Nanoscience and Nanotechnology Lecture 19

P. E. Russell: [email protected]

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Nanoscience and Nanotechnology Lecture 19



Only 7 lectures remaining!

(including this one)

Review of traditional microelectronics

Course notes are at:http://www.phys.appstate.edu/nanotech/NanoCourseLecturePPTs2008/


Review of basic semiconductor Physics

When considering the Physics and Engineering of traditional semiconductors, we are mainly concerned with ‘bulk’ properties; and consider surfaces as defects.

This approach is the opposite of the nanotechnology approach, where the goal is to use the special properties of materials and devices at the nm scale, where surface atoms control properties.


Material based on content of:

Principles of Electronic Materials and Devices,

by S. O. Kasap Third Edition

Text WEB site:

http://materials.usask.ca/textbook/




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

V a l e n c e B a n d ( V B )

F u l l o f e l e c t r o n s a t 0 K .

E c

E v

0

E c+ χ

(c)

B a n d g a p = E g

(b)

ψB

C o n d u c t i o n B a n d (C B )

E m p ty o f el ec t r o n s a t 0 K .

ψh yb

o r b i t a l s

S i i o n c o r e ( + 4 e )

V a l en c e

el ec t r o n(a)

Si crystal in 2-D

. 5.1: ( ) Fig a A simplified two dimensional illustration of a Si atom with four , hybrid orbitalsψhyb. . ( ) Each orbital has one electron b A simplified two

. ( ) dimensional view of a region of the Si crystal showing covalent bonds c The .energy band diagram at absolute zero of temperature

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Recall the Si band structure

P. E. Russell: [email protected] Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Fig. 5.2: A two dimensional pictorial view of the Si crystal showingcovalent bonds as two lines where each line is a valence electron.

Simplified 2 dim Si showing 4 bonds per atom


e–

h o l e

C B

V B

Ec

Ev

0

Ec+ χ

Eg

F r ee e≠h υ > Eg

H o le

E lec t r o n en er gy

h υ

( a) (b )


Fig. 5.3: (a) A photon with an energy greater than Eg can excite anelectron from the VB to the CB. (b) When a photon breaks a Si-Sibond, a free electron and a hole in the Si-Si bond is created.

Electron excitation across the forbidden gap


h+e–


Fig. 5.4: Thermal vibrations of atoms can break bonds and therebycreate electron-hole pairs.

Thermal carrier (electron-hole pair) generation


h+ E g

V B

C Bh

+

e–

(a)

h+

h+

h+

e–

h+

h+

h+ (d)

h+

e– e

–

h+

(f)

(b)

(c)

(e)

Fig. 5.5: A pictorial illustration of a hole in the valence bandwandering around the crystal due to the tunneling of electrons fromneighboring bonds.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Concept of hole motion in valence band and annihilation


.

V B

C B

V B

C B

x

x = 0 x = L

V ( x )E l e c t r o s t a t i c P E ( x )

( a ) ( b )

Ex

Ex

Fig. 5.6: When an electric field is applied, electrons in the CB andholes in the VB can drift and contribute to the conductivity. (a) Asimplified illustration of drift in Ex. (b) Applied field bends the energybands since the electrostatic PE of the electron is -eV(x) and V(x)decreases in the direction of Ex whereas PE increases.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Both electrons and holescan drift in an electric field


E

g(E)

g (E ) ∝ (E –Ec) 1 /2

f(E)

EF

nE(E) orpE(E)

E E

F or

electr ons

F or holes

[ 1 – f(E ) ]

nE(E )

pE(E )

A r ea = p

A r ea = ∫ nE(E )dE = n

Ec

Ev

Ev

Ec

0

Ec+ χ

EF

V B

C B

(a) (b ) (c) (d )

. 5.7: ( ) . ( ) ( Fig a Energy band diagram b Density of states number of states ). ( ) - per unit energy per unit volume c Fermi Dirac probability function

( ). ( ) probability of occupancy of a state d The product ofg(E) andf(E) is ( the energy density of electrons in the CB number of electrons per unit

). energy per unit volume The area undernE(E) . vsE is the electron .concentration in the conduction band


Ec

Ev

EF i

C B

EF p

EF n

Ec

Ev

Ec

Ev

( a ) ( c )( b )

V B


Fig. 5.8. Energy band diagrams for (a) intrinsic (b) n-type and (c) p-typesemiconductors. In all cases, np = ni

2

Mass action law np=ni2

Mass action law np=ni2 = NcNvexp(-Egap/kT)

Where Nc is the density of states near conduction band edgeNv is the density of states near valence band edge


A s+

e–


Fig. 5.9: Arsenic doped Si crystal. The four valence electrons of Asallow it to bond just like Si but the fifth electron is left orbiting theAs site. The energy required to release to free fifth-electron into theCB is very small.

Adding impurities or dopants


x

As+ As+ As+ As+

Ec

Ed

Ev

As atom sites every 106 Si atoms

Distanceintocrystal

~0.03 eV

CB

Electron Energy


Fig. 5.10: Energy band diagram for an n-type Si doped with 1 ppmAs. There are donor energy levels just below Ec around As+ sites.

Dopant levels for As impurities (valence V, n-type))

Each as provides an extra loosely bound electron


B –

h +

F r e e

B –

(a )

(b )


Fig. 5.11: Boron doped Si crystal. B has only three valence electrons.When it substitutes for a Si atom one of its bonds has an electronmissing and therefore a hole as shown in (a). The hole orbits aroundthe B- site by the tunneling of electrons from neighboring bonds asshown in (b). Eventually, thermally vibrating Si atoms providesenough energy to free the hole from the B- site into the VB as shown.

Boron as an example of n type doping


x

B–

Ev

Ea

B a t o m s i t e s e v e r y 1 06

S i a t o m s

D i s t a n c e

i n t o c r y s t a l

~ 0 .0 5 e V

B– B

–B

–

h+

V B

Ec



Fig. 5.12: Energy band diagram for a p-type Si doped with 1 ppm B.There are acceptor energy levels just above Ev around B- sites. Theseacceptor levels accept electrons from the VB and therefore create holesin the VB.

Band diagram for b doped (p-type) Si

Small ionization energy allows thermal ionization at room temperature


x

B–

Ev

Ea

B a t o m s i t e s e v e r y 1 06

S i a t o m s

D i s t a n c e

i n t o c r y s t a l

~ 0 .0 5 e V

B– B

–B

–

h+

V B

Ec



Fig. 5.12: Energy band diagram for a p-type Si doped with 1 ppm B.There are acceptor energy levels just above Ev around B- sites. Theseacceptor levels accept electrons from the VB and therefore create holesin the VB.

Band diagram for b doped (p-type) Si

Small ionization energy allows thermal ionization at room temperature

Note that the electron from the valance band becomes fixed charge (in energy and position)

Dopants are far apart


.

V

n - T y p e S e m i c o n d u c t o r

Ec

EF

A

B

V(x)

x

Electrostatic PE(x) = -eV

Ex

EFi

Ed

Ev


Fig. 5.13: Energy band diagram of an n-type semiconductor connectedto a voltage supply of V volts. The whole energy diagram tilts becausethe electron now has an electrostatic potential energy as well

For an n-type semiconductor (e.g. Arsenic doped) only the electrons are mobile. The ionized dopant atoms do not move.

Thus we have added both ‘mobile” and ‘fixed’ or ‘localized’ charge!


we have both ‘mobile’ and ‘fixed’ or ‘localized’ charge!

For the Case of an extrinsic (doped) semiconductor:

this is a very Important Concept!!!

Fixed charge: the ionized dopant atomsMobile charge: the extra electron or hole contributed by ionization


V B

C B

As As AsAs+E

F

Eg

T < Ts

Ts < T < T

i

(b ) T = T2

(a ) T = T1

T > Ti

( c ) T = T3

As+EF

As+ As+ As+ As+E

F

As+ As+ As+


Fig. 5.14: (a) Below Ts, the electron concentration is controlled bythe ionization of the donors. (b) Between Ts and Ti, the electronconcentration is equal to the concentration of donors since theywould all have ionized. (c) At high temperatures, thermally generatedelectrons from the VB exceed the number of electrons from ionizeddonors and the semiconductor behaves as if intrinsic.

Example of n-type material versus temperature


ln(n)

1/T

Ti

Ts

I N T R I N S I C

E X T R I N S I CI O N I Z A T I O N

ni( T )

l n ( Nd

) s l o p e = – Δ E / 2 k

sl o p e = – E g / 2 k


Fig. 5.15: The temperature dependence of the electron concentrationin an n-type semiconductor.

Temperature dependency of conductivity: critical concept!

From this data we can measure:Band gap energy Egap, and

ln(n)

1/T

Ti

Ts

I N T R I N S I C

E X T R I N S I CI O N I Z A T I O N

ni( T )

l n ( Nd

) s l o p e = – Δ E / 2 k

sl o p e = – E g / 2 k


Fig. 5.15: The temperature dependence of the electron concentrationin an n-type semiconductor.

Dopant ionization energy ΔEionization


Conductivity of a Semiconductor

= conductivity, e = electronic charge,

n = electron concentration in the CB, e = electron drift mobility,

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

= ene + eph

Note: this expression is valid for both intrinsic and extrinsic semiconductors!

We now see that n≠p for extrinsic semiconductors.


E

g(E)

g (E ) ∝ (E –Ec) 1 /2

f(E)

EF

nE(E) orpE(E)

E E

F or

electr ons

F or holes

[ 1 – f(E ) ]

nE(E )

pE(E )

A r ea = p

A r ea = ∫ nE(E )dE = n

Ec

Ev

Ev

Ec

0

Ec+ χ

EF

V B

C B

(a) (b ) (c) (d )

. 5.7: ( ) . ( ) ( Fig a Energy band diagram b Density of states number of states ). ( ) - per unit energy per unit volume c Fermi Dirac probability function

( ). ( ) probability of occupancy of a state d The product ofg(E) andf(E) is ( the energy density of electrons in the CB number of electrons per unit

). energy per unit volume The area undernE(E) . vsE is the electron .concentration in the conduction band

(The intrinsic case is shown here.)

We can determine n and p (the electron and hole concentrations) if we know g(E) and f(E) and…


Ec

Ev

EF i

C B

EF p

EF n

Ec

Ev

Ec

Ev

( a ) ( c )( b )

V B



2

Mass action law np=ni2: in equilibrium

Valid for both intrinsic and extrinsic semiconductors!

Mass action law np=ni2 = NcNvexp(-Egap/kT)

Where Nc is the density of states near conduction band edgeNv is the density of states near valence band edgeIf more electrons are added by doping, then fewer holes will exist due to recombination, keeping the product np constant!

Ec

Ev

EF i

C B

EF p

EF n

Ec

Ev

Ec

Ev

( a ) ( c )( b )

V B



2


Carrier recombination

In addition to carrier creation, we will have carrier recombination, i.e. if an electron excited to the CB is located (spatially) near a hole in the VB, the electron will reduce its energy by filling the hole (allowed electron state).

The electron and hole annihilate each other (as free charge carriers) via recombination.

.

Ev

Ec

CB

VB

ψcb(k

cb)

ψvb(k

vb)

h υ = Eg

Distance


Fig.5.22: Direct recombination in GaAs. kcb = kvb so that momentumconservation is satisfied


Carrier RecombinationThe recombination rate R will be proportional to both the electron and hole concentrations (and hence to their product) Rnp

What about the carrier generation rate G?G will depend on the number of electrons available for excitation at Ev i.e. Nv and empty states at Ec i.e. Nc.And the probability of an electron making the transition i.e. exp[-Eg/kT] so: G NvNc exp[-Eg/kT] In equilibrium G = R, generation and recombination rates are equal. (more on this later)

Next: Where is Ef within the band gap?

.

Ev

Ec

CB

VB

ψcb(k

cb)

ψvb(k

vb)

h υ = Eg

Distance




Eg

(eV)

χ ( )eV Nc (cm–3) N

v (cm–3) n

i

(cm–3)

e

(cm2 V–1 s–1)

h

(cm2 V–1 s–1)

me*/m

e m

h*/m

e ε

r

Ge 0.66 4.13 1.04×1019 6.0×1018 2.4×1013 3900 1900 0.12 a

0.56 b

0.23 a

0.40 b

16

Si 1.10 4.01 2.8×1019 1.04×1019 1.45×1010 1350 450 0.26 a

1.08 b

0.38 a

0.56 b

11.9

GaAs 1.42 4.07 4.7×1017 7×1018 1.8×106 8500 400 0.067 ,a b 0.40 a

0.50 b

13.1

Use effective mass related to conductivity, rather than for density of states.

Table 5.1 Selected typical properties of Ge, Si and GaAs at 300 K. Effective mass related to conductivity (labeled a) is different than that for density of states (labeled b).


.

Fig. 5.20: Temperature dependence of electrical conductivity for adoped (n-type) semiconductor.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

log(n)

Intrinsic

Extrinsic

Ionization

log()

log() ∝ T - 3 / 2 ∝ T 3 / 2

L a t t i c e

sc a t t e r i n g

I m p u r i t y

sc a t t e r i n g

1/TL o w T e m p e r a t u r eH i g h T e m p e r a t u r e

T

M e t a l

S e m i c o n d u c t o r

The combined temperature effects of carrier creation and carrier scattering.

KEY CONCEPT!!


We have been assuming that there are many more empty states than there are electrons in the conduction band, and/or many more filled states than holes in the valence band; i.e. that Boltzmann statistics rather than Fermi-Dirac apply.

This is the non-degenerate semiconductor case;

i.e when n<<Nc and p<<Nv

What is the other possibility? Degeneracy!Occurs when doping concentration is very large; usually 1019 or 1020 cm-3 (out of 1023 cm-3)

Non-degenerate semiconductors (the normal case)


C B

g ( E )

E

I m p u r i t i e s

f o r m i n g a b an d

( a ) ( b )

E Fp

E v

E c

EF n

E v

E c

C B

V B


Fig. 5.21: (a) Degenerate n-type semiconductor. Large number of donorsform a band that overlaps the CB. (b) Degenerate p-type semiconductor.

What happens with very high doping?: Degeneracy

Degenerate semiconductors behave as metals; this is very useful!


.

Ev

Ec

CB

VB

ψcb(k

cb)

ψvb(k

vb)

h υ = Eg

Distance



Direct gap recombination; are there other options?


Ek

kπ/a–π/a

Ec

Ev

CB

VB

Ec

Ev

The E-k Diagram The Energy BandDiagram

Empty ψk

Occupied ψk

h+

e–

Eg

e–

h+

hυ

VB

hυ

CB

Fig. 5.49: The E-k diagram of a direct bandgap semiconductor such asGaAs. The E-k curve consists of many discrete points each pointcorresponding to a possible state, wavefunction ψ k(x), that is allowedto exist in the crystal. The points are so close that we normally draw theE-k relationship as a continuous curve. In the energy range Ev to Ecthere are no points (ψk(x) solutions).From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca


E

CB

VB

k–k

Direct Band Gap

(a) GaAs

Eg PhotonEc

Ev

E

CB

VB

Indirect Band Gap, Eg

k–k

kcb

(b) Si

Ec

Ev

kvb

E

k–k

Phonons

(c) Si with a recombination center

VB

CB

Er Ec

Ev


Fig. 5.50: (a) In GaAs the minimum of the CB is directly above the maximumof the VB. GaAs is therefore a direct band gap semiconductor. (b) In Si, theminimum of the CB is displaced from the maximum of the VB and Si is anindirect band gap semiconductor. (c) Recombination of an electron and a holein Si involves a recombination center.

Si is not direct gap!This is most important in optoelectronic applications (laser, LED, photo…)More on this later.


Ec

Recombinationcenter

C B

V B

Phonons

( a ) R e c o m b i n a t i o n

Trappingcenter

Et

C B

V B

Ec

Ev

Et

Et

Ev

Er

Er

Er

( b ) T r a p p i n g


Fig. 5.23: Recombination and trapping. (a) Recombination in Si via arecombination center which has a localized energy level at Er in thebandgap, usually near the middle. (b) Trapping and detrapping ofelectrons by trapping centers. A trapping center has a localizedenergy level in the band gap.

Cases where midgap states exist (due to impurities and/or defects)

Be sure you understand that midgap states can act as Trap sites or Recombinationsites

In this caseEemission

is less than Egap


How can there be allowed states in the forbidden gap?

Allowed states (such as the charge carrier trap states and recombination centers of figure 5.23) are localized states within the forbidden band gap energy range of a defect-free material. The states are created by the perturbation of the normal periodic lattice by a point (or other) defect.

The trap states are localized in space to the area of the defect, and in energy to a specific energy within the band gap; determined by the type of defect creating the state.

Examples of defects creating midgap states include vacancies, interstitial and substitutional impurities, dislocations, stacking faults, etc.


How can there be allowed states in the forbidden gap?

These allowed individual states created by the perturbation of the normal periodic lattice act as:

Recombination sites if near midgap,Or asTrapping sites for electrons if near the CB edge, Or asTrapping sites for holes if near the CB edge.

Note that trapping just removes the charge carrier from the free carrier pool for a brief time; with the overall effect of having many traps being a reduction in charge carrier density available for conduction at any given time.


Ec

Recombinationcenter

C B

V B

Phonons

( a ) R e c o m b i n a t i o n

Trappingcenter

Et

C B

V B

Ec

Ev

Et

Et

Ev

Er

Er

Er

( b ) T r a p p i n g


Fig. 5.23: Recombination and trapping. (a) Recombination in Si via arecombination center which has a localized energy level at Er in thebandgap, usually near the middle. (b) Trapping and detrapping ofelectrons by trapping centers. A trapping center has a localizedenergy level in the band gap.

Charge carrier pair is lost permanently due to recombination.

Trapping causes a temporary loss of a single charge carrier.

Note: phonon assistance is usually required!


How long do free carriers exist?

How long do free carriers exist before being trapped or recombined?

This depends on the number of traps and recombination centers; which depends on the number (actually density) of defects.

If we measure carrier lifetime, we get an idea of the perfection of the semiconductor: which is very useful in developing processing.

How can we measure carrier lifetime?


Ec

Ev

Ed

C B

V B

Fig. 5.24: Low level photoinjection into an n-type semiconductor inwhich Δnn < nno


Carrier creation: Photoinjected charge carriers

If we shine light on a semiconductor, we will generate new charge carriers (in addition to those thermally generated) if Ephoton>Egap.

If the light is always on and of constant intensity, some steady state concentration of electrons and holes will result.

Ec

Ev

Ed

C B

V B

Fig. 5.24: Low level photoinjection into an n-type semiconductor inwhich Δnn < nno



Carrier creation: Photoinjected charge carriersLet’s consider the case of n-type material

Consider an n-type semiconductor with a doping concentration of 5 x 1016 cm-3.

What are the carrier concentrations?Let’s define some terms;nno majority carrier concentration in the n-type semiconductor in the dark (only thermally ionized carriers) (i.e. the electron concentration in n-type)

pno minority carrier concentration in the n-type semiconductor in the dark (only thermally ionized carriers) (i.e. the hole concentration in n-type)

Note: the no subscript implies that mass action law is valid!


When we have light:

With light of Ephoton>Egap hitting the semiconductor, we get photogeneration of excess charge carriers.

Δnn excess electron concentration such that::Δnn = nn-nn0

And Δpn excess hole concentration such that::Δpn = pn-pn0Note that photogenerated carriers excited across the gap can only be created in pairs i.e. Δpn = Δnn and now (in light) nnpn≠ni

2 i.e. mass action not valid!


If the temperature is constant, nn0 and pn0 are not time dependent, so

and

Consider the case of ‘weak’ illumination, which creates a 10% change in nn0 i.e. Δnn = 0.1nn0

Or if the doping level is nno=5 x 1016cm-3, then

Δnn = 0.1nn0= 0.5 x 1016cm-3

And Δpn =Δnn = 0.5 x 1016cm-3

Which change is more important? Majority or minority?

€

dnn

dt=

dΔnn

dt

€

dpn

dt=

dΔpn

dt

Carrier density change under illumination


.

S i

G e

G a A s

0 ° C2 0 0 ° C4 0 0 ° C6 0 0 ° C 2 7 ° C

LLLL

1018

1 1.5 2 2.5 3 3.5 4

1000/T (1/K )

1015

1012

103

106

1091.45×1 0 1 0 cm -3

2 .4×1 0 1 3 cm -3

2 .1×1 0 6 cm -3


Fig. 5.16: The temperature dependence of the intrinsic concentration.

Recall the intrinsic carrier concentration

For Sini is roughly 1.5x1010cm-3

At room temperature

Since pno=ni2/nn0

= (1.5x1010)2/5x1016

pno =4500 cm-3


.

n n o

n i

p n o

5 × 1 01 6

1 . 5 × 1 01 0

4 . 5 × 1 03

( a ) I n th e d a r k : n p = ni

2

nn = n

n o+ ∆ n

n

p n o

0 . 5 × 1 01 6

pn = p

n o+ ∆ p

n

5 . 5 × 1 01 6

∆ pn = 0 . 5 × 1 0 1 6

n i

( b ) I n l i g h t : n p ≠ ni

2

. 5.25: Fig Low level injection in ann- type semiconductor does not affectnn but drastically affects the minority carrier

concentrationpn.

From , Principles of Electronic Materials and Devices Second Edition, . . (© - , 2002)S O Kasap McGraw Hill:// . .http Materials Usask Ca

An extremely important concept!

Minority carrier concentration can be controlled over many orders of magnitude with only a small change in majority concn.

Δpn =Δnn = 0.5 x 1016cm-3


–

–

–

–

–

–

–

–

–

–

– ––

–

–

– –––

–

–

––

–

––

–

–

––

–+

+

–

–

–

–

–

–

––

–

–

– ––

–

–

––

––

–

–

– ––

–

–

–

–

–

–

–+

+

+

+

++

+ +

++

+

+

++

++

+

+

–

–

–

–

–

–

––

–

–

– ––

–

–

––

––

–

–

– ––

–

–

–

–

–

–

–

++

+ +

+

+

++

++

n - t y p e s e m i c o n d u c t o r i n

t h e d a r k .

pn

= pn o

< < nn o

A

I l l u m i n a t i o n w i t h h υ > Eg

c reates ex cess ho les :

pn = p

n o+ Δ p

n = Δ n

n

B

I l l u m in a t i o n

I n dark af ter

i l lum inatio n . E x cess

ho les are d isappearing

by reco m binatio n .

C


Fig. 5.26: Illumination of an n-type semiconductor results in excesselectron and hole concentrations. After the illumination, therecombination process restores equilibrium; the excess electronsand holes simply recombine.

Carrier creation followed by recombination


–

–

–

–

–

–

–

–

–

–

– ––

–

–

– –––

–

–

––

–

––

–

–

––

–+

+

–

–

–

–

–

–

––

–

–

– ––

–

–

––

––

–

–

– ––

–

–

–

–

–

–

–+

+

+

+

++

+ +

++

+

+

++

++

+

+

–

–

–

–

–

–

––

–

–

– ––

–

–

––

––

–

–

– ––

–

–

–

–

–

–

–

++

+ +

+

+

++

++

n - t y p e s e m i c o n d u c t o r i n

t h e d a r k .

pn

= pn o

< < nn o

A

I l l u m i n a t i o n w i t h h υ > Eg

c reates ex cess ho les :

pn = p

n o+ Δ p

n = Δ n

n

B

I l l u m in a t i o n

I n dark af ter

i l lum inatio n . E x cess

ho les are d isappearing

by reco m binatio n .

C


Fig. 5.26: Illumination of an n-type semiconductor results in excesselectron and hole concentrations. After the illumination, therecombination process restores equilibrium; the excess electronsand holes simply recombine.

Carrier creation followed by recombination

Mostly majority carriers in the dark

Almost equalCarrier concnsIn light

The extra minority carriers recombine once the generation source is removed.

How quickly do the carriers recombine?


Minority carrier lifetime h for n-type

h = average time a hole exists in the valance band from its generation until its recombination

And so 1/ h is the average probability (per unit time) that a hole will recombine with an electron.

h depends on impurities, defects and temperature.

The recombination process in a real semiconductor usually involves a carrier being localized at a recombination center.

can be short (nanoseconds) allowing fast response (e.g. switch)or slow (seconds) for a photoconductor or solar cell


Excess Minority Carrier Concentration

Δpn = excess hole (minority carrier) concentration,

t = time,

Gph = rate of photogeneration,

h = minority carrier lifetime (mean recombination time)

h = average time a hole exists in the valance band from its generation until its recombination

dΔpn

dt=Gph −

Δpn

τh


I l l u m i n a t i o n

0Time, t

p no

G ph

t off

pno+ Δ pn(∞)

t'

hG ph

0

G and pn(t)

Δ p n( t' ) = Δ p n( 0 ) e x p ( - t' / h)

Fig. 5.27: Illumination is switched on at time t = 0 and then off at t = toff.

The excess minority carrier concentration, Δ pn(t) rises exponentially to itssteady state value with a time constant h. From toff, the excess minoritycarrier concentration decays exponentially to its equilibrium value.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Carrier concentration versus time with pulsed illumination

t’ is time after illumination is removed


W

D

L

V Ip h

L i g h t


Fig. 5.28: A semiconductor slab of length L, width W and depth D isilluminated with light of wavelength λ. Iph is the steady statephotocurrent.

Continuous illumination provides increased conductivity

Often used as a switchor motion detector


Carrier motion in a concentration gradient

How can we describe how charge carriers move is response to a concentration gradient?

It is just simple diffusion; i.e Ficks first law; but of course it is with charged ‘particles’

Diffusion current density = (charge) x fluxWhere Flux = ΔN/A Δt or #of charge carriers passing unit area per unit time

Ficks first law relates flux to concentration gradientFlux =e= -De dn/dx for electronsWhere De = diffusion coefficient for electrons



Diffusion current density = (charge) x fluxWhere Flux = ΔN/A Δt or #of charge carriers passing unit area per unit time

Ficks first law relates Flux to concentration gradientFlux =e= -De dn/dx for electrons

So the current density (due to diffusion) becomes: Diffusion current density = (charge) x flux

note: diffusion is away from high concentrationJDe= -ee for charge carriers due to diffusion, orJDe= eDe dn/dx for electrons and JDh= -eDe dp/dx for holes

( is gamma)



So, now for the total current we have both drift and diffusion currents for both types of charge carriers!

Jtotal = Jdrift + Jdiffusion Jtotal = Jdrift + Jdiffusion

Je= eneEx + eDe dn/dx for electrons, and

Jh= ephEx - eDh dp/dx for holes.

We relate the diffusion coefficient to temperature and mobility by the Einstein relationship:De = (kT/e) e or De/e = (kT/e) and Dh/h = (kT/e)

We also find that D=l2/ where l is the mean free path, and is the mean time between scattering events.


Definition of Particle Flux

= particle flux, ΔN = number of particles crossing A in Δt, A = area, Δt = time interval

Γ =ΔNAΔt

Definition of Current Density

J = electric current density,

Q = charge of the particle, = particle flux

J =QΓ


n1

n2

xo

xo

+ lxo

- l

n ( x , t )

x

n(x,t)

x

N e t e l e c t r o n d i f f u s i o n f l u x

E l e c t r i c c u r r e n t

xo+ lx

o- l

xo


Fig. 5.29: (a) Arbitrary electron concentration n(x,t) profile in asemiconductor. There is a net diffusion (flux) of electrons fromhigher to lower concentrations. (b) Expanded view of twoadjacent sections at xo. There are more electrons crossing xocoming from left (xo-l ) than coming from right (xo+l ).

(a) (b)


Fick’s First Law: relating diffusion current to carrier concentration gradient.

e = electron flux, De = diffusion coefficient of electrons, dn/dx = electron concentration gradient

Γe =−De

dndx

JD, e = electric current density due to electron diffusion,

e = electron flux, e = electronic charge,

De = diffusion coefficient of electrons,

dn/dx = electron concentration gradient

Electron Diffusion Current Density

J D,e =−eΓe =eDe

dndx

Where:


p(x,t)

x

N e t h o l e d i f f u s i o n f l u x

E l e c t r i c c u r r e n t

xo

+ lxo- l

xo


Fig. 5.30: Arbitrary hole concentration p(x,t) profile in asemiconductor. There is a net diffusion (flux) of holes from higherto lower concentrations. There are more holes crossing xo comingfrom left (xo-l ) than coming from right (xo+l ).

Carrier diffusion away from high concentration

holes in this p-type example


Ex

n - T y p e S e m i c o n d u c t o r

L i g h t x

H o l e D r i f t

H o l e D i f f u s i o n

E l e c t r o n D r i f t

E l e c t r o n D i f f u s i o n

S e m i t r a n s p a r e n t e l e c t r o d e


Fig. 5.31: When there is an electric field and also a concentrationgradient, charge carriers move both by diffusion and drift. (Ex is theelectric field.)

Carrier motion: via diffusion (due to concn gradient) and drift (due to electric field)

Both diffusion and drift occur in semiconductors.

Note here that holes (minority carriers) drift and diffuse in the same direction; but electrons (majority carriers) do not!

With light we alter minority carrier concn


Hole Diffusion Current Density

JD, h = electric current density due to hole diffusion, e = electronic charge, h = hole flux, Dh = diffusion coefficient of holes, dp/dx = hole concentration gradient€

JD,h = eΓh = −eDh

dp

dx

Total Electron Current Due to Drift and Diffusion

Je = electron current due to drift and diffusion, n = electron concentration, e = electron drift mobility, Ex = electric field in the x direction, De = diffusion coefficient of electrons, dn/dx = electron concentration gradient

J e =enμeEx +eDe

dndx


Total Currents Due to Drift and Diffusion

Jh = hole current due to drift and diffusion, p = hole concentration,

h = hole drift mobility, Ex = electric field in the x direction,

Dh = diffusion coefficient of holes,

dp/dx = hole concentration gradient

J h =epμhEx −eDh

dpdx

J e =enμeEx +eDe

dndx

Je = electron current due to drift and diffusion, n = electron concentration

e = electron drift mobility, Ex = electric field in the x direction,


dn/dx = electron concentration gradient Jtotal = Jh+Je



e = electron drift mobility,

Dh = diffusion coefficient of the holes,

h = hole drift mobility

De

μe

=kTe

andDh

μh

=kTe

Einstein Relation: diffusion coefficient and mobility are related!


E x p o s e d

A s+

D o n o r

n2

n1

D i f f u s i o n F l u x

D r i f t

N e t c u r r e n t = 0

Ex


Fig. 5.32: Non-uniform doping profile results in electron diffusiontowards the less concentrated regions. This exposes positively chargeddonors and sets up a built-in field Ex . In the steady state, the diffusion ofelectrons towards the right is balanced by their drift towards the left.

Vo

Carrier diffusion due to doping level gradient.This is a common device fabrication step.

Diffusion occurs until an electric field builds up!

We call this the built-in potential.

Note: the As+

are fixed,non-mobile charges!

· represents electrons

(majority carriers in this case)


Built-In Potential and Concentration

V2 = potential at point 2, V1 = potential at point 1,

n2 = electron concentration at point 2,

n1 = electron concentration at point 1

V2 −V1 =kTe

lnn2

n1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Built-In Field in Nonuniform Doping

Ex = electric field in the x direction,

b = characteristic of the exponential doping profile,

e = electronic charge .

Ex =

kTbe

E x p o s e d

A s+

D o n o r

n2

n1

D i f f u s i o n F l u x

D r i f t

N e t c u r r e n t = 0

Ex


Fig. 5.32: Non-uniform doping profile results in electron diffusiontowards the less concentrated regions. This exposes positively chargeddonors and sets up a built-in field Ex . In the steady state, the diffusion ofelectrons towards the right is balanced by their drift towards the left.

Vo


The SUNY buffalo Applets on conventional semiconductors:http://jas.eng.buffalo.edu/

QuickTime™ and aTIFF (LZW) decompressor


http://jas.eng.buffalo.edu/education/system/cdrw/index.html


What’s next

Optoelectronics as examples of nanoelectronics Light emission from quantum well devices Light absorption in solar cells/photovoltaics

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Nanoscience and Nanotechnology Lecture 19