Schriemer Pres

8/11/2019 Schriemer Pres

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Towards Modelling Semiconductor

Heterojunctions

Ronald Millett, Jeff Wheeldon, Trevor Hall and Henry Schriemer

Centre for Research in Photonics

University of Ottawa

Presented at the COMSOL Users Conference 2006 Boston



Introduction

Motivation

Semiconductor heterojunctions occur in numerous important

devices, including heterojunction bipolar transistors (HBTs), high-

electron mobility transistors (HEMTs), photodiodes and laser diodes.FEMlab contains models such as the “Semiconductor Diode” but

this model requires continuous dopant concentrations between P-n

junctions.

In this presentation we demonstrate a 2D multiphysics model using

FEMlab/Matlab to simulate a heterojunction separating abruptly

doped semiconductor layers of different dopant concentrations.

This carrier transport model will be included in a larger overall

separately-confined heterostructure laser simulation that will include

optical and thermal effects.

To model abrupt heterojunction structures, other groups are usingeither custom code or very expensive programs such as Crosslight.



Semiconductor Electronic Theory

To solve our heterojunction 9 equations must be solved self consistently.

nq n qD nμ ϕ = − ∇ + ∇n

J

Poisson’s Equation (1)

1n

dn Rdt q

= ∇ ⋅ −n

J

( ) ( ) D Aq p n N N ε ϕ + −∇ ⋅ − ∇ = − − + −

Current Continuity Equations (2)

pq p qD pμ ϕ = − ∇ − ∇p

J

Carrier Transport Equations (6)

1 p

dp R

dt q= − ∇ ⋅ −

pJ 0

dp dn

dt dt = =

Given boundary conditions we solve for ( ) ( ) ( ), ,n p ϕ ⎡ ⎤⎣ ⎦r r r



Semiconductor Electronic Theory

The recombination term in the drift-diffusion equations is assumed to be

Shockley-Read-Hall (trap-assisted) recombination:

( ) ( )11

00,

p pnn

pnnp R R

n p

SRH pn+++

−==

τ τ

This model can be easily modified to include other recombination terms

such as optical injection or optical gain.



Numerical Model

p-N Heterojunction Diode

First example is a simple p-NHeterojunction Diode

Five application modes requiredwith five coupled variables to be

solved: cn1,cp1,cn2,cp2,V



The upper and lower boundaries are

assumed to form ideal ohmic contacts,

meaning that the carrier concentrations will be

at their thermal equilibrium values at these

points.

Thermionic emission, drift-diffusion, and

tunneling are three important carrier transport

mechanisms in semiconductorheterojunctions; this model thermionic

emission is assumed to be the dominant

mechanism at the junction.

0n n=0 aV ϕ ϕ = −

0 p p=0ϕ =

Numerical Model

p-N Heterojunction Diode



Thermionic emission current is calculated by assuming that all carriers

with a kinetic energy, sufficiently high to classically traverse the potential

boundary at the heterojunction, form the current.

( ) ( ) ( ),F F F F n N n N j j jTn Tn TN

+ −= −

Thermionic Emission

( )

( )( )

,2 1,

, 22

E k x j dk d f k x xTn p

k k x xπ

∞ ∂±= ∫ ∫

± ∂Δ

⎛ ⎞⎜ ⎟⎝ ⎠

k τ

k k τ τ

h

( ) ( ) ( ),

2*,

ln 1 exp, 2 3

2F n p

m k T Be h j d x xTn p

x

η ξ η π η

± ∞± ±= + −∫

±Δ

⎡ ⎤⎣ ⎦

h

( )2

,2 e h

E m∗±

⋅=

k k k

h

T k

E

B

x x =η

n

B

F

k T ξ =

2 2

,2

x Boundary

e h

k E

m

Δ∗±

= h

( )F N jTN − ( )F n j

Tn+

( ) ( ) ( ),F F F F p P p P j j jTp Tp TP+ −

= −

p N



Numerical Model

Solve Drift Diffusion and

Poisson Equations

Calculate the Thermionic

Emission based on solved

carrier concentrations at

boundary.

j ATn

= j BTp

Thermionic Emission Boundary Conditions

( juncn x

nF

( ) junc p x pF

( )F n j ATn

= ( )F p j BTp

=

Thermionic Emission Current must be Self Consistent for a given

A and B are Unique to Be Self Consistent!!!!!!

0n n=

0 aV ϕ ϕ = −

0 p p=0ϕ =

p side and N side Boundary Conditions

0 0 0, , , a p n V ϕ

[ ],n p

[ ], A B



Numerical Model

Initial Values of Bulk Semiconductor

0 0 A Dn N p N − ++ = +

( ) ( ) ( )0 ,n n n e

n F f F E E dE ρ ∞

−∞

= ∫ ( ) ( ) ( )0 , p p p h

p F f F E E dE ρ ∞

−∞

= ∫

0

N pF F

qϕ

−= p n

F F = P N F F =

Contact Potential

0qϕ

p-N Heterojunction Before Contact

p N

Charge Neutrality Condition

E

l e c t r o n E n e r g y (

e V )

p nF F =

For Each Bulk Material before

contact, solve the above 4 equationsto find

0 0, , , p n

F F n p

Fermi Levels Equal in a bulk

material at Thermal Equilibrium

Distance x along Heterojunction



Numerical Model

p-N Heterojunction Zero-bias solution

At Zero-bias the thermionic emission current is zero

Solve Drift Diffusion andPoisson Equations



carrier concentrations at

boundary.

Is the Thermionic Emission

zero? If not, increase the

contact potential.End

0, j j jTn p T T =

+ −

= −

Fixed Boundary Conditions

Initial Guess start withContact Potential set to zero

0ϕ

Variable Boundary

Condition

Contact Potential

0 0, p n

Highly Nonlinear Problem!!!!

Ideal Ohmic Contact Carrier Concentrations



p-GaAs/N-Al0.25Ga0.75As heterojunction with doping of N ap

=1017 cm-3, N dN

=1017 cm-3 and dimensions of

w N

=wn=0.5μm

p N

Zero Bias p-N Heterojunction

E l e c t r o n E n

e r g y ( e V )


Conduction Band

Valance Band

Fermi Energy

Discontinuous



Numerical Model

p-N Heterojunction Forward-bias solution

Beginning with zero-bias solution, increase the voltage on upper contact by Va.

Zero-Bias solution,

initial guess of

Thermionic Emission

Current and set the Bias

Voltage V A

Solve Drift Diffusion and

Poisson Equations



carrier concentrations atboundary.

Calculate Difference

between values and then

using root-finding routine,

find better guess of

Theremal emission current

End

0 aV ϕ +

Variable BoundaryCondition

Contact Potential and Bias Voltage

, j jTn Tn⎡ ⎤⎣ ⎦

Thermionic Emission

Current

Fixed Boundary Conditions

0 0, p n Ideal Ohmic Contact Carrier Concentrations



p N

Forward Bias p-N Heterojunction

Va=1.2V

p-GaAs/N-Al0.25Ga0.75As heterojunction with doping of N ap

=1017 cm-3, N dN

=1017 cm-3 and dimensions of

w N

=wn=0.5μm

E l e c t r o n E

n e r g y ( e V )


Conduction Band

Valance Band

Fermi Energy

Electrons

Discontinuous

Fermi Energy

Holes



Results

p-N Heterojunction

Current-voltage characteristic of

p-N heterojunction diode

p N

+ − AV

Solving for multiple values of V a



Numerical Model

P-p-N heterostructure

A narrow bandgap material

between two doped wider bandgap

materials

This structure is useful for laser

diodes as the narrow bandgapmaterial can be used as the active

region

The electrical characteristics,

such as level of carrier injection

and leakage currents, are useful

parameters in laser modelling

0n n=0 aV ϕ ϕ = −

0 p p=0ϕ =



Numerical Model

Forward Bias P-p-N heterostructure

p N PFixed

Ohmic

Fixed

Ohmic

Thermionic

Emission

Variable

Thermionic

Emission

Fixed

p N PFixed

Ohmic

Fixed

Ohmic

Thermionic

Emission

Variable

Thermionic

Emission

Fixed

Solve Self consistentlySolve Self consistently

Switch the Fixed

and Variable

Thermionic

Emission

Switch the Fixed

and Variable

Thermionic

Emission



p N P

Zero Bias P-p-N Heterojunction

E l e c t r o

n E n e r g y ( e V )


Conduction Band

Valance Band

Fermi Energy

Discontinuous



p

N

P

Forward Bias Va=0.5V P-p-N Heterojunction

E l e c t r o

n E n e r g y ( e V )


Conduction Band

Valance Band

Fermi EnergyElectrons

Discontinuous

Fermi Energy

Holes



Results

Test of P-p-N heterostructure model

p N

p N

P

P-p-N and p-N junctions

are identical

Large p region



Conclusions

A finite-element semiconductor heterojunction simulation

with abrupt changes in dopant concentrations has been

created, which allows us to predict device characteristics andoptimize their design.

The results for p-N, and P-p-N heterostructures have beendemonstrated and the current method can be expanded to m

layers.

Future work will focus on adding tunneling effects,

quantum wells, and integrating this carrier transport model

into a larger laser simulation.

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