semiconductor Physics ~ Dr. Jena University of Notre Dame Department of Electrical Engineering Conclusions Hydrogenic impurity binding energy in a quantum wire: Decreases as wire radius increases Maximum when impurity is on the wire axis The electron carrier density: Increases when wire radius, temperature, and doping density increase Incomplete ionization at room temperature Percentage of ionization decreases as doping density increases and temperature decreases surface roughness limited momentum relaxation time: Mobility varies as a function of d 6 Mobility first decreases, then increases as roughness variation Λ is increased, and reaches a maximum at the Fermi wavelength Questions???

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