Solid State Device Fundamentals - City University of … - Lectures... · Working temperature range...

Solid State Device Fundamentals

ENS 345

Lecture Course

by

Alexander M. Zaitsev

[email protected]: 718 982 2812

4N101b

College of Staten Island / CUNY Department of Engineering Science and Physics

Solid State Device Fundamentals 4. Doped Semiconductors

mailto:[email protected]

Department of Engineering Science and Physics


Doping semiconductors

Doped semiconductors are semiconductors, which contain impurities, foreign atomsincorporated into the crystal structure of the semiconductor. Either these impurities canbe unintentional, due to lack of control during the growth of the semiconductor, or theycan be added on purpose to provide free carriers in the semiconductor.

Ionization of a shallow donor and a shallow acceptor.



2



Dopants for doping A4 semiconductors


Acceptors DonorsDopants in silicon and germanium

3



Dopants for A3B5 semiconductors


Dopants in GaAs

Acceptors Donors

As AsGa

Ga GaAs

AsGa Ga

n-type: tellurium, sulphur (substituting

As), tin, silicon, germanium (substituting

Ga).

p-type: zinc, chromium (substituting

Ga), silicon, germanium (substituting

As).4



Ionization energy of dopants in semiconductors

Ionization energy of shallow donors and acceptors can be evaluated using hydrogenic model:

5

Trajectory of an electron bound to a

donor ion within a semiconductor crystal.

𝑟𝐷,𝐴 =4𝜋𝜀𝜀0ℏ

2

𝑚∗𝑒2~𝟏 𝒏𝒎

EHion

m0 e4

13.6 eV==8e0

2h2

Ionization energy EHion and orbital radius a0 of hydrogen atom

Ionization energy Eion and orbital radius 𝒓𝑫,𝑨 of donors and acceptors

Eion

m* e4

=8e2e0

2h2

~𝟓𝟎 𝒎𝒆𝑽

Energy levels of donors and acceptors

Conduction Band Ec

EvValence Band

Donor Level

Acceptor Level

Ed

Ea

Donor ionization energy

Acceptor ionization energy

Ionization energy of selected donors and acceptors in silicon

Acceptors

Dopant Sb P As B Al In

Ionization energy, E c–E d or E a–E v (meV) 39 44 54 45 57 160

Donors



6

7

Dielectric constant of silicon is e = 11.7.Dielectric constant of germanium is e = 16.2.mn*/m0 = 0.26mp*/m0 = 0.39

Calculate the ionization energies and radii of donors and acceptors in Si and Ge.

Homework

Hydrogenic model of donors and acceptors



2

innp =

Charge neutrality

2/1

2

2

22

= i

dada nNNNN

p

2/1

2

2

22

= i

adad nNNNN

n



Since doped semiconductor, as a whole, is electroneutral, concentration of negative charges equals the concentration of positive charges.

DionizedAionized NpNn =

8

9College of Staten Island / CUNY Department of Engineering Science and Physics


Homework

Concentration of electrons and holes

1. Calculate concentrations of electrons and holes in Si and Ge with donor concentration of 3x1017 cm-3 and acceptor concentration of 8x1016 cm-3 at room temperature.

2. Will these concentrations change much with temperature increase to 100°C?

Charge carrier concentration in doped semiconductor



AD NNn =

nnp i

2=

iAD nNN

if , AD NN di Nnp

2=andn = ND

n-type

ida nNN da NNp =

pnn i

2=

if , da NN aNp =

ai Nnn2

=and

p-type

10

Dopant compensation: Example

What are n and p in Si with (a) ND = 61016 cm-3 and NA = 21016 cm-3 and (b) additional 61016 cm-3 of NA?

(a)

(b) Na = 21016 + 61016 = 81016 cm-3 > Nd

316cm104 == ad NNn

3316202cm105.2104/10/ === nnp i

3161616 cm102106108 === da NNp

3316202cm105102/10/ === pnn i

+ + + + + + . . . . . .

. . . . . . . . . . .

Nd = 61016 cm-3

Na = 21016 cm-3

n = 41016 cm-3

+ + + + + +

- - - - - - - -

. . . . . .

. . . . . .

Nd = 61016 cm-3

Na = 81016 cm-3

p = 21016 cm-3



The resulting carrier density in compensated material

is approximately equal to the difference between the

donor and acceptor concentration:

11



Homework

Charge carrier concentration in n-type and p-type semiconductor

1. Calculate concentrations of electrons and holes in Si and Ge containing 3x1017 cm-3 donors and 8x1016 cm-3 acceptors at room temperature with assumption NA, ND >> ni.

2. Compare the obtained values with those calculated in Homework on slide 9.

Position of Fermi level versus dopant concentration

kTEE

CFCeNn

/)( =

nNkTEE CCF ln=



kTEE

VVFeNp

/)( =

pNkTEE VVF ln=

With concentration of donors, the Fermi level approaches the conduction band.With concentration of acceptors, the Fermi level approaches the valence band.

13

Fermi energy of doped silicon (n-type and p-type) vary with doping density. As the doping increases, Fermi level shifts towards conduction band (Ec) in the case of n-type doping and shifts towards valance band (Ev) in p-type doping. Note where Fermi level crosses Conduction (valance) band).

The Fermi level and carrier concentration



kTEE

CFCeNn

/)( =

eV 614.010/108.2ln026.0

ln

1719 ==

== nNkTEE CFC

Where is EF for n =1017 cm-3?ECEF

EC

0.146 eV

eV 31.010/1004.1ln026.0

ln

1419 ==

== pNkTEE VVF

Where is EF for p = 1014 cm-3 ?

kTEE

VVFeNp

/)( =

0.31 eV

EC

EF

EV

14



Homework

Position of Fermi level in n-type and p-type semiconductor

1. Calculate position of Fermi level in n-type silicon with donor concentration of 3x1017 cm-3 at room temperature.

2. Calculate position of Fermi level in p-type silicon with acceptor concentration of 8x1016 cm-3 at room temperature.



As the temperature increases Fermi level shifts towards the middle of the bandgap.

Position of Fermi level in doped semiconductor versus temperature

16

ND = 1017 cm-3. What fraction of the donors are not ionized?

Solution: First assume that all the donors are ionized.

Probability of not being ionized:

04.05.01

1

5.01

1meV26/)meV)45146((/)(

=

=

eekTEE FD

It is reasonable to assume that at room temperature the complete ionization occur, i.e., n = ND .

Is assumption n = ND valid for low doped semiconductors?

meV146cm10 317 ===

CFD EENn Ec

Ef

Ev

146 meV

Ed

45meV

Complete ionization of dopants



17

Example: Carrier concentrations

What is the hole concentration in an n-type semiconductor with 1015 cm-3 of donors?

n = 1015 cm-3.

After increasing temperature by 60C, n remains the same at 1015 cm-3

while p increases by about a factor of 2300 because of thermal activation..

What is n if p = 1017cm-3 in a p-type silicon wafer?

3-5

315

-3202

cm10cm10

cm10==

n

np i

kTE

igen

2/2

3-3

317

-3202

cm10cm10

cm10==

p

nn i



18



Homework

Position of Fermi level in n-type and p-type semiconductor at elevated temperature

1. Calculate position of Fermi level in n-type silicon with donor concentration of 3x1017 cm-3 at a temperature of 200°C.

2. Calculate position of Fermi level in p-type silicon with acceptor concentration of 8x1016 cm-3 at a temperature of 200°C.

Temperature dependence of charge carrier concentration

kTEEDC DCeNN

n2/)(

2/1

2

=Low T:



Electron density as a function of temperature in silicon with:ND = 1016 cm-3

NA = 1014 cm-3

EC - ED = EA - EV = 50 meV.

RT300°CLNT

n = Nd

High T:kTE

VCigeNNnpn

2/===

20

Working temperature range of semiconductor device



At high temperatures, the electrical difference between the n and p regions disappears and the p-n junction becomes ineffective in controlling carrier movement.

The basic upper temperature limit of semiconductor material is determined by its bandgap energy. A rule-of-thumb is that the maximum temperature (in K) is approximately 500 times the bandgap energy in eV. For Si this rule gives Tmax ~ 290°C.

The lower temperature limit is determined by the thermal ionization energy of the dopants. If the temperature is too low, the dopants are not sufficiently ionized and there are no insufficient charge carriers. The result is a condition called "freeze-out." For example, silicon freezes out at about 40 K.

21

Dependence of the temperature range on doping



The higher dopant concentration the wider the temperature range

22



Homework

Working temperature range

1. High Temperature Limit: For n-type silicon with 1016 cm-3 phosphorous donors, at what temperature the concentration of holes is 10% of that of electrons?

2. Low Temperature Limit: For n-type silicon with 1016 cm-3 phosphorous donors, at what temperature the concentration of electrons is 10% of that of donors?

Solid State Device Fundamentals - City University of … - Lectures... · Working temperature range...

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