1 TUTORIAL I DEVICE PHYSICS, CHARGE TRANSPORT, APPLICATIONS AND PROCESSING IN ORGANIC ELECTRONICS Nir Tessler Devin Mackenzie March 28, 2005 1:30 – 5:00.

1

TUTORIAL I

DEVICE PHYSICS, CHARGE TRANSPORT, APPLICATIONS AND PROCESSING IN ORGANIC

ELECTRONICS

Nir TesslerDevin Mackenzie

March 28, 2005 1:30 – 5:00 PM

2

MRS SPRING 2005 TUTORIAL I

DEVICE PHYSICS, CHARGE TRANSPORT, APPLICATIONS AND PROCESSING IN ORGANIC

ELECTRONICS

PART 1

DEVICE PHYSICS and CHARGE TRANSPORT

Nir Tessler EE Dept. Technion

www.ee.technion.ac.il/nir

3

Organic Semiconductors

1. Semiconductor

2. High band gap

3. Low mobility

4. Molecular

You are holding ~60 slides but we will look together only at part of them. Watch for the slide number

Nir Tessler, EE Dept. Technionwww.ee.technion.ac.il/nir

4

If the band-gap is high Insulator

EFi

EF

EFi

EC

EV

EF=EFi

Intrinsic N-type P-type

EF

EFM

Metal

Reminder

5

Semiconductor

6

Semiconductor

Metal

Metal

EFM

E0- Vacuum level

sM

EF

EC

EV

= work function

The (average) energy required to extract an electron.

Isolated Material(not in equilibrium)

0B C ME

The energy required to “lift” an electron from the metal to the semiconductor

7

Semiconductor

Metal

EFM

E0

sM

EF

EC

EV

Making contact(creating equilibrium)

Is there any electronic interaction?

8

Semiconductor

Metal

Making contact

EF

EV

EC

E0

EFM

Charging a capacitor to a voltage of: M sV

V

9

What assumptions did we use?

0

+Q

-Q

a. There exist equilibrium between M and SC Fermi level is continuous.

b. The metal is an infinite reservoir (attaching the SC is a small perturbation)

c. The potential is continuous (no dipoles )

10

Semiconductor

Metal

Metal

EFM

E0- Vacuum level

sM

EF

EC

EV

Isolated Materials

What will occur after making contact?

Will the semiconductor become metallic?Will the entire volume be chemically reduced?

11

Metal

EFM

E0

sM

EF

EC

EV

Isolated Material(no equilibrium)

Metal

EFM EF

EC

EV

Connected(equilibrium)

ultra-thin

Ultra-thin ~ nm scale

12

Metal

EFM EF

EC

EV

Ultra-thin

Metal

EFM EF

EC

EV

Interface dipole

Since the ultra-thin region is negligible in size it is not drawn:

Conclusion: the metal workfunction can not be above (below) the conduction (valence) band.

13

5.2 5.2

3.5

2.7

What will happen after making contact

5.2-3.5=1.7

14

E

X

ECBarrier

Ec

Thermionic Emission

Basic Assumptions:1. Emission from A to B does not depend on emission from B to A but

only on the concentration in A and B respectively (there doesn’t have to be equilibrium across the interface).

2. The charge density in the metal is fixed (infinite reservoir)

* 2 /kT qM SJ A T e

Current

15

In Low mobility semiconductors (organics) the emission rates from metal to organic and back are much larger then the current flowing in the device:

1. There is equilibrium at the contact interface2. The thermionic emission process is not important but for ensuring

equilibrium.

What may change the above?

What may slow the emission across the interface?

The presence of a thin insulating layer will make the crossing from the metal to the organic (tunneling) very slow and it will become the rate limiting factor (i.e. break the equilibrium).

22

Now the charges are in

• How do they move?

23

Molecular Localization

• Conjugated segments “States”• Charge conduction non coherent hopping

x

24

What are the important factors?

1. Energy difference

2. Distance

3. Similarity of the Molecular structures

1. What is the statistics of energy-distribution?

2. What is the statistics of distance-distribution?

3. Is it important to note that we are dealing with molecular SC? Do we need to use the concept of polaron?

25

Detailed Equilibrium

1 1i j ij j i jif E f E f E f E

exp

j iij

ji kT

, 1 1 exp /i if E E kT

exp /

1

j i j iij j it

E E kT E EE E

else

0

exp / exp

1

j i j iij

E E kT E E

else

ijR

Anderson:

Ei

Ej

26

Molecular Nature of the Envelope Function

exp /

1

j i j iij j it

E E kT E EE E

else

The polaron picture:

27

-2000

0

2000

4000

6000

8000

1 104

1.2 104

-40 -20 0 20 40

E

Q

Elastic energy:

2elastE BQ

cc

c

c

cc

cc

cc

cc

c

c

cc

cc

cc

Equilibrium

Stretched

Squeezedc

cc

c

cc

cc

cc

cc

c

c

cc

cc

cc

cc

c

c

cc

cc

cc

cc

c

c

cc

cc

cc

cc

c

c

cc

cc

cc

cc

c

c

cc

cc

cc

2elastE BQ

Simplistic approach

Q is a molecular configuration coordinate

28

Q0

E0spring

E=E0+B(Q-Q0)2

Elastic Energy (spring)

-1000

0

1000

2000

3000

4000

5000

6000

-30 -20 -10 0 10 20 30 40

E

Q Configuration coordinates

30

Q0

E0spring

-1000

0

1000

2000

3000

4000

5000

6000

-30 -20 -10 0 10 20 30 40

E

Q

A

A*

A

Q

A*

E0t= E0spring+

mgQ0

E0= E0t +

BQ2 - mgQ

31

2 22

22n

e

E nm L

2 22

32

2n

e

dE n dL F dLm L

L L + dL

Stretch modeEn

En +dEn

For small variations in the “size” of the molecule the electron phonon contribution to the energy of the electron is linear with the displacement of the molecular coordinates.

For -conjugated the atomic displacement is ~0.1A and F=2-3eV/A.

The general formalism: Ee-ph=-AQ

The electronic equivalent

32

-1000

0

1000

2000

3000

4000

5000

6000

-30 -20 -10 0 10 20 30 40

E

Q

Linear electron-phonon interaction:

e phE AQ

0

20

elast e phE E E E

E E BQ AQ

2 22min min 2 2 4b

A A AE BQ AQ B A

B B B

bE

minQ

The system was stabilized by E through electron-phonon interaction Polaron binding energy

0E

Q

0 0 _ 0 _ 0 _ ; elast e e nE E E E E

0 elast e phE E E E 0 elastE E E

34

2

exp exp2 2 8

j i j ibi j i j

b

ER

kT kT kTE

Accounts for difference between the equilibrium energies

If initial and final energies are different:(In disordered materials E0_e is not identical for the two molecules)

35

AqW

kTphononR e P

Average attempt frequency Activation of the

molecular conformation

Probability of electron to move (tunnel) between two molecules that are in their “best” conformation

Requires the “presence” of phonons.Or the occupation of the relevant phonons should be significant

36

Bosons: 1 1( )

1 1Bose Einstein E h

kT kT

f E

e e

What will happen if T<Tphonon/2 ?

1 1( )

11effective phononeffective h T

kT T

f h

ee

The relevance to our average attempt frequency:

The molecules will not reach the “best” conformation that was accessible at higher (room) temperature

New activation energy

New attempt frequency

Typical temperature at which the transport mechanism changes is 150-200k

37

Sites-Energy Statistics(the most popular ones)

• Gaussian DOS

• Exponential DOS

• Completely ignore the issue

38

What is the statistical Energy-Distribution?

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0 0

exptNgkT kT

2

0exp2 2

VNg

Gaussian

Exponential

3910-5

10-4

10-3

10-2

10-1

100

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

0.2

0.4

0.6

0.8

1

1.2

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

The two look very different

BUT – a single experiment (typically) samples only a small region of the DOS

40

Power LawT0=450k =5.5kT

M. Vissenberg and M. Matters, "Theory of the field-effect mobility in amorphous organic transistors," Physical Review B, vol. 57, pp. 12964-12967, 1998

Y. Roichman, Y. Preezant, and N. Tessler, "Analysis and modeling of organic devices," Physica Status Solidi a-Applied Research, vol. 201, pp. 1246-1262, 2004

Charge Mobility & Charge Density

0 0

exptNgkT kT

2

0exp2 2

VNg

0 1T

Tn

For low enough density:

0FE kT

(Nt=1020cm-3 -> n<5x1018)

=0.73-1.17 exp1.65kT kT

n

For high enough density:

10 kTVn N

(=5kT, Nt=1020cm-3 -> n>1x1015)

42

hhhhh ndx

dDEnJ

dx

dnDEn h

hhh dV

dn

nD

Edx

dn

nD h

hh

h

hhh

11

Current continuity Eq.

In the absence of external force (J=0)

Derivation of the Generalized Einstein Relation

f

h

hhh dE

dn

nD

1

Equilibrium conditions(existence of a Fermi level + constant temperature)

GeneralizedEinstein-Relation(Ashcroft, solid state physics)

43

1

2

3

4

5

1014 1015 1016 1017 1018 1019

0

0.5

1

1.5

2

10-6 10-5 10-4 10-3 10-2 10-1

=5kT

=4kT

(A) =7kTE

nhan

cem

ent

of E

inst

ein

Rel

atio

n

T0=400kT0=500k

T0=600k

Relative Charge Density

0 0

exptNgkT kT

2

0exp2 2

VNg

Generalized Einstein Relation

Y. Roichman and N. Tessler, "Generalized Einstein relation for disordered semiconductors - Implications for device performance," Applied Physics Letters, vol. 80, pp. 1948-1950, 2002

Charge Diffusion & Charge Density

44

1

1.5

2

2.5

3

3.5

430 40 50 60 70 80

=7

=4

=5

Enh

ance

men

t of

Ein

stei

n R

ela

tion

1/kT

1

1.5

2

2.5

3

3.5

4

30 40 50 60 70 80

=5

T0=400

T0=500

T0=600

Calc for DOS Filling at RT = 0.01

1

1.5

2

2.5

3

3.5

430 40 50 60 70 80

=7

=4

=5

Enh

ance

men

t of

Ein

stei

n R

ela

tion

1/kT

1

1.5

2

2.5

3

3.5

4

30 40 50 60 70 80

=5

T0=400

T0=500

T0=600

Calc for DOS Filling at RT = 0.01

1

1.5

2

2.5

3

3.5

4

30 40 50 60 70 80

=5

T0=400

T0=500

T0=600

Calc for DOS Filling at RT = 0.01

0 exp 1qV

J JnkT

0 exp 1qV

J JkT

Or:

Diode

=4kT T0=450k

45

Extracting Mobility

• Analysis of LEDs

• Analysis of FETs

The disorder parameter is an established important featureCan we extract it?

46

LEDs

2

0 exp 2.25E c EkT

Gaussian DOS at low density limit

H. Bassler, Phys. Stat. Sol. (b), vol. 175, pp. 15-56, 1993

2

3

9

8SCL

VJ

d

20

1 3( )

2

d VP P x dx

d ed

( )2

3

4d

VP

ed

Average Density

Density at the exit contact

Need a formalism that accounts for both electric field and density

47

W

L

Vg

Si

SiO2

W

L

Vg

Conductor

Insulator

- conjugated

Source Drain

W

L

Vg

Si

SiO2

W

L

Vg

Conductor

Insulator

Source Drain

- conjugated

y

xz

CI-FET

48

Mo

bili

ty (

a.u

.)

=4kT=7kT

Charge Density & Electric Field Dependence(Gaussian DOS)

The exponential prefactor depends on as well as Charge Density

Y. Roichman, Y. Preezant, N. Tessler, Phys. Stat. Sol. 2004

10-5

10-4

10-3

10-2

10-1

0 200 400 600 800

(Electric Field)0.5

2x1014

1015

4x1016

1018

1019

10-5

10-4

10-3

10-2

10-1

0 200 400 600 800

1014

7x10161018

1019

(Electric Field)0.5

51

Extracting Mobility - FETs

2

2DS

DS ins GS T DS ins DSGS GS

VW WI C V V V C V

V V L L

But 100% is not always critical

10-11

10-10

10-9

10-8

10-7

0

5 10-6

1 10-5

1.5 10-5

2 10-5

2.5 10-5

3 10-5

0 5 10 15 20 25

Gate Voltage

Cur

rent

Mob

ility

(cm

2 V-1

s-1)

10-11

10-10

10-9

10-8

10-7

0

5 10-6

1 10-5

1.5 10-5

2 10-5

2.5 10-5

3 10-5

0 5 10 15 20 25

Gate Voltage

Cur

rent

Mob

ility

(cm

2 V-1

s-1)

0 100

2 10-10

4 10-10

6 10-10

8 10-10

1 10-9

1.2 10-9

1.4 10-9

4 6 8 10 12 14 16 18 20

V

Gate VoltageC

urre

nt D

eriv

ativ

e0 100

2 10-10

4 10-10

6 10-10

8 10-10

1 10-9

1.2 10-9

1.4 10-9

4 6 8 10 12 14 16 18 20

V

Gate VoltageC

urre

nt D

eriv

ativ

e0 100

2 10-10

4 10-10

6 10-10

8 10-10

1 10-9

1.2 10-9

1.4 10-9

4 6 8 10 12 14 16 18 20

V

Gate VoltageC

urre

nt D

eriv

ativ

e

52

s g GS Tn x C q V V v x

DS s

dv xI Wqn x x

dx

0 0

L L

DS s

dv xI dx Wqn x x dx

dx

0

L

DS g GS T

dv xWI C V V v x x dx

L dx

_

0

_

0

GS T

GS T

DS

GS T

V

DS Lin g GS T

V V

DS Sat g GS T

V V v

V V v

WI C V V v dv

L

WI C V V v dv

L

Deriving the expressions for charge density dependent mobility

53

_

0

_

0

GS T

GS T

DS

GS T

V

DS Lin g GS T

V V

DS Sat g GS T

V V v

V V v

WI C V V v dv

L

WI C V V v dv

L

0

Nn

GS T n GS Tn

V V v V V v

2 2

_0 2

Nn nn

DS Lin g GS T GD Tn

WI C V V V V

L n

2

_0 2

Nnn

DS sat g GS Tn

WI C V V

L n

By making the best fit one finds: 1. Density dependent mobility

2. Threshold voltage (+-)

This procedure does NOT assume a given DOS

Shape(i.e. general procedure)

54

10-6

10-5

10-4

0.1110

Mob

ility

(cm

2v-1

s-1)

|VG-V

T-V(y)|

10-3

10-2

10-1

100

101

102

-10 -8 -6 -4 -2

So

urce

Cu

rre

nt (

nA

)

Gate-Source Bias (V)

-1-2-4-8

-1-2

-4-8

-32

( )k

G T yV V V

2k=0.85±0.1

=0.73-1.17 exp1.65kT kT

˜ 5kT=130meV

VDS

VDS

55

Einstein relation is larger then 1 and depends on the charge density

Accounting for it:

1. Charge Density can not exceed the DOS

2. Channel depth does not go below 1-2 monolayer

To evaluate charge density transfer Vg to density in cm-2 and then to cm-3

It is too fundamental to be ignored!

Vg

W

LSiO2

W

L

Insulator

Source Drain

kT kT

Simple to implement

VG-VT

( )N

1

1015

1016

1017

1018

1019

1020

1021

0.1

1

10

100

0 5 10 15 20

Ch

arge

De

nsity

(cm

-3)

Ch

ann

el D

epth

(n

m)

VG-VT

( )N

1

1015

1016

1017

1018

1019

1020

1021

0.1

1

10

100

0 5 10 15 20

Ch

arge

De

nsity

(cm

-3)

Ch

ann

el D

epth

(n

m)

1015

1016

1017

1018

1019

1020

1021

0.1

1

10

100

0 5 10 15 20

Ch

arge

De

nsity

(cm

-3)

Ch

ann

el D

epth

(n

m)

56

EC

EV

EF

EC

EV

Intrinsic

Gate Voltage

Threshold Voltage

57

EC

EV

EF

EC

EV

Intrinsic

EF

VGGate Voltage

Linear

Threshold Voltage

58

EC

EV

EF

EC

EV

IntrinsicEF

VGGate Voltage

Exp

onen

t

Linear

Sub-Threshold

Threshold Voltage – disordered material

59

Models for Contact injection:[1] V. I. Arkhipov, E. V. Emelianova, Y. H. Tak, and H. Bassler, "Charge injection into light-emitting

diodes: Theory and experiment," Journal of Applied Physics, vol. 84, pp. 848-856, 1998.[2] V. I. Arkhipov, U. Wolf, and H. Bassler, "Current injection from metal to disordered hopping system. II. Comparison between analytic theory and simulation," Phys. Rev. B, vol. 59, pp. 7514-7520, 1999[3] M. A. Baldo and S. R. Forrest, "Interface-limited injection in amorphous organic semiconductors - art. no. 085201," Physical Review B, vol. 6408, pp. 5201-+, 2001.[4] M. A. Baldo, Z. G. Soos, and S. R. Forrest, "Local order in amorphous organic molecular thin films," Chemical Physics Letters, vol. 347, pp. 297-303, 2001[5] Y. Preezant and N. Tessler, "Self-consistent analysis of the contact phenomena in low- mobility semiconductors," Journal of Applied Physics, vol. 93, pp. 2059- 2064, 2003.[6] Y. Preezant, Y. Roichman, and N. Tessler, "Amorphous Organic Devices - Degenerate Semiconductors," J. Phys. Cond. Matt., vol. 14, pp. 9913–9924, 2002.[7] Y. Roichman, Y. Preezant, and N. Tessler, "Analysis and modeling of organic devices," Physica

Status Solidi a-Applied Research, vol. 201, pp. 1246-1262, 2004[8] J. C. Scott and G. G. Malliaras, "Charge injection and recombination at the metal-organic interface," Chemical Physics Letters, vol. 299, pp. 115-119, 1999.[9] T. van Woudenbergh, P. W. M. Blom, M. Vissenberg, and J. N. Huiberts, "Temperature dependence of the charge injection in poly-dialkoxy-p-phenylene vinylene," Applied Physics Letters, vol. 79, pp. 1697-1699, 2001 [10] J. H. Werner and H. H. Guttler, "Barrier Inhomogeneities at Schottky Contacts," Journal of Applied Physics, vol. 69, pp. 1522-1533, 1991

60

Transport models[1] W. D. Gill, "Drift mobilities in amorphous charge-transfer complexes of trinitrofluorenone and poly-n-vinylcarbazole," J. Appl. Phys., vol. 43, pp. 5033, 1972.[2] M. Van der Auweraer, F. C. Deschryver, P. M. Borsenberger, and H. Bassler, "Disorder in Charge-Transport in Doped Polymers," Advanced Materials, vol. 6, pp. 199-213, 1994.[3] R. Richert, L. Pautmeier, and H. Bassler, "Diffusion and drift of charge-carriers in a random potential - deviation

from einstein law," Phys. Rev. Lett., vol. 63, pp. 547-550, 1989.[4] V. I. Arkhipov, P. Heremans, E. V. Emelianova, G. J. Adriaenssens, and H. Bassler, "Weak-field carrier hopping in

disordered organic semiconductors: the effects of deep traps and partly filled density-of-states distribution," Journal of Physics-Condensed Matter, vol. 14, pp. 9899-9911, 2002.[5] M. Vissenberg and M. Matters, "Theory of the field-effect mobility in amorphous organic transistors," Physical Review B, vol. 57, pp. 12964-12967, 1998.[6] D. Monroe, "Hopping in Exponential Band Tails," Phys. Rev. Lett., vol. 54, pp. 146-149, 1985.[7] H. Scher, M. F. Shlesinger, and J. T. Bendler, "TIME-SCALE INVARIANCE IN TRANSPORT AND RELAXATION,"

Physics Today, vol. 44, pp. 26-34, 1991.[8] H. Scher and E. M. Montroll, "Anomalous transit-time dispersion in amorphous solids," Phys. Rev. B, vol. 12, pp.

2455–2477, 1975.[9] E. M. Horsche, D. Haarer, and H. Scher, "Transition from dispersive to nondispersive transport: Photoconduction

of polyvinylcarbazole," Phys. Rev. B, vol. 35, pp. 1273-1280, 1987.[10] Y. Roichman, Y. Preezant, and N. Tessler, "Analysis and modeling of organic devices," Physica Status Solidi a-

Applied Research, vol. 201, pp. 1246-1262, 2004.[11] Y. Roichman and N. Tessler, "Generalized Einstein relation for disordered semiconductors - Implications for device performance," Applied Physics Letters, vol. 80, pp. 1948-1950, 2002.[12] Y. N. Gartstein and E. M. Conwell, "High-Field Hopping Mobility in Molecular-Systems with Spatially Correlated

Energetic Disorder," Chemical Physics Letters, vol. 245, pp. 351-358, 1995.[13] H. C. F. Martens, P. W. M. Blom, and H. F. M. Schoo, "Comparative study of hole transport in poly(p- phenylene

vinylene) derivatives," Physical Review B, vol. 61, pp. 7489-7493, 2000 [14] S. V. Rakhmanova and E. M. Conwell, "Electric-field dependence of mobility in conjugated polymer films," Applied Physics Letters, vol. 76, pp. 3822-3824, 2000[15] R. A. Marcus, "Chemical + Electrochemical Electron-Transfer Theory," Annual Review of Physical Chemistry, vol.

15, pp. 155-&, 1964.[16] R. A. Marcus, "Theory of Oxidation-Reduction Reactions Involving Electron Transfer .5. Comparison and Properties

of Electrochemical and Chemical Rate Constants," Journal of Physical Chemistry, vol. 67, pp. 853- &, 1963.[17] D. Emin, "Small polarons," Phys. Today, vol. 35, pp. 34-40, 1982

61

Transport in FETs

[1] S. M. Sze, Physics of Semiconductor Devices. New York: Wiley, 1981.[2] A. A. Muhammad, A. Dodabalapur, and M. R. Pinto, "A two-dimensional simulation of organic transistors," IEEE trans. elect. dev., vol. 44, pp. 1332-1337, 1997.[3] G. Horowitz, P. Lang, M. Mottaghi, and H. Aubin, "Extracting parameters from the current-voltage characteristics of field-effect transistors," Advanced Functional Materials, vol. 14, pp. 1069-1074, 2004.[4] G. Horowitz, M. E. Hajlaoui, and R. Hajlaoui, "Temperature and gate voltage dependence of hole mobility in polycrystalline oligothiophene thin film transistors," J. Appl. Phys., vol. 87, pp. 4456-4463, 2000.[5] Y. Roichman and N. Tessler, "Structures of polymer field-effect transistor: Experimental and numerical analyses," Applied Physics Letters, vol. 80, pp. 151-153, 2002.[6] Y. Roichman, Y. Preezant, and N. Tessler, "Analysis and modeling of organic devices," Physica Status Solidi a- Applied Research, vol. 201, pp. 1246-1262, 2004.[7] S. Shaked, S. Tal, Y. Roichman, A. Razin, S. Xiao, Y. Eichen, and N. Tessler, "Charge density and film morphology dependence of charge mobility in polymer field-effect transistors," Advanced Materials, vol. 15, pp. 913-+, 2003.[8] N. Tessler and Y. Roichman, "Two-dimensional simulation of polymer field-effect transistor," Applied Physics

Letters, vol. 79, pp. 2987-2989, 2001.[9] L. Burgi, R. H. Friend, and H. Sirringhaus, "Formation of the accumulation layer in polymer field-effect transistors," Applied Physics Letters, vol. 82, pp. 1482-1484, 2003.[10] L. Burgi, H. Sirringhaus, and R. H. Friend, "Noncontact potentiometry of polymer field-effect transistors," Applied Physics Letters, vol. 80, pp. 2913-2915, 2002.[11] S. Scheinert and G. Paasch, "Fabrication and analysis of polymer field-effect transistors," Physica Status Solidi

a-Applied Research, vol. 201, pp. 1263-1301, 2004.[12] E. J. Meijer, C. Tanase, P. W. M. Blom, E. van Veenendaal, B. H. Huisman, D. M. de Leeuw, and T. M. Klapwijk, "Switch-on voltage in disordered organic field-effect transistors," Applied Physics Letters, vol. 80, pp. 3838-3840, 2002.[13] C. Tanase, E. J. Meijer, P. W. M. Blom, and D. M. de Leeuw, "Unification of the hole transport in polymeric field-effect transistors and light-emitting diodes," Physical Review Letters, vol. 91, pp. 216601, 2003.[14] G. Paasch and S. Scheinert, "Scaling organic transistors: materials and design," Materials Science-Poland, vol. 22, pp. 423-434, 2004

62

10-6

10-5

10-4

0.1110

Mo

bilit

y (c

m2v-1

s-1)

|VG-V

T-V(y)|

-1-2

-4-8

-3

2( )G T yP V V V

=0.73-1.17 exp1.65kT kT

2k=0.85±0.1

≈5kT=130meV

VDS

kP

Our Model (for low field limit):

In FETs:

It is very important to measure down to very low charge densityAND not force a single power law

10-6

10-5

10-4

0.1110

Mob

ility

(cm

2v-1

s-1)

|VG-V

T-V(y)|

0.85 --> =5kT

0.4 --> =3.3kT

63

-1000

0

1000

2000

3000

4000

5000

6000

E

Q*

A system that is made of two identical molecules

As the molecules are identical it will be symmetric (charge on 1 is equivalent to charge on 2)

Wa

A B A B

Reactants Products

(Room Temperature)

64

-1000

0

1000

2000

3000

4000

5000

6000

E

Q

A system that is made of two identical molecules

At low temperature the probability to acquire enough energy to bring the two molecules to the top of the barrier is VERY low.In this case the electron may be exchanged at “non-ideal” configuration of the atoms or in other words there would be tunneling in the atoms configuration (atoms tunnel!).

Wa

A B A B

Would the electron transfer rate still follow exp(-qWa/kT)

(Low Temperature)

65

transit

Channel

t

LWQ

time

eChI

arg#

DSDStransit V

L

LVL

E

L

v

Lt

2

TGTGinschannel VVVVCQ ;

DSTGins

DS

TGins

VVVCL

WI

VL

LWVVCI

2

1

TGins

DS

DSON VVC

L

W

I

VR

G

S D

Trans-Resistor = Transistor

Assumed channel depth is negligible compared to insulator thickness so that C=COX (and VDS is small).

Assumed is constant

66

DSTGoxDS VVVCL

WI

1

TGox

DS

DSON VVC

L

W

I

VR

G

S D

B

IDS

VDS

Vg1>VT

Vg2>Vg1

Vg3>Vg2

Vg4>Vg3

Trans-Resistor = Transistor

67

0V 0V0V 0V - 3V -1.5V

0V - 5V 0V -7V

Region with no charge where all voltage beyond VG drops upon.

- 5V - 5V

- 5V(a) (b)

(c) (d)

- 5V Gate

Source Drain

y

x

-2.5V -2.5V

68

2

2D

DToxDS

VVVVgC

L

WI TD VVgV 0

2

2 ToxDS VVgCL

WI DT VVVg

IDS

VDS

Vg>VT

TD VVgV

Ranges

Saturation

Linea

r