Lecture 13 - Stanford University

Lecture 13Lecture 13

Doped Semiconductors

Suggested reading: 5.2-5.4

Why doping is good for semiconductor devices

Planar transistor Intel’s new 3D transistor (May 2011)( y )

gatep-drain

h l

gate

d i

p-source

p-source

n-channelp-drain

SiO2

n-channelSiO2

= en + eph

Semiconductor conductivity

= conductivity, e = electronic charge, n = electron concentration in the CB

ene eph

e = electron drift mobility, p = hole concentration in the VB, h = hole drift mobility

= en + eph

Semiconductor conductivity

= conductivity, e = electronic charge, n = electron concentration in the CB

ene eph

e = electron drift mobility, p = hole concentration in the VB, h = hole drift mobility

Intrinsic semiconductors

Intrinsic Semiconductor n in the CB & VB

kTEENn Fc

c)(exp

kTEENp vF

v)(exp

Nc & Nv = effective electron/hole density of states at the band edge, Ec = conduction band edge, Ev = valence band edge , EF = Fermi

energy k Boltzmann constant T temperatureenergy, k = Boltzmann constant, T = temperature

The product of n and p is a temperature-dependent constant:

2exp g nE

NNnp

exp ivc nkT

NNnp

ni = intrinsic carrier concentration

Recall for a metal that all levels up to the Fermi energy are filled with

Fermi energy of intrinsic semiconductors

Recall for a metal that all levels up to the Fermi energy are filled with electrons. For an intrinsic semiconductor:

E

kTE

NNn gvci exp2pn and

EENp vF )(exp

Substitute p in for ni:

kTNp v exp

kTE

NNkT

EENpn gvc

vFivi 2

exp)(exp

We can rearrange this expression to find the Fermi level for an intrinsic semiconductor, EFi

Fermi energy of intrinsic semiconductors

c

gvFi NNkTEEE ln

21

21

EFi = Fermi energy in the intrinsic semiconductor, Ev = valence band edge, Eg = Ec - Evis the bandgap energy, k = Boltzmann constant, T = temperature, Nc = effective

vN22

density of states at the CB edge, Nv = effective density of states at the VB edge

Substituting our expressions for Nc and Nv:

E E1

E3

kT lme*

b g p c v

EFi Ev 2Eg 4

kT ln e

mh*

me* = electron effective mass (CB), mh

* = hole effective mass (VB)

Fermi energies: intrinsic & doped semiconductors

Intrinsic(n=p)

What happens to the hole concentration if I double the concentration of electrons (i.e., n-doping)?


Intrinsic(n=p)

n-type

In all cases, np = ni2


Intrinsic(n=p)

n-type p-type

In all cases, np = ni2

Work function φ (for intrinsic & doped) Work function: energy difference between vacuum and the Fermi energygy gy

Vacuum φ φ φ

Intrinsic(n=p)

n-type p-type

**The energy required to remove an electron, even though there are no electrons at EF in a semiconductor. **

Average electron energy in the CB (intrinsic)

3 kTEE c 23

CB 2

Recall: (3/2)kT is also the average kinetic energy per atom in a monatomic gas (kinetic energy per atom in a monatomic gas (kinetic

molecular theory) in which the gas atoms move around freely and randomly inside a container.

The electron in the CB behaves as if it were “free” with a mean kinetic energy that is (3/2)kT and an effective mass me*.

Very different from a metal, where the average energy is 3/5 EF and is practically temperature independent.

Doped (extrinsic) semiconductors

Modifying the carrier concentration (electrons or holes) Modifying the carrier concentration (electrons or holes) with impurities

1. By adding impurities to Si that had a valency of more than 4 (i.e., the pentavalent elements of group V on the periodic table such as As) we can obtain a semiconductor with table, such as As), we can obtain a semiconductor with

n>>p.

n-type doping n type doping

2. By adding trivalent impurities (like B), with a valency of less than 4, we can find an excess of holes,

p-type doping

Doped (extrinsic) semiconductors

Arsenic-doped Si crystal.The four valence electrons of As allow it to bond just like Si, but the j

fifth electron is left orbiting the As site. The energy required to release the free fifth electron into the CB is very small.

n-type Doping (i.e., adding As, P, or Sb to Si)

For As: ε=11.9, me* ~ 1/3 me

ED=0.032 eVRoom temp vibrations ~

0.07 eV

Add a small amount of impurity atoms (1 impurity atoms for every million host atoms) want to maintain diamond crystal structure

To ionize the impurity (i.e., release the electron into CB, free from it’s As host), we require energy:) q gy

)1)((6.13*e

DmeVE ))((6.3

reD m

eV

Energy band diagram for an n-type Si doped with 1 ppm As. There are donor energy levels just below Ec around As+ sites.

At room temperature, n ~ ND (the donor atom concentration)

are donor energy levels just below Ec around As sites.

n-type Conductivity at room temperature

n22innp DNn ~

d

i

Nnp

he epen

eDhi

eD eNNneeN

2

eDhD

eD N

n-type Conductivity at LOW temperature

At low temperatures, not all the donors will be ionized fewer electrons in CB than at room temperature.

NDT=ionized donor concentration at temperature T

N T N probabilit of not finding an electron at ENDT=ND x probability of not finding an electron at ED

NDT=ND x [1-fD(ED)]

EEEf

FDdD )(11

1)(

kTFD )(exp

21

Can only take one Can only take one electron (spin up or spin

down, not both)

p-type Doping (i.e., adding B, Al, Ga, In to Si)

B has only three valence electrons. yWhen B substitutes for a Si atom, one of its bonds has an electron

missing and therefore a hole. The hole orbits around the B- site by the tunneling of electrons from neighboring bonds tunneling of electrons from neighboring bonds.

Eventually, thermally vibrating Si atoms provide enough energy to free the hole from the B- site into the VB

Energy band diagram for a p-type Si doped with 1 ppm B. There are acceptor energy levels E just above E around B- sites These There are acceptor energy levels EA just above Ev around B sites. These

acceptor levels accept electrons from the VB and therefore create holes in the VB.

p-type Conductivity at room temperature

n22innp ANp ~

A

i

Nnn

he epen

hAhAei eNeN

Nne

2

hAhAeAN

Example: Resistance of a doped Si crystal

Let’s calculate the resistance of a 1 cm3 pure Si crystal given:

Atomic concentration = 5 x 1022 cm-3

ni= 1 x 1010 cm-3

μe=1350 cm2/(Vs)μh=450 cm2/(Vs)μh 450 cm /(Vs)

R=347 kΩ

Now, let’s dope this crystal with As, such that one atom in every billi i A Wh t i th i ti it t t t ? billion is As. What is the resistivity at room temperature?

R=92.6 Ω

Compensation Doping (at room temperature)

To precisely control a semiconductor’s properties we can dope with To precisely control a semiconductor s properties, we can dope with both donors (contribute electrons) and acceptors (contribute holes)

nn 22

iad nNN More donors than acceptors

ad NNn ad

ii

NNn

nnp

More acceptors than donors id nNN More acceptors than donors ida nNN

NN ii nn 22

da NNp da

ii

NNpn

Temperature Dependence of Conductivity

• Main application: Thermoelectric coolers (move heat away • Main application: Thermoelectric coolers (move heat away from a source) & thermoelectric switches/modulators


Depends on two properties: Depends on two properties: 1. The temperature dependence of the carrier concentration

2. The temperature dependence of the mobility

Carrier concentration versus T

Low temperature: some donors are being ionized. p gIonization continues with increasing temperature until

all donors have been ionized.


Intermediate temperature: nearly all donors have been p yionized. At a temperature Ti, ni becomes equal to ND.


High temperature: The concentration of electrons g pgenerated by thermal excitation of electrons across Egexceeds ND. The semiconductor behaves as if it were

intrinsic intrinsic.

n=ni(T) and p=n (“intrinsic range”)

Carrier concentration versus T: Low temperature

Combine:

1. Charge carrier statistics: n=NCexp[-(EC-EF)/kT]1. Charge carrier statistics: n NCexp[ (EC EF)/kT]

2. Mass action law: np=ni2

3. Electric neutrality of the crystal: p+ND+=n

4. Statistics of ionization of dopants: NDT=ND x [1-fD(ED)]4. Statistics of ionization of dopants: ND ND x [1 fD(ED)]

)2/(2/1)21( kTE

DC eNNn DC EEE

Carrier concentration versus T: Medium and high temperatureg p

Medium temperature: n ~ ND

High Temperature: n=ni(T))2/(2/3~ kTEgeTn

High Temperature: n ni(T)

Low Temperature:

)2/(2/1)21( kTE

DC eNNn


High temperatures

Low temperatures

h d d f h lThe temperature dependence of the electron concentration in an n-type semiconductor.

Mobility

Two temperature ranges:p g

1 High temperatures: limited by scattering from lattice vibrations1. High temperatures: limited by scattering from lattice vibrations

2. Low temperatures: limited by Coulombic attraction between d dcarriers and ionized impurities

High temperature Mobility

Electron drift mobility depends on mean time between scattering Electron drift mobility depends on mean time between scattering events:

*de

NS1

e

d m sthNSvlCross-sectional

area of the scatterer

# scatterers per unit l

Mean speed of electrons (thermal

volume

electrons (thermal velocity)

High temperature Mobility

1

sthNSv1

TaS 2

Also, since an electron in the conduction band is free:

31 kTvm the 23

21 2* 2/1Tvth

2/31 T TTT

Low temperature Mobility

El d if bili li i d b i b b Electron drift mobility limited not by scattering events, but by interactions with ionized impurities:

Scattering of electrons by an

dionized impurity.

Low temperature Mobility

S i i Scattering cross section: 2

crS

Critical radius for scattering given by KE = |PE(rc)|

ekT23

crr

kT042

2

1T

S # ionized impurities

2TT 2/311

IIIth NNTTNSv 2/12

Lattice-Scattering-Limited Mobility (High Temp)(High Temp)

L T 3 / 2

L = lattice vibration scattering limited mobility, T = temperature

Ionized Impurity Scattering Limited Mobility (Low Temp)(Low Temp)

IT 2/3

I = ionized impurity scattering limited mobility, NI = concentration of the ionized i iti ( ll i i d i iti i l di d d t )

II N

impurities (all ionized impurities including donors and acceptors)

Overall Mobility

1

1

1

= effective drift mobility

e I L

e effective drift mobility

I = ionized impurity scattering limited mobility (low temperature)p )

L = lattice vibration scattering limited mobility (high temperature)p )

Th i h i h l bili The scattering process having the lowest mobility determined the overall effective mobility.

Log-log plot of drift mobility versus temperature for n-type Ge and n-type Si samples. Various donor concentrations for Si are shown. Nd are in cm-3.

The upper right inset is the simple theory for lattice limited mobility, whereas the lower left inset is the simple theory for impurity scattering limited mobility.

The variation of the drift mobility with dopant concentration in Si for electrons and holes at 300 K.




(n-type semiconductor)

Lecture 13 - Stanford University

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