introduction to electronic devices - jacobs university … diodes.pdf · the drift current is equal...

1

Introduction to Electronic Devices, Fall 2006, Dr. D. Knipp

Diodes

Source: Apple

Ref.: IBM

101110-110-210-310-410-510-610-8Critical

dimension (m)

Ref.: Apple

Ref.: Palo Alto Research Center

10-7

Introduction to Electronic Devices(Course Number 300331) Fall 2006

Diode

Dr. Dietmar Knipp

Assistant Professor of Electrical Engineering

Information:http://www.faculty.iu-bremen.de/dknipp/

2


Diodes

3 Diodes3.1 Introduction

3.2 Pn-junctions at zero bias3.2.1 Semiconductor in Thermal Equilibrium3.2.2 Pn-junctions in thermal equilibrium3.2.3 The Space Charge Region3.2.4 The Built-in voltage3.2.5 Field and Potential Distribution

3.3 Pn-junctions under bias conditions3.3.1 Diode under forward bias3.3.2 The law of the Junction3.3.3 Splitting of Quasi Fermi levels3.3.4 The Shockley model of the pn-junction

3.3.4.1 The abrupt pn-junction3.3.4.2 Carrier concentration at the boundaries3.3.4.3 Low Injection3.3.4.4 Constant current densities in the depletion region

3.3.5 Minority carrier in the neutral region of a pn-junction3.3.6 Ideal diode equation3.3.7 The asymmetric pn-junction / p+n diode

3


Diodes

3.4 Deviations from the ideal Diode3.4.1 Influence of Generation3.4.2 Avalanche Breakdown3.4.3 Influence of Recombination3.4.4 Influence of High Injection3.4.5 Series resistance and bulk region effects3.4.6 Experimental behavior under forward bias

3.5 Equivalent circuits of a pn-junction3.5.1 Series resistance3.5.2 Shunt / leakage resistance 3.5.3 Differential Resistance3.5.4 Differential Capacitance

3.5.4.1 Depletion capacitance3.5.4.2 Diffusion capacitance

3.6 Hetero junctions

3.7 Metal Semiconductor Contacts3.7.1 The Schottky Diode3.7.2 The Ohmic Contact

References

4


Diodes

3.1 IntroductionThis chapter will deal with pn-junctions, metal-semiconductor junctions (Schottky diodes) and ohmic contacts. Pn-junctions are of importance for several modern electronic devices like diodes (e.g. Optical detectors, solar cells, light emitting diodes, laser diodes), bipolar transistors, thyristor and field effect transistors. In the following the operating principle of pn-junctions will be discussed starting from the semiconductor materials. The ideal diode equation will be derived. Furthermore, an equivalent circuit for pn-junctions will be presented.The main characteristic of a pn-junction or a diode is that for positive voltages (forward bias) the diode is conduction, whereas for negative voltages (reverse bias) the current flow is blocked. Due to this properties a diode can be used for several applications like rectifiers, sampling circuits, demodulators, logic gates and so on.

5


Diodes

3.1 Introduction

A silicon pn-junction can be realized by the formation of a junction between an n-type and a p-type doped region. The junction can be formed by the implantation or diffusion of dopants in an wafer. For example a silicon pn-junction can be formed by the implantation of boron atoms in an n-type silicon wafer.

Silicon pn-junction: (left) Fabricated by the implantation or diffusion of boron atoms into an n-type (phosphorus doped) silicon wafer. (middle) Concentration of donors and acceptors in the wafer. (right) Cross section of a pn-junction.

n-type Silicon

Dopant concentration

Dep

th

Donors

Acceptorsp-type Silicon

n

p+

6


Diodes

3.2 Pn-junctions at zero bias3.2.1 Semiconductor in Thermal equilibriumIn order to understand the operating principle of a pn-junction we will start with two separate n-type and p-type semiconductors. For the p-type semiconductor the Fermi level is close to the valence band, whereas for the n-type semiconductor the Fermi level is close to the conduction band.

Separated and uniformly doped p-type and n-type semiconductors in thermal equilibrium. Ref.: M.S. Sze, Semiconductor Devices

In our case we assume that the semiconducting materials are silicon and the bandgaps for both pieces of materials are equal. In this case we speak about a homo-junction.

As the Fermi level in both pieces of silicon is constant, the semiconductors are in thermal equilibrium.

7


Diodes

3.2.2 Pn-junctions in thermal equilibriumWhen the p-type and the n-type semi-conductors are jointed together a carrier concentration gradient is observed at the interface of the n- and p- region. The gradient leads to the diffusion of carriers. The diffusion occurs because the concentration of electrons in the n-type semiconductor is much higher than the concentration of electrons in the p-type semiconductor. The opposite behavior is observed for the holes. The hole concentration in the p-type semiconductor is much higher than the hole concentration in the n-type semiconductor. Consequently, holes from the p-side diffuse towards the n-side and vice versa. Despite the flow of electrons and holes the device is still in thermal equilibrium. Ref.: M.S. Sze, Semiconductor Devices

F

Formation of a space charge region and energy band diagram of a pn-junction in thermal equilibrium.

8


Diodes

Chargeρ

SCR

-xp xn

p+ n- +

-

+ Position

x

0

Charges in the space charge region of a pn-junction at zero bias.

3.2.2 Pn-junctions in thermal equilibriumSince holes leave the p-type region the negative acceptor ions near the junction are left uncompensated and a space charge region is formed. This means that the acceptor ions are localized, whereas the holes are mobile in the valence band. The same applies for the electrons in the conduction band. The electrons diffuse towards the interface and the localized donor states are uncompensated in the p-type semiconductor. As a consequence a space charge region is formed. In this particular case free carriers get removed from the space charge region and the region gets depleted (depletion region).

9


Diodes

3.2.2 Pn-junctions in thermal equilibrium

The localized charges (acceptor and donor ions) create a potential barrier that prevents more electrons and holes from moving towards the space charge region. As a consequence of the distribution of carriers throughout the pn-junction an electric field is created that maintains drift fluxes of electrons and holes in opposite direction. Therefore, a drift current and a diffusion current is flowing.

Ref.: M.Shur Introduction to Electronic Devices

Energy band diagram of a pn-junction (silicon pn-junction) at zero bias.

10


Diodes

3.2.2 Pn-junctions in thermal equilibriumAn equilibrium is reached, when the Fermi level throughout the entire sample becomes constant. Since the semiconductor is still in thermal equilibrium (no additional excitation of carriers and no temperature gradient throughout the sample is assumed) the overall current flow through the sample is zero, meaning the current flow for each position in the pn-junction is zero. Therefore, the electron and the hole current has to be zero. This can only be achieved if the drift current is equal to the diffusion current for each spatial position throughout the device. This means that the current caused by the electric field is compensated by the current caused by the carrier gradient. This relationship can be derived from the current density equation for the electrons and the holes (Homework).

diffn

driftnn jjj −== 0 The drift current is equal to the diffusion current

for electrons in thermal equilibrium

The drift current is equal to the diffusion current for holes in thermal equilibrium

diffp

driftpp jjj −== 0

11


Diodes

3.2.2 Pn-junctions in thermal equilibriumAt zero bias the drift current (caused by the electric field) and the diffusion current (caused by the gradient of the carrier concentration) compensate each other. Therefore:

The Fermi level is constant throughout the sample independent of the spatailposition.

0==

+⋅=

dxdEnq

dxdnDFnqj

nF

nnnn µµ

0==

−⋅=

dxdEpq

dxdpDFpqj

pF

ppnp µµ

12


Diodes

D

inDn N

npNn2

, =≈

A

ipAp N

nnNp2

, =≈

Ref.: M.Shur, Introduction to Electronic Devices

3.2.3 The Space Charge region

Let’s now have a closer look at the carrier concentrations in the different regions of the device. Far away from the pn-junction the carrier concentration in the n-region is given by:

The carrier concentration in the p-region is given by:

Energy band diagram and carrier concentration for a silicon pn-junction at zero bias.

13


Diodes

3.2.3 The Space Charge regionIn our model an abrupt transition from a p-type to a n-type semi-conductor (step junction) is assumed. Further, it is assumed that the charges in the space charge region are determined by the localized ions. The influence of free charges on the charge distribution are negletable. Therefore,

Due to charge neutrality the charge per unit area in the p-region has to be equal to the charges per unit area in the n-region, so that

nDpA xNxN = Charge neutrality

Dn

n

Ap

p

qNxx

qNxx

=<<

−=<<−

ρ

ρ

0

0

Chargeρ

SCR

-xp xn

p+ n- +

-

+ Position

xSpace charge ρ=0

0

Charges in the space charge region of a pn-junction at zero bias.

14


Diodes

3.2.4 The Built-in voltage

bpbnbiV ϕϕ −=

Built-in voltage

EC

EV

EiEF0

EnergyE

SCR

Vbi

-xp xn

p+ n- +

Position

x0

Potentialϕ

ϕbn

ϕbp0

Vbi

In the next step the built-in voltage is introduced. The built-in voltage is the internal voltage of a diode as a consequence of the formation of a space charge region. The built-in voltage is the difference between the bulk potential in the n-type and the p-type regions. Further, the built-in voltage can be extracted from the bending of the bands in the space charge region.

15


Diodes

3.2.4 The Built-in voltageBy using the expression previously derived for the bulk potential,

the build-in voltage can be described as a function of the doping concentrations.

Therefore, the built-in voltage is determined by the product of the donor and acceptor concentration. The intrinsic carrier concentration can be substituted by

⋅≈

i

Dthbn n

NV lnϕ

⋅−≈

i

Athbp n

NV lnϕ

⋅≈ 2ln

i

ADthbi n

NNVV Built-in voltage

−⋅=kTE

NNn gVCi 2exp Intrinsic carrier concentration

16


Diodes

so that the built-in voltage can be described by the following equation,

where ND is the Donor and NA is the acceptor concentration, NC and NV are the effective densities of states in the conduction and the valence band. In order to increase the build-in voltage the doping concentration can be increased. This is for example of interest for specific applications like solar cells.

⋅+≈

VC

ADth

gbi NN

NNVqE

V ln

3.2.4 The Built-in voltage


Influence of the acceptor concentration on the build-in potential of a pn-junction.

Built-in voltage

17


Diodes

np xxandxx >−<

ρεε

ϕ

rdxd

dxdF

02

2 1=−=

3.2.5 Field and Potential DistributionThe diode is still unbiased. In the next step the Poisson equation will be used to calculate the electric field and the potential distribution for a silicon diode in thermal equilibrium. The Poisson equation is given by

Let’s start with the semiconductor material outside of the space charge region. The charges outside of the space charge region are zero, ρ=0, so that the electric field distribution is zero

and subsequently the potential distribution for this region of the device results to

Within the space charge region the space charge density is given by

Poisson Equation

0== pn FF

0=ϕ

for

np xxandxx >−<for

( )AD NNnpq −+−=ρ np xxx <<−for

18


Diodes

3.2.5 Field and Potential DistributionThe space charge density for the p-type and the n-type regions of the depletion region is given by

It is assumed that the pn-junction is an abrupt junction, so that the acceptor and donor concentrations are constant throughout the p-region and the n-region of the space charge region. This approximation is called depletion approximation. Based on the depletion approximation the electric field can be calculated by

Here, x=0 corresponds to the boundary between the p-type and the n-type region and xp and xn are the depletion widths on both sides of the junction.

Space charge density

<<

<<−−=

nS

D

pS

A

xxforqN

xxforqN

dxdF

0

0

ε

ε

<<<<−−

=nD

pA

xxforqNxxforqN

00

ρ

19


Diodes

<<

−⋅−

<<−

+⋅−

=

nn

m

pp

m

xxforxxF

xxforxxF

F01

01

3.2.5 Field and Potential DistributionAfter the integration the electric field is given by

where Fm is the maximum electric field. Further, it is assumed that F(x=-xp)=0 and F(x=xn)=0. The maximum electric field is observed for Fm=F(x=0), where

so that electric field distribution results to

Electric field distribution

S

pA

S

nDm

xqNxqNFεε

==

( )

( )

<<−⋅−

<<−+⋅−=

nnS

D

ppS

A

xxforxxqN

xxforxxqN

F0

0

ε

ε Electric field distribution

20


Diodes

3.2.5 Field and Potential DistributionThe potential distribution is obtained by the integration of the electric field.

where ϕ(-xp)=Vbi and ϕ(xn)=0. Due to charge neutrality, xpNA=xnND, the built-in voltage can be calculated by

( ) ( )

( ) ( )

<<−

−

<<−+

+−−=

nS

nDn

pS

pAp

xxforxxqNx

xxforxxqN

x

02

02

2

2

εϕ

εϕ

ϕ

S

pA

S

nDbi

xqNxqNVεε 22

22

+=

Potential distribution

21


Diodes

3.2.5 Field and Potential DistributionCombining the expression for the charge neutrality with the built-in voltage leads to the following terms for the width of the depletion region:

The overall width of the depletion region is given by:

Again, it is obvious that these equations only apply for zero bias. Otherwise the width of the depletion region has to be a function of the applied bias.

DAA

Dbi

Sp NNN

NVq

x+

⋅⋅⋅=12ε

DAD

Abi

Sn NNN

NVq

x+

⋅⋅⋅=12ε

pn xxw +=

DA

DAbi

S

NNNNV

qw

⋅+

⋅⋅=ε2

Width of the space charge region in the p-type semiconductor

Width of the space charge region in the n-type semiconductor

Width of the overall space charge region

22


Diodes

ρ=0

ρn

ρp

-xP xn

-xP xn

-xP xn

3.2.5 Field and Potential DistributionSummary of the Field and Potential Distribution

<<<<−−

=nD

pA

xxforqNxxforqN

00

ρ

Space charge density

( )

( )

<<−⋅−

<<−+⋅−=

nnS

D

ppS

A

xxforxxqN

xxforxxqN

F0

0

ε

ε

Electric field distribution

( )

( )

<<−

<<−+

+−=

nS

nD

pS

pAbi

xxforxxqN

xxforxxqN

V

02

02

2

2

ε

εϕ

Potential distribution

23


Diodes

3.3 Pn-junctions under bias conditionsWhat happens if we apply a voltage to a pn-junction? Before deriving a model for a pn–junction under bias voltages the phenomenological behavior of the diodes will be discussed. Under positive bias the diode operates in forward operation. The build-in voltage of the diode is lowered and the width of the space charge region is reduced. The opposite behavior is obtained for reverse bias. Here, the internal electric field distribution is increased and the space charge region is widened.

Ref.: M.S. Sze, Semiconductor Devices

Forward and reverse biasing of a pn junction: A positive (negative) voltage is applied to the pn-junction which decreases (increases) the potential barrier between the n- and the r-region.

Forward bias

Reversebias

24


Diodes

3.3.1 Diode under forward biasDue to the reduced electric field under forward bias the depletion region is getting smaller and the diffusion and drift currents are reduced. The total current is not zero anymore. The diffusion current is not completely compensated by the drift current. The diffusion current is getting the dominate current. Under non-thermal equilibrium more electrons flow from the n-region to the p-region and more holes flow from the p-region to the n-region. As a consequence a net current flow can be observed, which is a diffusion current. There is an injection of electrons in the p-region and an injection of holes in the n-region. The carrier concentration in the space charge region is not in thermal equilibrium anymore. The quasi Fermi levels start to split up.

Depletion layer width and energy diagram of a pn-junction under two different biasing conditions. (left) Thermal equilibrium, (right) Forward bias.


25


Diodes

3.3.2 The law of the JunctionAn external voltage violates therium balance between the drift and the diffusion current in thermal equilibrium. However, for small currents the net current is still much smaller than the drift current and the diffusion current in the depletion region. Therefore, the carrier concentration is still close to the equilibrium state and the electron and hole densities can be described by the quasi Fermi levels. Under forward bias the quasi Fermi levels shift towards the bands (injection of carriers). Under such conditions the carrier concentration is given by

Under thermal equilibrium the quasi Fermi levels for electrons and holes are identical, so that the Fermi potential is identical for electrons and holes. Under non-equilibrium conditions EF(n) is not equal to EF(p) and both might be a function of the position and the time.

( )

−⋅=

kTEENn C

nF

C exp Quasi electron concentration

( )

−⋅=

kTEENppFV

V exp Quasi hole concentration

26


Diodes

3.3.2 The law of the JunctionThe splitting of the quasi Fermi levels can be expressed in terms of the applied bias.

Using the pn product

The following expression is calculated for the product of electrons and holes under quasi-thermal equilibrium

pF

nF EEqV −=

( ) ( )

−⋅=⋅

kTEEnnp

pF

nF

i exp2

Law of the junction

⋅=⋅kTqVnnp i exp

2


Energy band diagram for a silicon pn-junction under zero bias (top) and forward bias (bottom). –xp and xn definethe boundary conditions between the depletion and the neutral region. EFnand EFp are nearly constant within the depletion region.

27


Diodes


3.3.2 The law of the Junction The splitting of the Fermi level occurs not only in the depletion region of the diode. The splitting exceeds into the n-and the p-region of the pn-junction.

As a consequence of the splitting of the Fermi levels the concentration of carriers in the band is changed. The carrier concentration is given by the law of the junction.

Energy band diagram for a silicon pn-junction under forward bias (bottom). Note that the length scale for the two band diagrams is different.

−⋅=

kTEEnnpF

nF

pp exp0

28


Diodes

3.3.3 Splitting of Quasi Fermi levelsThe majority carrier concentration in the p- and the n-region is given by:

The minority carrier concentration at the boundary x=-xp is given by

⋅=

⋅=

kTqV

Nn

kTqVnn

A

i

pp

exp

exp

2

0

Energy band diagram and carrier concentration for a forward and reverse biased diode.Ref.: M.S. Sze, Semiconductor Devices

App Npp ≈≈ 0

Dnn Nnn ≈≈ 0

29


Diodes

3.3.3 Splitting of Quasi Fermi levelsand the minority carrier concentration at the boundary x=xn is given by

Under forward bias conditions V>0, so that the minority carrier concentration is higher than the minority carrier concentration in thermal equilibrium. Under reverse bias conditions V<0, so that the minority carrier concentration is lowerthan the minority carrier concentration in thermal equilibrium.

⋅=

⋅=

kTqV

Nn

kTqVpp

D

inn expexp

2

0

30


Diodes

3.3.4 The Shockley model of the pn-junctionIn order to derive an analytical model for a pn-junction the following assumptions are introduced in order to simplify the description:

(1) An abrupt pn-junction is assumed. The semiconductor is neutral outside of the space charge region.

(2) The carrier densities at the boundaries are related to the potential distribution.

(3) Low injection conditions are assumed, which means that the injected minority carrier density is small in comparison to the majority carrier density. The changes of the majority carrier concentration at the boundaries of the neutral regions can be ignored.

(4) Neither generation nor recombination takes place in the depletion region and the electron and hole current densities are constant throughout the depletion region.

31


Diodes

3.3.4 The Shockley model of the pn-junctionThe assumptions of the Shockley model are illustrated in the figure.


32


Diodes

3.3.4.1 The abrupt pn-junctionMost of the applied voltage V will drop across the depletion region, because the number of carriers, which contribute to the current flow, is lowest in the depletion region. The remaining voltage will drop across the neutral region. The voltage drop across the neutral regions will lead to a band bending in the neutral regions.

As the majority carrier concentrations in the neutral regions are very high, it can be assumed that the voltage drop across the neutral regions is small in comparison to the voltage drop across the depletion region.

For forward bias the applied voltage Vf is opposing the built-in voltage of the diode. The potential across the space charge region is lowered by the applied voltage. Therefore, the voltage drop across the space charge region of the pn-junction in forward bias is given by VSCR=Vbi-Vf. In thermal equilibrium it is VSCR=Vbi.

For reverse bias the applied voltage Vr the potential across the space charge region is increased by the applied voltage. Therefore, the voltage drop across the space charge region of the pn-junction in reverse bias is given by VSCR=Vbi+Vr.

33


Diodes

3.3.4.2 Carrier concentration at the boundariesThe carrier concentration at the boundary is given by the previously derived equations. The important equations are here the expressions for the minority carrier concentration.

The minority carrier concentration at the boundary x=-xp is given by

The minority carrier concentration at the boundary x=xn is given by

Therefore, the minority carrier concentration under bias (non-thermal equilibrium) is expressed in terms of the carrier concentration under thermal equilibrium.

⋅=kTqVnn pp exp0

⋅=kTqVpp nn exp0

34


Diodes

3.3.4.3 Low InjectionIn the following low injection is assumed. Under such conditions the quasi Fermi levels will be very close to the Fermi level under thermal equilibrium. Based on the assumption of quasi charge neutrality we introduced the quasi Fermi levels.

Low injection assumes now that the change of the minority carrier concentration is smaller than the majority carrier concentration in thermal equilibrium. As a consequence Fermi level for the majority carriers is assumed to be identical with the Fermi level under thermal equilibrium.

Under the assumption of high injection the concentration of minority carriers is getting that larger, so that the Fermi level of the majority carriers can not be described by the Fermi level in thermal equilibrium anymore.

00 nnnn ppnn −≈− Quasi Charge neutrality

FnFnn EEnp ≈<∆ 0

Low injection

pn ∆≈∆

0nn np >∆ High injection

35


Diodes

3.3.4.4 Constant current densities in the depletion regionFurthermore, Shockley assumed in his model of the pn-junction that generation and recombination of carriers in the space charge region can be ignored. As a consequence the electron and hole current densities are constant in the depletion region. This assumption is reasonable if the diffusion length is much larger than the width of the depletion region. As a consequence the current densities throughout the device can be described by the current densities at the boundaries between the neutral region and the space charge region.

36


Diodes

3.3.4.5 Minority carriers in the neutral region of a pn-junctionShockley assumed in his model (4. Shockley assumption) that the current density in the space charge region is constant. As a consequence the current densities in the space charge region can be described by the current densities at the boundary between the neutral region and the space charge region.

In a first step the carrier concentration in the neutral regions will be calculated. The carrier concentration can be calculated by solving the continuity equation.

The continuity equation can be simplified by assuming the diffusion current to be the dominant current contribution. The drift current plays a minor role. Therefore, it can be assumed that the contributions of the applied electric field is ignored. Furthermore, it can be assumed that the generation rate G is zero, so that the equation results to.

02

2

=−+∂∂

+∂∂

+∂∂ RG

xpD

xpF

xFp n

nn

nnn µµ

02

2

=−∂∂ RxpD n

n

Continuity equation for holes

37


Diodes

3.3.5 Minority carriers in the neutral region of a pn-junctionTo solve the differential equation we need a description of the recombination rate. In the following we will assume band-to-band recombination, which is the dominate recombination mechanism for indirect semiconductors like silicon. In general band-to-band recombination is described by the following recombination equation.

The recombination rate is determined by the minority carrier concentrations, which in our case is given by the concentration of the holes in an n-type semiconductor. If holes are minority carriers (n=np<< p ≈ NA=pp, where NAis the concentration of shallow acceptors (complete ionization)) the recombination rate is given by:

The parameter tpl is the hole lifetime. The resulting equation is called the Diffusion equation.

Recombination equation

pl

nn ppRτ

0−= Recombination rate

( ) ( )002

pnnni pnpnAnnpAR −⋅=−⋅=

38


Diodes

3.3.5 Minority carriers in the neutral region of a pn-junctionThe diffusion equation is usually stated for the minority carrier concentration.

The Diffusion equation can be solved by using the following general solution.

To simply the equation the Diffusion Length for minority carriers is introduced.

002

2

=−

−∂∂

pl

nnnp

ppxpD

τDiffusion equation for holes (minority carriers)

002

2

=−

−∂

∂

nl

pppn

nnxn

Dτ

Diffusion equation for electrons (minority carriers)

( )

−−⋅+

−⋅=−

plp

n

plp

nnn D

xxBDxxApxp

ττexpexp0

39


Diodes

3.3.5 Minority carriers in the neutral region of a pn-junction

So that the general solution of the Diffusion equation results to

The following boundary conditions can be used to determine the constants A and B:

plpp DL τ=

nlnn DL τ=

Hole diffusion length

Electron diffusion length

( )

−−⋅+

−⋅=−

p

n

p

nnn L

xxBLxxApxp expexp0

( )

⋅==kTqVpxxp nnn exp0

( ) 0nn pxp =∞→

40


Diodes

3.3.5 Minority carriers in the neutral region of a pn-junctionThe following solution can be obtained by considering the boundary conditions:

( )

−−⋅

−

⋅=−

p

nnnn L

xxkTqVppxp exp1exp00

The solution states that the minority carrier concentration depends on the applied bias voltage and the distance from the boundary. The applied bias voltage directly controls the minority carrier concentration at the boundary between the neutral region and the space charge region.

Car

rier C

once

ntra

tion

(cm

-3)

Electron (thin line) and hole (thick line) concentration in a silicon pn-junction for zero bias (solid lines ) and forward bias (dashed lines). Ref.: M.Shur, Introduction to Electronic Devices

41


Diodes

3.3.5 Minority carriers in the neutral region of a pn-junctionThe variation of the carrier concentration is not noticeable on the scale of the space charge region. The carrier concentrations decays on a scale determined by the diffusion length.

3.3.6 Ideal diode equationIn the next step the current density for the holes in the neutral region is calculated by

The drift contribution can be neglect due to the dominance of the diffusion current. Accordingly the following expression is obtained:

−−⋅

−

=−=≈

P

n

p

npnppDp L

xxkTqV

LpqD

dxdpqDjj exp1exp0

dxdpqDpFqj ppp ⋅−= µ Current density for holes

42


Diodes

3.3.6 Ideal diode equationThe 3. Shockley assumption can be used to calculate the overall current follow in the neutral region. If a quasi neutral semiconductor is assumed the quasi charge neutrality condition applies:

Quasi charge neutrality leads to the following relationship between the gradients of the minority carrier concentration.

By using this simple relationship the electron diffusion current can be correlated with the hole diffusion current.

So that the total diffusion current jD in the n-type neutral region becomes

pn ∆≈∆ Quasi Charge neutrality

xp

xn np

∂∂

≈∂∂

and 2

2

2

2

xp

xn np

∂∂

≈∂

∂

pDp

nnp

nnnD j

DD

xpqD

xnqDj −=

∂∂

≈∂∂

=

( )

−−⋅

−

⋅−⋅−≈+=

p

nnp

p

nnDpDD L

xxkTqVDD

Lqpjjj exp1exp0

43


Diodes

3.3.6 Ideal diode equationThe total current density in the neutral region is equal to the sum of the electron diffusion current density and the hole diffusion current density.

We still get a function that varies with the distance from the boundary due to recombination. However, the overall current of a diode is independent of the position. Furthermore, the current density scales with the electron and the hole diffusion length. The diffusion length is typically much larger than the width of the depletion region

Therefore, the hole and the electron diffusion current densities remain nearly constant throughout the depletion region. The total current density is given by

where jpD and jnD are the hole and the electron diffusion current densities at the boundaries.

pnn xxL +>> pnp xxL +>>

pn xxnDxxpD jjj−==

+=

44


Diodes

Injected minority carrier distribution and current density distribution for a forward biased (a) and a reverse biased diode (b). The figures are drawn based on the Shockley assumptions. In reality the current density is not constant throughout the depletion region.

3.3.6 Ideal diode equation


45


Diodes

3.3.6 Ideal diode equationThe current is finally given by

The equation can be rewritten in the form:

Leading to the Shockley / ideal diode equation

Ideal diode equation / Shockley equation

−

⋅

+= 1exp00

kTqV

LnqD

LpqD

jn

pn

p

np

−

⋅

+⋅= 1exp

22

thAn

in

Dp

ip

VV

NLnD

NLnD

qj

−

⋅

+⋅= 1exp2

thAn

n

Dp

pi V

VNLD

NLD

qAnI

46


Diodes

−

⋅= 1exp

thS V

VII

+⋅=

An

n

Dp

piS NL

DNLD

qAnI 2

Ideal diode equation / Shockley equation

Saturation current


3.3.6 Ideal diode equationThe current is finally given by

Current voltage curve using a linear (a) and a semilog (b) plot.

47


Diodes

3.3.6 Ideal diode equation

Adding

Equations width of the space charge region

Carrier, electric field and potential distribution under biasing conditions

48


Diodes

Short pn-junction with a highly doped p-region.

3.3.7 Asymmetry pn-junction / p+n diodeIn the case of a asymmetric pn-junction the analytical description of the diode has to be modified. The example of a short p+-n diode will be discussed in the following. P+ indicates that the doping concentration in the p-region is much higher than the doping concentration in the n-region.

Furthermore, the overall diode is very short and the width of the n-type depletion region is limited by the extension of the n-region. The overall width of the n-region is given by Xn. The description of a short pn-diode is important for the analysis of the bipolar transistor.

n-type Siliconp-type Silicon

Insulator

Xn

49


Diodes

3.3.7 Asymmetry pn-junction / p+n diodeIn such a case the solution of the diffusion equation is given by

and the hole concentration in the n-type region can be approximated by a linear function of x. The overall current is given by

The electron diffusion current can be ignored, because the hole concentration is much higher due to the highly doped p-region.

( )

−−

⋅

−

⋅≈−

nn

nnnn xX

xXkTqVppxp 1exp00

−

⋅

−=

∂∂

≈=

1exp0

kTqV

xXpqD

xpqDj

nn

np

xx

np

n

50


Diodes

3.4 Deviations from the ideal DiodeIn many cases the ideal diode equation does not accurately describe the current-voltage characteristic of a pn-junction diode in forward and reverse bias. The deviations from the ideal diode equation will be discussed in the following. The reasons for the deviations are the following:

Reverse Bias:

•Generation

•Avalanche Breakdown

Forward bias:

•Recombination

•High injection

•Series resistance and bulk region effects Comparison of the ideal diode equation with a

non-ideal model of a diode.Ref.: T. Fjeldly, T. Ytterdal, M. S. Shur, Introduction to device modeling and circuit simulation

51


Diodes

3.4.1 Influence of GenerationThe generation current becomes important under reverse bias conditions. In such a case the carriers in the depletion region are extracted due to the applied reverse electric field. The carrier concentration can be described by:

In thermal equilibrium the thermal generation of electron-hole pairs is balanced by the thermal recombination (G=R) of carriers. Under reverse bias, the concentration of holes and electrons in the depletion region is low. The recombination rate is close to zero (R≈0). Due to thermal activation electron-hole pairs are continuously created.

The generation rate can be described by

where τgen is the effective generation time of an electron-hole pair in the depletion region.

1exp2 <<

−=

⋅kTV

qnnp

i

gen

ithermal

nGτ

= Generation rate

52


Diodes

3.4.1 Influence of Generation

As the saturation velocity of the carriers is in the order of 105 m/s and the width of the depletion region is in the order of a few microns, the transit time of the carriers under reverse bias is in the order of 10ps.

The generation time varies between ms and ns, so that generationtime is the limiting the generation current and not the transit time.

The width of the depletion region under bias can be calculated by:

( )( )ADD

bisn NNqN

VVx+

−=

12ε ( )

( )ADA

bisp NNqN

VVx+

−=

12ε

53


Diodes

3.4.1 Influence of Generation The generation current in the depletion region is given by

where w is the width of the depletion region w=xp+xn.

The generation current scales with the intrinsic carrier concentration, which depends on the bandgap of the material. As a consequence the generation current for a silicon diode is much higher than the generation current of a GaAs diode.

gen

iGen

nwqjτ⋅⋅=

Generation current

Generation current (per µm3) versus the energy gap.

Gen

erat

ion

curr

ent (

A)

Energy Gap (eV)


54


Diodes

3.4.1 Influence of Generation The charge is swept out of the depletion region by the electric field. The time to sweep out the generated electrons and holes is given by

where vs is the saturation velocity of the carriers. As the saturation velocity of the carriers is in the order of 105 m/s and the width of the depletion region is in the order of a few microns, the transit time of the carriers under reverse bias is in the order of 10ps. The generation time varies between ms and ns, so that generation time is the limiting the generation current and not the transit time.

The reverse current of the diode is calculated by:

The reverse current is the sum of the saturation and generation.

str v

wt ≈ Transit time

( )genSgensR jjAIII +⋅=+= Reverse current

55


Diodes

3.4.2 Avalanche BreakdownWhen a high reverse voltage is applied to the diode, the junction breaks and conducts a very large current. Two mechanism lead to the breakdown: Tunneling and the avalanche multiplication. Here we will concentrate only on avalanche multiplication. Avalanche breakdown is caused by impact ionization of electron-hole pairs. When applying a high electric field to the diode, carriers gain kinetic energy and generate additional electron-hole pairs through impact ionization.


Energy band diagrams under junction-breakdown conditions. (a) Tunneling effect (b) Avalanche multiplication.

56


Diodes

3.4.2 Avalanche BreakdownThe number of electron/hole pairs generated by an electron per unit distance is called the ionization rate. The ionization rate, α, is defined for electrons and holes. The ionization rate for the electrons and holes can be experimentally determined.


Measured ionization rate for silicon and GaAs.

57


Diodes

3.4.2 Avalanche BreakdownIn the next step the break down condition has to be derived for a pn-junction. The break down is observed, if the multiplication of carriers in the space charge region due to ionization reaches infinity. Accordingly a current can be defined for which the break down is observed.

Based on this equation for the multiplication current the condition for breakdown can be derived.

( )dxqI

qId n

nn α⋅

=

0

Depletion region in a p-n junction with multiplication

of an incident current.

10

=∫w

dxα

Multiplication currrent

Breakdown condition


58


Diodes

3.4.2 Avalanche BreakdownBased on the breakdown condition and the electric field dependence of the ionization rate the critical electric field can be calculated for which the diode breaks down. The break down voltage can be determined by solving the Poisson’s equation.

Tunneling of charges is observed for high doping concentrations.

B

CSCB qN

FwFV22

2ε==

Critical electric field at breakdown as a function of the doping concentration for silicon and GaAs (one-sided abrupt junctions).


Breakdown voltage

59


Diodes

3.4.3 Influence of RecombinationUnder forward bias conditions (V>0) excess carriers are injected in the depletion region, where a certain fraction recombines. The recombination current is equal to the total charge per unit area recombining in the depletion region per second. If recombination via impurities or traps near the middle of the bandgap is assumed the recombination current can be described by

In reality traps levels may not only exist in the middle of the bandgap, which leads to the following empirical equation:

So that the overall forward current can be described by:

=kTqVjj recsrec 2

exp Recombination current

=

kTmqVjjr

recsrec exp Recombination current

⋅+

⋅=

kTmqVj

kTqVjj

rrecssF expexp Forward bias current

60


Diodes

3.4.3 Influence of RecombinationThe diffusion current is proportional to ni

2, whereas the recombination current is proportional to ni, so that the recombination current is getting more important for large bandgap materials (GaAs, GaN, SiC), where ni is getting small.For large forward voltages the diffusion current dominates the overall current. Instead of using the two terms to described the current of a pn-diode in forward direction the equation can be simplified by

where Is_eff is the effective saturation current and n is the ideality factor. Therefore, the ideality factor varies between 1 and 2 depending on the material properties and the applied bias voltage. If the ideality factor shifts towards 2 for higher applied voltages we can conclude that the properties materials are reduced (materials with higher defect density).

⋅=nkTqVII effsF exp_ (Empirical) Forward bias current

61


Diodes

3.4.4 Influence of High Injection

At high forward bias (the voltage drop across the space charge region is getting close to the build-in voltage) the injected minority carrier concentration becomes comparable to the majority carrier concentration. As a consequence the assumption of low injection is not valid any more. This means that the majority carrier concentration at the boundary under high forward bias is not equal to the majority carrier concentration in thermal equilibrium and/or low injection. To derive an expression for the current we can start with charge neutrality:

Carrier concentration at low-level injection (a) and high-level injection (b).

Ref.: G.W. Neudeck, The pn Junction diode

AnDn NnNp +=+ Charge neutrality

AnnDnn NnnNpp ++∆=++∆ 00Charge neutrality

62


Diodes

3.4.4 Influence of High InjectionFor low injection the following expressions can be derived:

For high injection the following expressions can be derived:

( )

⋅=+⋅=kTqVnpNpnp inDnnn exp2

0nn np >∆

0nn np <∆ Low injection

⋅≈kTqVnp in 2

exp

⋅=⋅=kTqVnNpnp iDnnn exp2

⋅≈kTqV

NnpD

in exp

2

∝kTqVI exp

∝kTqVI2

exp

Diode under forward bias.


High injection

63


Diodes

3.4.5 Series resistance and bulk region effectsThe ideal diode model assumes that the voltage drop in the neutral (bulk) regions is ignored. Furthermore, the voltage drops across the metal semiconductor interfaces are neglected. This is a very good assumption for low levels of current. For higher levels of current these assumptions do not apply anymore. Under high forward bias conditions the voltage drop across the neutral region has to be considered. Also, the metal semiconductor interface leads to a small additional voltage drop.

Both effects can be combined in a single series resistance. The I/V curve of a diode can be described by:

−

−⋅= 1exp

th

SS V

IRVII

64


Diodes

3.4.5 Series resistance and bulk region effects

For small series resistance the diode behaves like an ideal diode.

Influence of the series resistance on the I/V curve under forward bias.


65


Diodes

3.4.6 Experimental behavior under forward bias

Experimental forward bias currents of silicon and GaAs pn-diodes are shown in the graph. For lower bias voltages the recombination current is dominant for both diodes, whereas for intermediate bias levels the experimental results follow the ideal diode equation. The diode current is a pure diffusion current. For higher bias voltages the overall current of the diode is determined by high injection and the series resistance.

As expected the non-ideal behavior is more pronounced for the GaAs diode, because the intrinsic carrier concentration is lower due to the higher bandgap.


Silicon and GaAs diode under forward bias.

66


Diodes

3.5 Equivalent circuits of a pn-junctionSo far we tried to describe the pn-junction diode by deriving the diode equations under ideal and non-ideal conditions. This description is essential to understand the electronic transport of pn-junctions. In order to implement the diode as part of a complex electronic circuit the diode has to be described in a different way. In order to implement the diode in a circuit simulator like SPICE the diode has to be described in the form of an equivalent circuit, which consists of current and voltage sources, resistors and capacitors. The diode can be described by the given equivalent circuit, which consists of the following components:

Small signal equivalent circuit of a pn diode.

Internal diode

Rshunt

I(V)

Rd(V)

Rseries

+ -

Cd(V)

• Series Resistance

• Shunt resistance

• Differential resistance

• Differential capacitance

67


Diodes

3.5.1 Series resistanceThe series resistance is caused by a voltage drop across the neutral regions of the diode plus the voltage drop cross the contacts and the interconnects. See as a reference section 3.4.7.2. The series resistance only influences the admittance or impedance of a diode under high forward voltages. The series resistance can be extracted from the following equation:

3.5.2 Shunt / leakage resistance For practical devices the parallel leakage plays an important role. The leakage or shunt resistance can be extracted by

The leakage current is caused by the generation current and leakage effects. The shunt resistance influences the I/V characteristic under reverse bias.

shuntleakage R

VI = Leakage current

shunt

s

th

SS R

IRVVIRVII −

+

−

−⋅= 1exp Diode current

68


Diodes

3.5.3 Differential resistanceIn the next step the differential resistance/admittance can be calculated. In general the differential resistance is given by:

Under reverse bias the differential admittance is

Under forward bias the differential admittance is

leading to

===

thth

SFd

d VV

VI

dVdIG

Rexp1

Differential admittance under reverse bias 0≈dG

dVdIG

R dd

==1

Differential resistance

th

Fd V

IG = Differential admittance under forward bias

69


Diodes

3.5.4 Differential CapacitanceThe differential capacitance of a diode can be divided in two different contributions: A depletion capacitance and a diffusion capacitance

The depletion capacitance is formed by the majority charges in the pn junction. The depletion capacitance can be treated as a plate capacitor, where the distance between the plates corresponds to the width of the depletion region. Only majority carrier contribute to the depletion capacitance.

The second contribution is caused by the diffusion of carriers. The diffusion capacitance caused only by the minority carriers. A diffusion capacitance is observed only for positive voltages, when the minority carriers in the space charge region exceeds the concentration for thermal equilibrium. In the following expressions for the depletion and the diffusion capacitors will be derived.

70


Diodes

3.5.4.1 Depletion capacitanceThe depletion capacitance of a diode is caused by the majority carriers, which form the space charge region. The depletion capacitance can be treated as a capacitance caused by a plate capacitor. The capacitance can be calculated by

where A is area of the diode, w is the width of the depletion region and εs is the dielectric constant of the semiconductor. The width of the depletion region is the sum of the depletion region in the n- and the p-region.

np

ssdep xx

Aw

AC+

⋅=⋅=εε Depletion capacitance of a

symmetric and abrupt pn-junction

Schematic sketch of a pn-junction including the distribution of charges and the equivalent depletion capacitor.


71


Diodes

3.5.4.1 Depletion capacitanceThe width of the depletion region can be calculated by

( )DAA

Dbi

Sn NNN

NVVq

x+

⋅⋅−⋅=12ε

( )DAD

Abi

Sp NNN

NVVq

x+

⋅⋅−⋅=12ε

pn xxw +=

( )DA

DAbi

S

NNNNVV

qw

⋅+

⋅−⋅=ε2

Width of the space charge region in the n-type region

Width of the space charge region in the p-type region

Width of the overall space charge region

72


Diodes

( ) ( )VVNNNNqA

dVdQC

biDA

DAsddep −⋅+⋅

==2

ε Depletion capacitance

3.5.4.1 Depletion capacitanceLeading to the following expression of the depletion capacitance

The depletion capacitance is usually written in the following form

where Cj is the junction capacitor, which is independent of the applied voltage. Therefore, the depletion capacitance of an abrupt pn-junction is proportional to

( ) bi

j

bibiDA

DAsdep VV

CVVVNN

NNqAC−

=−

⋅⋅+

=11

12

ε

biVV−11

73


Diodes

3.5.4.1 Depletion capacitanceFor large variations of the doping profile the voltage dependence of the depletion capacitance has to be considered, which leads to the following empirical equation of the depletion capacitance, where m is the grading coefficient.

( )mbi

jdep VV

CC

−=1

0 Depletion capacitance

Calculated depletion capacitance of a silicon diode.


74


Diodes

3.5.4.2 Diffusion capacitanceThe diffusion capacitance can be calculated by solving the Diffusion equation in a complex form (will not be discussed here). Here we will concentrate on a short p+n-diode (much simpler description of the diffusion capacitance). The diffusion capacitance is caused by an increase of the minority carrier concentration in the neutral and the depletion region. The increase of the minority carrier concentration is caused by the applied forward bias.

The diffusion capacitance in combination with the differential resistance can be treated as an equivalent RC circuit. The corresponding time constant is in this case the recombination time.

Formation of the diffusion capacitance in the neutral region under forward bias voltage.

Ref.: M. Shur, Introduction to Electronic Devices

Depletion Region

75


Diodes

3.5.4.2 Diffusion capacitanceThe RC time constant is given by

where the factor ½ correspond to the ideality factor in the empirical diode equation.

If the transit time of a carrier is much smaller than the recombination time (ttr << τpl), the recombination time can be expressed in terms of the transit time for holes.

This leads to the following expression for the diffusion capacitance

( )tr

p

nn

p

ppl t

DxX

DL

222

=−

⇒=τ

trth

Ftrddif tVItGC ==

Diffusion capacitancepldd

pldif G

RC τ

τ21

21

=⋅=

Diffusion capacitance

76


Diodes

3.5.4.2 Diffusion CapacitanceBased on the equation for the diffusion capacitance it would be obvious to conclude that the diffusion capacitance increases as the forward diode current increases. However, for high forward currents the diffusion capacitance is limited by the maximum diffusion capacitance. The maximum diffusion capacitance can be calculated by

where LDn and LDp are the Debye lengths for holes and electrons. The Debye length defines the minimum width of the depletion region.

DpDn

sd LL

AC+

=ε

max_

D

thsDn qN

VL ε=

Maximum Diffusion capacitance

A

thsDp qN

VL ε=

D

bisn qN

Vx ε2=

A

bisp qN

Vx ε2=

Debye length

Width of the depletion region in thermal equilibrium

77


Diodes

3.5.4 Differential capacitanceThe differential capacitance of a pn junction can now be described in the form

The differential capacitance of a diode is given in the graph.

( ) max_111

ddifdepd CCCC

++= Differential capacitance

Depletion (Cdep), diffusion (Cdif), maximum diffusion (Cd_max) and differential capacitance (Cd).


78


Diodes

3.5.4 Differential capacitanceThe maximum of the differential capacitance is observed under forward bias conditions. Under reverse bias conditions the differential capacitance is determined by the depletion capacitance, which is more or less constant. Under forward bias the depletion capacitance gets larger, because the width of the depletion region is reduced, so that the differential capacitance is increased. The diffusion capacitance is zero for reverse bias. Only under forward bias the diffusion capacitance contributes to the differential capacitance. The diffusion capacitance increases as the minority carrier concentration increases in the neutral regions. The differential capacitance of the diode is limited by a maximum diffusion capacitance. The maximum diffusion capacitance is determined the Debye length for the electrons and holes. The Debye length defines a lower limit for the width of the depletion region.

79


Diodes

3.6 Hetero junctionsSo far we discussed the behavior of homo junctions. A pn junction is a homo junctions, if the bandgap for the n-types and the p-type semiconductors is the same. The situation for a hetero junction is different. Here two semiconductor of different bandgap form a junction. Heterojunctions occur in different areas because of - Technological reasons- Electrical properties

The electric field in the space charge region is increased.Solar cells: Higher open circuit voltage and better extraction of carriersHigh mobility transistors: Faster transistors

- Optical propertiesThe layers with the higher bandgap can be used as an „optical window“.

A heterojunction is defined as a junction between two dissimilar semiconductors.The two semiconductors are assumed to have a different energy bandgap Eg. Furthermore, the two different semiconductors will have different dielectricpermittivity εs, different work function qΦs, and a different electron affinity qχ.

80


Diodes

3.6 Hetero junctionsThe work function qΦs is defined as as the energy required to remove an electron from the Fermi level EF to a position outside the material (vacuum level).

The electron affinity qχ is the energy required to remove an electron from the bottom of the conduction band to the vacuum level.

Energy diagram of two isolated semiconductors

( )12 χχ −=∆ qEC

( )Cg

ggV

EE

qEqEE

∆−∆=

+−+=∆ 2211 χχ

M.S. Sze, Semiconductor Devices

81


Diodes

3.6 Hetero junctionsIf the two semiconductors are joint the following band diagram is formed. We assume that the lattice constant of the two materials is matched. Otherwise, the number of traps at the interface is not neglectable.

Energy diagram of two isolated semiconductors

( )12 χχ −=∆ qEC

( )Cg

ggV

EE

qEqEE

∆−∆=

+−+=∆ 2211 χχ

M.S. Sze, Semiconductor Devices

82


Diodes

3.6 Hetero junctionsThere are two basic requirements in the construction of the energy

band diagram:

(a) the Fermi level must be the same on both sides of the interface in thermal equilibrium,

(b) the vacuum level must be continuous and parallel to the band edges.

The discontinuity is unaffected by the doping level as long as the affinities and the bandgap are not a function of the doping level.

Vb1 and Vb2 are the electrostatic potentials at equilibrium.

Assumption: Depletion approximation: ε1F1= ε2F2 and F1(x=0)=F2(x=0).

21 bbbi VVV += Built-in Voltage

83


Diodes

3.6 Hetero junctions

N1 and N2 are the doping concentrations in the semiconductor region 1 andregion 2. The width of the depletion region can be calculated by

( )2211

221 NN

VVNV bib εε

ε+

−=

Width of the depletion region in semiconductor 1

( )2211

112 NN

VVNV bib εε

ε+

−=

( )( )22111

2211

2NNqNVVNx bi

εεεε

+⋅−⋅

=

Built-in Voltage semiconductor 1

Built-in Voltage semiconductor 2

( )( )22112

2212

2NNqNVVNx bi

εεεε

+⋅−⋅

= Width of the depletion region in semiconductor 2

84


Diodes

3.7 Metal Semiconductor ContactsMetal Semiconductor contacts are very important for all electronic devices, because these contacts form the interconnection between the semiconductor device and the outside world. In general it can bedistinguished between two different kinds of metal semiconductor contacts: Schottky barriers or Schottky diodes on one side and ohmic contacts on the other side. In the following both of these very important contacts will be described. We will start with the Schottky diode.

Schematic energy diagram for the electrons in a metal and a semiconductor.


85


Diodes

3.7.1 The Schottky DiodeThe energy diagram of a metal and a semiconductor exhibits the major difference between the two materials. For the metal the Fermi level is in the conduction band, whereas the Fermi level for the semiconductor material is in the bandgap depending on the distribution of carriers in the band. If the metal and the semiconductor are brought together a transfer of electrons from one material to the other and vice versa is observed. If the metal semiconductor contact is under thermal equilibrium the Fermi level will be constant through the entire contact.

Schematic energy diagram for the electrons in a metal and a semiconductor including the vacuum energy level.


86


Diodes

3.7.1 The Schottky DiodeA more detailed picture of the two materials in terms of the energy levels is given in the next graph. In this diagram a new energy level is introduced, which is the macro potential or the vacuum energy level. The vacuum energy level corresponds to the energy needed to remove an electron from the system. It means that an electron is free to move in a surrounding (vacuum), so that the electron can move away from the solid. The vacuum energy level can be seen as a reference energy level. The vacuum energy level can be relatively easy determined for the different materials. The vacuum energy level allows a comparison of the energy levels for different materials.

The energy required to remove an electron from the Fermi level EF to a position outside the material (vacuum level) is called the work function Φ. The work function can be defined for the metal ΦM and the semiconductor ΦS.

The energy required to move an electron from the bottom of the conduction band to the vacuum energy level is called the electron affinity χ.

87


Diodes

3.7.1 The Schottky DiodeConnecting the two materials leads to a band bending. There are two basic requirements in the construction of the energy band diagram, which have to be fulfilled:

(a) The Fermi level has to be constant throughout the contact in thermal equilibrium.

(b) The vacuum level has to be continuous and parallel to the band edges.

In this case the right part of the energy diagram (semiconductor) is pulled down, so that the bands bend upwards to keep the vacuum level continuous.

Metal Semiconductor barrier: (a) physical contact, (b) energy bands before forming a contact between the metal and the semiconductor, (c) in thermal equilibrium Ref.: G.W. Neudeck, The pn Junction diode

88


Diodes

3.7.1 The Schottky DiodeThe bending of the bands leads to a formation of a barrier for electrons. The electrons have to overcome this energy barrier ΦB, which is the energy difference between the metal work function ΦM and the electron affinity χ.If ΦM > ΦS a Schottky barrier is formed at the interface between the metal and the semiconductor. The work function of the semiconductor is controlled by the doping level of the material. The work function of the metal is a material inherent property. For example silver has a work function of 4.25eV and gold has work function of 5.2eV. Both materials are high work function materials. Chromium and Aluminum has a lower work function of <4eV. Therefore, the barrier height can be controlled by the materials involved and the doping concentration of the semiconductor. The formation of the barrier limits the current flow from the metal to the semiconductor, whereas the current flow from the semiconductor to the metal is not hindered. The Schottky barrier behaves like a diode or a rectifier.

89


Diodes

3.7.1 The Schottky DiodeA space charge region is formed at the metal-semiconductor interface. The charge distributions of a metal-semiconductor and pn-junction are similar. Based on the charge distribution in the device the electric field distribution can be calculated by solving the Poisson equation. The second integration of the charge distribution leads to the potential distribution in the device. A built-in voltage can be defined, which is equal to Vbi=(ΦM -ΦS)/q.

Energy bands, charge distribution, electric field and potential distribution of a Metal Semiconductor barrier.


90


Diodes

3.7.1 The Schottky DiodeThe work function differences between different materials. The energy difference varies with the given metal and the doping concentration.


Work function difference as a function of the dopantconcentration for aluminum and poly silicon (gate).

91


Diodes

3.7.1 The Schottky DiodeApplying a positive voltage (forward bias) to the Schottky barrier leads to the lowering of the barrier, so that more electrons can overcome the barrier. For negative bias (reverse voltage) the bands are bended more and the barrier height is increased. The current flowing through a Schottky Barrier can be described by an expression similar to a pn-junction.

Schottky diode under forward bias (a) and reverse bias (b).


⋅=nkTqVII exp0 Schottky barrier under (Empirical)

Forward bias

92


Diodes

3.7.2 The Ohmic ContactA perfect Ohmic contact would be a contact which does not leads to a voltage drop. A good ohmic contact would lead to a very small voltage drop across the metal semiconductor interface. Furthermore, the contact exhibits a linear behavior (I/V curve). In general two ways can be distinguished how to achieve an ohmic contact. In the first case a current is tunneling through a very thin barrier. In the other case the work function of the metal is smaller than the work function of the semiconductor (ΦM < ΦS).

93


Diodes

ReferencesMichael Shur, Introduction to Electronic Devices, John Wiley & Sons; (January 1996).(Price: US$100), Audience: under graduate students

Simon M. Sze, Semiconductor Devices, Physics and Technology, John Wiley & Sons; 2nd Edition (2001). (Price: US$115), Audience: under graduate students

R.F. Pierret, G.W. Neudeck, Modular Series on Solid State Devices, Volumes in the Series: Semicondcutor Fundamentals, The pn junctiondiode, The bipolar junction transistor, Field effect devices,(Price: US$25 per book), Audience: under graduate students

T. Fjeldly, T. Ytterdal, M. S. Shur, Introduction to devicemodeling and circuit simulation, Wiley (1997)

introduction to electronic devices - jacobs university … diodes.pdf · the drift current is equal...

Documents