Advanced Semiconductor Physics ~ Dr. JenaUniversity of Notre Dame

Department of Electrical Engineering

SIZE DEPENDENT TRANSPORT IN

DOPED NANOWIRES

Qin ZhangAnubhav Khandelwal

Jeffrey Bean

December 13, 2004

Advanced Semiconductor Physics ~ Dr. JenaUniversity of Notre Dame

Department of Electrical Engineering

OUTLINE

Introduction and MotivationBandgap variation in 1D wiresImpurity binding energyCarrier concentration in 1D wiresRoughness scattering limited momentum relaxation time

Mobility in nanowires

Conclusion

Advanced Semiconductor Physics ~ Dr. JenaUniversity of Notre Dame

Department of Electrical Engineering

Introduction and Motivation

In low dimensional structures, such as nanowires, quantum effects change electrical properties:

electronic band gapimpurity binding energycarrier concentrationcarrier mobility

Doping in nanowiresMobility in nanowires

Advanced Semiconductor Physics ~ Dr. JenaUniversity of Notre Dame

Department of Electrical Engineering

Bandgap Variation

For a quantum wire, confinement energy is given by:

in nm

Advanced Semiconductor Physics ~ Dr. JenaUniversity 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 Eg is small

Advanced Semiconductor Physics ~ Dr. JenaUniversity of Notre Dame

Department of Electrical Engineering

Impurity Binding EnergyThe binding energy was calculated using the expression1:

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(t0=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)

Advanced Semiconductor Physics ~ Dr. JenaUniversity 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.

174.1310.20.13CdSe

5210.7812.90.063GaAs

9780.6311.70.98Si

R0*(meV

)a0

*(nm)

ε/ε0me*/m0

Binding Energy

Binding energy vs. Wire radius for CdSe Binding energy vs. Wire radius for GaAsBinding energy vs. Wire radius for Si

Advanced Semiconductor Physics ~ Dr. JenaUniversity of Notre Dame

Department of Electrical Engineering

Carrier Concentration

For the 1-D case, the total electron concentration in the conduction band is:

Where: gc1D(E) is the density of states (DOS) for 1-D

f(E) is the Fermi distribution function.

Under non-degenerate conditions,

where:

Advanced Semiconductor Physics ~ Dr. JenaUniversity of Notre Dame

Department of Electrical Engineering

Carrier Concentration

Using the charge-neutral relationship for n-type material:

Since Eb 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.

Advanced Semiconductor Physics ~ Dr. JenaUniversity 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 Nd

Electron concentration in the conduction band of CdSe: n vs. d with T=300K and different doping densites

n as d and Nd

Advanced Semiconductor Physics ~ Dr. JenaUniversity of Notre Dame

Department of Electrical Engineering

Carrier ConcentrationElectron concentration for CdSe, GaAs, and Si vs. wire radius with doping density of 5*105 cm-

1 at 300K

Advanced Semiconductor Physics ~ Dr. JenaUniversity 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.

Advanced Semiconductor Physics ~ Dr. JenaUniversity of Notre Dame

Department of Electrical Engineering

Roughness Scattering Limited Momentum Relaxation Time

Momentum relaxation time is given by:

where:

total carrier density N1V = 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:

Advanced Semiconductor Physics ~ Dr. JenaUniversity of Notre Dame

Department of Electrical Engineering

Momentum relaxation time is calculated as:

where k10d=2.405 (first root of J1(k10d)), 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: kF is the Fermi wave vector given by n/2

Advanced Semiconductor Physics ~ Dr. JenaUniversity 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

Advanced Semiconductor Physics ~ Dr. JenaUniversity of Notre Dame

Department of Electrical Engineering

Conclusions

Hydrogenic impurity binding energy in a quantum wire:

Decreases as wire radius increasesMaximum when impurity is on the wire axis

The electron carrier density:Increases when wire radius, temperature, and doping density increaseIncomplete ionization at room temperaturePercentage of ionization decreases as doping density increases and temperature decreases

surface roughness limited momentum relaxation time:

Mobility varies as a function of d6

Mobility first decreases, then increases as roughness variation Λ is increased, and reaches a maximum at the Fermi wavelength

Questions???