TCAD ready density gradient calculation of channel charge for strained sistrained si 1− x ge x dual channel pMOSFETs on (001) relaxed si 1− y ge y

Title

TCAD ready density gradient calculation of channel charge for
Strained Si/Strained Si1−xGex dual channel pMOSFETs on
(001) Relaxed Si1−y Gey

C. D. Nguyen, A. T. Pham, C. Jungemann, and B. Meinerzhagen
Institut fă
ur Netzwerktheorie und Schaltungstechnik
Technische Universită
at Braunschweig

C. Jungemann

IWCE 2004

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Outline

ã Motivation

ã Schră
odinger/Poisson Solver for Strained Si and SiGe

ã Density Gradient Model

• Extraction of the heterojunction valence band offsets needed for TCAD
simulators

• Conclusion and Outlook

C. Jungemann

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Motivation

C. Jungemann

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Motivation

multi stacked strained structure
Vg

0

SiO2 (4.4nm)
 

✁

 

✁

 

✁

 

✁

 

✁

 

✁

 
 

✁

 

✁

 

✁

 

✁

 

✁

 

✁

 

✁

Strained Si0.4 Ge0.6
(5nm)

✁

Strained Si (3.3nm)

III

III/II

∆EV

II

II/I

∆EV

I

Relaxed Si0.7 Ge0.3

EV

EF

EC

Changes in the band structure and small
thickness of the strained layers
= Size Quantization
Solution: Schră
odinger equation (SE)
with a full band description using the
k · p-method
For TCAD use, directly solving the SE
is too CPU intensive.
=⇒ Density Gradient Method (DGM)
Problem: unknown model parameters
e. g. effective band offsets.

z

Vb

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Motivation

Effective band offsets can be determined by:

• Measurement: The effective band offsets can be extracted by inverse
modeling of CV measurements based on the DGM [1].
=⇒ Uncertainty due to incomplete knowledge of the investigated devices

• Simulation: Based on the self-consistent solution of the SE and Poisson
equations, the effective band offsets can be extracted and the errors of
the DGM approximation can be investigated.
[1] C. Ni Chleirigh et al., “Extraction of band offsets in Strained Si/Strained SiGe on relaxed
SiGe dual-channel enhanced mobility structures” to be presented at SiGe Materials, Processing
and Devices Symposium, Hawai, 2004.

C. Jungemann

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Schră
odinger/Poisson Solver for Strained Si and SiGe

C. Jungemann

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Schră
odinger/Poisson Solver for Strained Si and SiGe

6 ì 6 k · p SE for holes:
∂
n
+ ˆI · eV (z) Fn
k (z) = En (k)Fk (z)
∂z
ˆ =H
ˆ kp + H
ˆ so + H
ˆ str and
with k = (kx, ky ), H
ˆ k, kz = −i
H

V (z) = Ψ(z) + ∆Evav/e,
∆Evav [2]: “natural” valance band offset step of the Si/ SiGe heterostructure.
The quantum-mechanical charge density:
pqm(z) =

1
2
n (2π)

|Fkn|2f (En(k) + EF ) d2k ,

(1)

In contrast to nextnano3, a modified discretization scheme for the twodimensional k space is used in order to reduce the computation time and
to calculate (1) with high accuracy. Moreover, the CV characteristics for
mobility and band-offset extraction are determined by 1st order perturbation
theory. =⇒ About 30 times less CPU intensive than nextnano.
[2] C. G. van de Wall Phys. Rev. B, vol. 35, no. 15, pp. 8154–8165, 1987

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Schră
odinger/Poisson Solver for Strained Si and SiGe

New interpolation method and grid
0.378

0.40

0.376

0.35
k||=0.2 [π/a0,Si]

Energy [eV]

Energy [eV]

0.374
0.372
0.370
Nφ=45

0.368
0.366

φ=00

0.30
0.25
0.20
0.15

Nk||=45

Nφ=8, linear inter.

0.10

Nk||=8, linear inter.

Nφ=8, harmonic inter.

0.05

Nk||=8, cubic spline inter.

0.364

0
0

5

10

15

20

25

30

35

40

45

0

0.1

0.2

o

φ[ ]

=⇒

C. Jungemann

0.3
0.4
Si
k|| [π/a0 ]

0.5

0.6

CPU-time gain = 25-30

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Schră
odinger/Poisson Solver for Strained Si and SiGe

Band structure of first three subbands (ND = 5 × 1017 cm−3, VG = −2.5V,
φ = 0o) and the wave function of the first energy level.
1.0
1. subband
2. subband
3. subband

0.9
0.8

0.4
1,1

F0 (z)

strained Si0.4Ge0.6

strained Si

0.7
0.6

Energy [eV]

Energy [eV]

0.3

relaxed
Si0.7Ge0.3

0.5
0.4

0.2

0.3
0.1

0.2
a)

0.1
0

0.0

0.1

0.2

C. Jungemann

0.3

0.4 0.5 0.6
Si
k|| [ π /a0 ]

0.7

0.8

0.9

1

2

3

4

5

6

7

8

9

10

1.0

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Schră
odinger/Poisson Solver for Strained Si and SiGe

Hole density at room temperature for two gate biases evaluated by SE
50
FBSC (VG=-4V)
FBSC (VG=-2V)

Hole density [x10

18

-3

cm ]

40

30

20

10

0

C. Jungemann

0

1

2

3

4

5
z [nm]

6

7

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9

10

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Density Gradient Model

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Density Gradient Model

Approximate quantum correction by the density gradient model (DGM):

p

dg



(z) = Nv exp 



Ev + Φm + Λ − EF

.
kB T

Here, Φm = (3/2)kB T log(m∗) and Λ is obtained by solving a differential equation:

2γ 

¯ − EF
¯ − EF
1
Φ
Φ
∇·∇
+
∇
Λ=
12m 
kB T
2
kB T

C. Jungemann


2

¯ = Ev + Φm + Λ
, with Φ



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Density Gradient Model

What is new in strained material compared to relaxed material?

 

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Electrons:
∆Ec(y) known from literature.

 

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∆Ec(y)

 

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∆Ev (y, k||)

 

✁

✁

 

tSSi

C. Jungemann

Relaxed Si1−y Gey

Holes:
∆Ev (y, k||) depends on k||
¯v (y) independent from k||
but ∆E
required for TCAD (effective valence band offsets)

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Extraction of the band offsets for TCAD

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Extraction of the band offsets for TCAD

20

Cgd[pF]

15

10
I/II

∆Ev

I/II
∆Ev

+ 40meV
- 40meV

I/II

5

ana. ∆Ev
Ii/III

∆Ev

II/III

∆Ev

0
-4

C. Jungemann

-3

II/III

and ∆Ev

- 40 meV
+ 40 meV

-2
VG[V]

-1

0

• Based on the CV data calculated by
SE, the valance band offsets have
been extracted by matching the CV
data calculated by DGM.
• The conduction band offsets are
fixed during the fitting procedure.
• Note that in this version the effective mass of Si for DGM was used
because no values are available for
strained Si and strained SiGe.

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Extraction of the band offsets for TCAD

Gate capacitance with different thickness of strained Si region (T = 300 K)
20

20

DGM
FBSC

DGM

FBSC
15

Cgd[pF]

Cgd[pF]

15

10

5

0
-4

10

5

-3

-2
VG[V]

-1

0

0

-4

tSSi = 3.3 [nm]

C. Jungemann

-3

-2
VG[V]

-1

0

tSSi = 4.0 [nm]

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Conclusion and Outlook

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Conclusion and Outlook

• Conclusions
– Efficient evaluation of low frequency CV characteristic for multi stacked
strained Si structures with a complete description of the valance band
structure is now possible.
– Accurate calculation of CV-characteristics for strained Si/SiGe dual
channel pMOSFETs based on Density Gradient Method with the corresponding extracted valance band offsets.
• Outlook
– Improvement of the state of art Density Gradient Model for holes in
strained Si and strained Si1−xGex based on our SE/PE solver.
– Extraction of the heterojunction valence band offsets and other parameters for wide range of Ge contents.
– Verification of the extracted results by comparison with measured CVdata.
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IWCE 2004

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