iee5328 nanodevice transport theory and computational tools

IEE5328 Nanodevice Transport Theory and Computational Tools

Prof. Ming-Jer ChenDept. Electronics EngineeringNational Chiao-Tung UniversityMay 1, 2013

Lecture 7:

Effective Mobility in 2DEG and 2DHG of Long-Channel Thick Gate-Oxide Planar Bulk MOSFETs: Microscopic Calculation

(Advanced Device Physics with emphasis onhands-on calculations)

1IEE5328 Prof. MJ Chen NCTU

Two-Dimensional Hole Gas

IEE5328 Prof. MJ Chen NCTU 2


Thick Oxides

No Stress

Stress

National Chiao-Tung University Nano Electronics Physics Lab. 5

Kubo-Greenwood Formula

vxμ is the group velocity of subband μ along x-direction and f0 is the

equilibrium Fermi distribution.

The mobility formula in electron case

is no longer valid for the hole case due to the failure of the effective mass approximation.Thus, the hole mobility formula as derived from the Boltzmann transport equation (BTE) must be used:


Kubo-Greenwood Formula

Group Velocity (cm/sec)

k<100>

(cm-1)

k<010> (

cm

-1)

-4 -2 0 2 4

x 107

-4

-3

-2

-1

0

1

2

3

4x 10

7

-1

-0.5

0

0.5

1

x 108

k<100>

(1/cm)

k<

01

0> (

1/c

m)

-4 -2 0 2 4

x 107

-4

-3

-2

-1

0

1

2

3

4x 10

7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1st SubbandSi (001) @ 300K FS=1MV/cm


Physical Models② Acoustic Phonon, Optical Phonon, and Surface Roughness

Scattering

; ;

ω

q

Longitudinal Transverse

Acoustic

Optical

Si Phonon Dispersion

H’Hole-AC Phonon

H’Hole-OP PhononΔE≈1meV within 1/2 Brillouin zone

ΔE≈61.2meV

Phonon wave vector



Scattering

The acoustic deformation potential Dac, is strongly connected to Bir-Pikus deformation potentials. According to Lawaetz, Dac can be formulated as

; ;

c11, c12, and c44 are the elastic coefficients

H’Hole-AC Phonon

Small vibration termElastic scattering

Isotropic approximation



Scattering

According to Wiley and Costato and Reggiani, the optical deformation potential Dop can have the following formalism:

; ;

Average sound velocity

H’Hole-OP Phonon

Small vibration termInelastic scattering (61.2meV)

Isotropic approximation

ωop : optical phonon frequency; nop : Bose occupation factor of optical phonons



Scattering

; ;

H’Hole-SR

Small vibration term

Elastic scattering

p+ p+

Gate

Inversion Layern-type Substrate

H’Hole-SR

Anisotropic scattering



Scattering

; ;

Anisotropic surface roughness scattering rate

0.25 0.30 0.35 0.400

2x1014

4x1014

6x1014

8x1014

Unstressed

Surf

ace

Roughnes

s Sca

tter

ing R

ate

(sec

-1)

Energy (eV)

0o from k<100>

45o from k<100>

135o from k<100>

Long. -1 GPa

(001) 1st SubbandE

eff ~ 0.6 MV/cm

Unscreened

0.05 0.10 0.15 0.20 0.250

1x1014

2x1014

3x1014

4x1014

Unstressed

Long. -1 GPa

(110) 1st Subband E

eff ~ 0.6 MV/cm

Unscreened

Surf

ace

Roughnes

s Sca

tter

ing R

ate

(sec

-1)

Energy (eV)

0o from k<110>

45o from k<110>

90o from k<110>



Scattering

; ;

Isotropic SR scattering

0

50

100

150

200

250

0

45

90

135

180

225

270

315

0

50

100

150

200

250

(110) x'= <001>y'= <110>

Hole

Mobili

ty (cm

2 /Vse

c)

(001)------- (110) [Pham.] This work

(001) x'= <100>y'= <010>

0

50

100

150

200

250

0

45

90

135

180

225

270

315

0

50

100

150

200

250

(110) x'= <001>y'= <110>

Hole

Mobili

ty (cm

2 /Vse

c)

(001)------- (110) [Pham] This Work

(001) x'= <100>y'= <010>

Anisotropic SR scattering

*A. T. Pham, C. Jungemann, and B. Meinerzhagen, “Microscopic modeling of hole inversion layer mobility in unstrained and uniaxially stressed Si on arbitrarily oriented substrates,” Solid-State Electronics, vol. 52, pp. 1437-1442, May 2008.


; ;

Two-Dimensional Electron Gas



Under the momentum relaxation time approximation,

Thick Oxides


Thick Oxides

Gaussian model:

Exponential model:


Thin Oxides

20

Inversion layer mobility in thick oxide MOSFETs can be limited to three primary scattering mechanisms:

Total Mobility

Effec

tive

Mobility

Effective Field

Coulmbscattering phonon

scattering

Surface roughnessscattering

Coulomb-Limited Mobility Model due to Ionized Impurity Atoms in the Substrate Region- The scattering rate of ionized impurity in 3-D case can be presented by:

- However, it is not the 2-D electron gas inside the MOSFET.

Nano Electronics Physics Lab @ NCTU 21

Ionized Impurity Ionized Impurity Scattering ModelScattering Model

: the ionized impurity concentration

: the Debye length can be written as , where n0 is the 3-D density of the mobile carrier

34 222 2 2 22

2 22 2

81[ln(1 ) ] , 4 ( )

( ) 116 2I D

Dimp o si

N e mELr pr E r L

E rm

IN

DL 20

si o BD

k TL

e n

Coulomb-Limited Mobility Model due to Ionized Impurity Atoms in the Substrate Region at 2-D case- The momentum conservation in the z-direction of the 3-D case at the scattering process of 2-D carriers should be replaced by the integral as:

Therefore, the scattering rate of ionized impurity scattering in 2-D case

from mth subband to nth subband can be expressed as:


where H2D and H3D are the matrix elements for 2-D and 3-D scattering, respectively.

22 22 3 ( ) ( ) ( ) ( ) ( ), ziq z

D D z z z mn z m nH H I q dq I q I q z z e dz

34 2 1

2 2 222,, 22 2

, 2 / 4 3 ,

( )1 1[ln(1 ) ] , ( ) ( )

( ) 1 ( )16 2I D

m n m nm nimp D m no si

N e g Err E W z z dz

E r g E Wm

22 0

2 2

8, ox av BD

Dinv

Z k TmELr L

e N

• :The average inversion layer thickness

• :the density of states for two dimensions • :the density of states for three dimensions

avZ2 ( )Dg E

3 ( )Dg E

Ionized Impurity Ionized Impurity Scattering ModelScattering Model

- For intravalley phonon scattering model, the momentum-relaxation rate in the subband mth to nth:

The total scattering rate in mth subband is determined by summing up within all the subbands:


Phonon Scattering ModelPhonon Scattering Model

21

(2 / 4) (2 / 4) 2 2,2, 3

(2 / 4) ,

1 1, ( ) ( )

acv d ac B

m n m nm nac m nl

n m D k TW z z dz

Ws

Dac : deformation potential due to acoustic phononsSl : sound velocity ρ : crystal densityWm,n: the form factor determined by the wave-functions of the mth subband and nth subbands

,(2 / 4) (2 / 4)

( )1

( ) ( )m

m m nnac ac

U E E

E E

where ( ) 1( 0) 0( 0)U x x and x is a step function.

- For the intervalley phonon scattering model: (Incorrect Version) (1). From mth subband in twofold valleys to the nth subband in fourfold valleys:

(2). From mth subband in fourfold valleys to the nth subband in twofold valleys:

(3). From mth subband in fourfold valleys to the nth subband in fourfold valleys:

(4). From mth subband in twofold valleys to the nth subband in twofold valleys :


Phonon Scattering ModelPhonon Scattering Model

,

2{ } 14 ' 2 ' 2

, '2 ,

4 1 ( )1 1 1 1( ), ( ) ( )

( ) 2 2 2 1 ( ) m n

fd k k

k k n m nm nkINTER k m n

m D f E EN U E E E W z z dz

E E W f E

2{ } 1

2 ' ' 2 2,,

4 ,

2 1 ( )1 1 1 1( ), ( ) ( )

'( ) 2 2 2 1 ( )

fd k k

k k n m n m nm nkINTER k m n

m D f E EN U E E E W z z dz

E E W f E

2{ }4 '

, '4 ,

2{ }4 ' 1

',

2 1 ( )1 1 1 1( )

( ) 2 2 2 1 ( )

1 ( )1 1 1( ) , (exp( ) 1)

2 2 2 1 ( )

fd k k

k k nm nkINTER k m n

gd k k k

k k n kk k m n B

m D f E EN U E E E

E E W f E

m D f E E EN U E E E N

E W f E k T

where Ek and Dk are deformation energy and potential at kth intervalley phonon, and Nk is the occupation number of kth intervalley phonon.

2{ }2 ' 1

, '2 ,

1 ( )1 1 1 1( ) , (exp( ) 1)

( ) 2 2 2 1 ( )

fd k k k

k k n km nkINTER k m n B

m D f E E EN U E E E N

E E W f E k T

- The scattering rate for a Gaussian function is described as:


Surface Roughness Scattering Surface Roughness Scattering ModelModel

2 2( ) 2 2 2 2 2

4, 3

0

( )1( ) (1 cos )

( ) 2DOS eff

j ij q

ji jSR

m E e EU E E e d

E

2 2 2 2=2 (1 cos ) =4 sin2

q k k

( )2

2

2 (E-E ) = DOS

jjm

k

Δ : rms height of the amplitude of surface roughness

: correlation length of surface roughness

0( ) ( )

eff

ij j idVE z z dz

dz

Derivation of Two-Dimensional Mobility in the Universal Mobility Region- The scattering rates of the twofold and fourfold valley:

The electron mobility by using the average energy within the 2DEG in the relaxation time approximation can be given as

The total universal mobility averaged over the subband occupation is described by


Electron Mobility ModelElectron Mobility Model

2 2 2 4 4 4

1 1 1 1 1 1,

( ) ( ) ( ) ( ) ( ) ( )i i i i i iphonon SR phonon SRE E E E E E

'

'2 4

2 4'

2 4

( ) ( )( ) ( ) ( )( ),

( )( ) ( )( )

i i

i i

i ii iE Ei i

c i c iE E

f fe E E E dE e E E E dE

E Ef f

m E E dE m E E dEE E

'2 4 '

'

( )i ii i

i iuni

s

N N

N

The physical parameters for phonon and surface-roughness

electron mobility used for Si in this work



The Universal mobility Curve - Universal electron mobility includes phonon scattering and surface roughness

scattering, which is independent of process parameters, especially when plotted versus of high effective field (Eeff).



0

( ), 0.5inv dep

effsi

e N NE

• : total inversion layer charge densityinvN

• :the surface concentration of the depletion charge

depN

10-1 100101

102

103

104

Dk=11.5x108 eV/cm Dac=13 eV

=14.9x10-8 cm =2.9x10

-8 cm

Eff

ective

Mobility

(cm

2 / V

s)

Eeff

(MV/cm)

Nsub

=7.7X1017cm-3

Nsub

=3.0X1017cm

-3 N

sub=7.2X10

16cm-3

Nsub

=2.0X1016cm-3 N

sub=3.9X10

15cm-3

Dots:Exp.(Takagi,et al.[10])Lines:Sim.(This work)

Npoly

=1x1020cm-3

T=300KT

ox=25nm

0.1 1102

103

104

Lines:Sim. total mobility cosists of uni

&imp

(This work)

T=397 K T=347 K T=297 K T=242 K

Exp. T(K) 397

347

297

242

Nsub

=3.9X1015cm

-3

T=397 K T=347 K T=297 K T=242 K

Mobili

ty (c

m2/V

s)

Eeff

(MV/cm)

=14.9x10-8 cm =2.9x10

-8 cm

Dk=11.5x108 eV/cm Dac=13 eV

Dots:Exp. data (Takagi, et al [10] )

Electron Effective Mobility:Thick Oxides


Current NEP-electron-mobility simulator was developed under the parabolic band approximation, the isotropic scattering approximation, the elastic scattering approximation (surface roughness and impurity scattering), and the momentum relaxation approximation.

30

Ionized Impurity

Scattering

2

4D

r

Ls

si

qU e

r

2si B

D

k TL

q n

34 2 2 222

1[ln(1 ) ]

( ) 116 2I

imp si

N qE

E m

2

22

8 DmEL

The perturbing potential is the screened Coulomb potential:

r: the distance from the scattering centerLD: Debye length

n: 3-D density of the mobile carriers and equal to Ninv/Zav

Through the Fermi’s Golden Rule, the relaxation time due to ionized impurity

scattering is:

NI: 3-D impurity concentration, which is about equal to

Nsub

Here, q means the elementary charge.

31

Impurity Scattering

The momentum relaxation time for scattering of 2-D carriers from uth subband to vth subband:

34 22 2 222

, 22, 2/4 3

1 ( )[ln(1 ) ] ( ) ( )

( ) 1 ( )16 2I D

u vu vimp Dsi

N q g EE z z dz

E g Em

P.S. Only consider intra-subband

K. Hirakawa and H. Sakaki, “Mobility of the two-dimensional electron gas at selectively doped n-type AlxGa1-xAs/GaAs heterojunctions with controlled electron concentrations,” Phys. Rev. B, Condens. Matter, vol. 33, no. 12, 8291-8303, Jun. 1986.

M. Lundstrom, “Fundamentals of carrier transport,” Cambridge University Press, 2000.

32

Phonon Scattering

1. intravalley phonon scattering model: (acoustic phonon) the momentum-relaxation time for scattering from the uth subband to the vth subband

2, 2/4 2 2 1

, ( 2/4) ( 2/4), ( 2/4),, 3 2int ( 2/4) , ( 2/4)

1 1( ) , ( ( ) ( ) )

( )d ac B

v u v u vu vra l u v

m D k TU E E W z z dz

E s W

kB : Boltzmann constant , Dac : deformation potential due to acoustic phonons ρ : crystal density , Sι: sound velocityU(x): step function

S. Takagi, J. L. Hoyt, J. J. Welser, and J. F. Gibbons, “Comparative study of phonon-limited mobility of two-dimensional electrons in strained and unstrained Si metal-oxide-semiconductor field-effect transistors,” J. Appl. Phys., vol. 80, no. 3, pp. 1567-1577, Aug. 1996.

D.Esseni, A. Abramo, L. Selmi, and E. Sangiorgi, “Physically Based modeling of Low Field Electron Mobility in Ultrathin Single- and Double-Gate SOI n-MOSFETs”, IEEE Trans. Electron Devices, vol. 50, no.12, pp.2445-2455, 2003.

K. Uchida, A. Kinoshita, and M. Saitoh, “Carrier transport in (110) nMOSFETs: Subband structures, non parabolicity, mobility characteristics, and uniaxial stress engineering,” in IEDM Tech. Dig., pp.1019-1021, 2006.

33

Phonon Scattering model (Correct Version)

2. intervalley phonon scattering model: (optical phonon) (a) From uth subband in twofold valleys to the vth subband in twofold valleys:

(b) From uth subband in twofold valleys to the vth subband in fourfold valleys:

(c)From uth subband in fourfold valleys to the vth subband in twofold valleys:

(d)From uth subband in fourfold valleys to the vth subband in fourfold valleys:

,

2{ } 1,, 2 ' 2 2

, , 2, 2, 2,, 'int , 2 2 , ,

1 ( )1 1 1 1( ) ( ), ( ) ( )

( ) 2 2 2 1 ( ) u v

gk gd k

k g k g v u vu vker k g u v

f E Em DN U E E E W z z dz

E E W f E

2{ },, 4 2 2 1

, , 4, , 2, 4,,int , 2 4 , ,

1 ( )41 1 1 1( ) ( ), ( ( ) ( ) )

( ) 2 2 2 1 ( )

fk fd k

k f k f v u v u vu vker k f u v

f E Em DN U E E E W z z dz

E E W f E

2{ },, 2 2 2 1

, , 2, , 4, 2,,int , 4 2 , ,

1 ( )21 1 1 1( ) ( ), ( ( ) ( ) )

( ) 2 2 2 1 ( )

fk fd k

k f k f v u v u vu vker k f u v

f E Em DN U E E E V z z dz

E E V f E

2{ },, 4

, , 4,, 'int , 4 4 , ,

2{ },, 4 ' 2 2 1

, , 4, , 4, 4,', ,

1 ( )21 1 1 1( ) ( )

( ) 2 2 2 1 ( )

1 ( )1 1 1( ) ( ), ( ( ) ( ) )

2 2 2 1 ( )

fk fd k

k f k f vu vker k f u v

gk gd k

k g k g v u v u vk k g u v

f E Em DN U E E E

E E V f E

f E Em DN U E E E V z z dz

E V f E

34

2

4

4 4

4

f-type

f-type

f-typef-

typ

e f-typ

ef-type

f-typef-typ

e

Acoustic phonon

Acoustic phonon

Acoustic phonon

Acoustic phononAcoustic phonon

g-type

g-type

g-type

35

Surface Roughness Scattering2 22 2 2 2 2

4, 3

0

2 2 22

( )1( ) (1 cos )

( ) 2

2 ( )where 2 1 cos and

i qDOS eff

ii iSR

DOS j

m E e EU E E e d

E

m E Eq k ( θ) k

0( ) ( )i i i

eff

dVE z z dz

dz

ki

kjq

E1

E2

P.S. Only consider intra-subband

Yamakawa, H. Ueno, K. Taniguchi, C. Hamaguchi, K. Miyatsuji, K. Masaki, and U. Ravaioli, “Study of interface roughness dependence of electron mobility in Si inversion layers using the Monte Carlo method,” J. Appl. Phys., vol. 79, no. 2, pp. 911-916, Jan. 1996.

D.Esseni, “On the Modeling of Surface Roughness Limited Mobility in SOI MOSFETs and its Correlation to the Transistor Effective Field”, IEEE TED, Vol.51 NO.3, pp.394-401, 2004.

36

Three-Dimensional Stress Effect on Electron Mobility

-3000 -1500 0 1500 30000

1

2

(001)/<110> nMOSFETSimulation of DG-NEP at E

eff=1MV/cm

Long./Tran. Vert. Biax.

Mobility R

atio

Stress (MPa)-3000 -1500 0 1500 30000

1

2

3

4

5M

obility R

atio

Stress (MPa)

(110)/<1-10> nMOSFETSimulation of DG-NEP at E

eff=1MV/cm

Long./Vert. Tran. Biax.

-3000 -1500 0 1500 30000

1

2

3

Mobility R

atio

Stress (MPa)

(111)/<1-10> nMOSFETSimulation of DG-NEP at E

eff=1MV/cm

Long. Tran. Biax. Vert.

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