Lundstrom ECE-656 F15
ECE 656: Electrothermal Transport in Semiconductors Fall 2015 Landauer Approach:
Examples
Professor Mark Lundstrom
Electrical and Computer Engineering
Purdue University West Lafayette, IN USA
10/8/15
Low-field transport in metallic CNT
Lundstrom ECE-656 F15
Zhen Yao, Charles L. Kane, and Cees Dekker, “High-Field Electrical Transport in Single-Wall Carbon Nanotubes,” Phys. Rev. Lett., 84, 2941-2944, 2000.
G1D =
ΔIΔV
=22 µA1.0 V
= 22 µS
1L mµ≈
What is the mean-free-path for backscattering?
Low-field transport in metallic CNT
Lundstrom ECE-656 F15
G1D = 4q2
hλ EF( )
λ EF( ) + L
Zhen Yao, Charles L. Kane, and Cees Dekker, “High-Field Electrical Transport in Single-Wall Carbon Nanotubes,” Phys. Rev. Lett., 84, 2941-2944, 2000.
G1D =
ΔIΔV
=22 µA1.0 V
= 22 µS
1L mµ≈
λ EF( ) ≈167 nm << L
GB = gv
2q2
h= 154 µS
gV = 2( )
1D current
Lundstrom ECE-656 F15
Ix = −gv
2q2
hT (E) − ∂ f0
∂E⎛⎝⎜
⎞⎠⎟ dE
ε1
∞
∫⎧⎨⎪
⎩⎪
⎫⎬⎪
⎭⎪VD ≈ −gv
2q2
hT EF( )V
Ix ≈ gv2q2λ EF( )
h−VL
⎛⎝⎜
⎞⎠⎟
Ix ≈ gv
2q2
hλ EF( )E x⎡⎣ ⎤⎦
T EF( ) = λ(EF )
λ(EF )+ L≈ λ(EF )
L (diffusive)
In 1D, the diffusive current is the quantum conductance times the volt drop across one mean-free-path.
Graphene
Lundstrom ECE-656 F15 5
CNTBands 2.0 (nanoHUB.org)
T = 10KB = 0
L ≈ 5000 nmtox = 300 nm
k
E k( )
f1 E( )gV = 2 EF
E k( ) = ±!υFk
υF ≈ 1×108 cm/s
D(E) =
2 Eπ!2υF
2
M (E) =
W 2 Eπ!υF
υ k( ) = υF
Fig. 30 in A. H. Castro, et al.,“The electronic properties of graphene,” Rev. of Mod. Phys., 81, 109, 2009.
Graphene
Lundstrom ECE-656 F15 6
VG V( )→
↑GS
mS
Fig. 30 in A. H. Castro, et al.,“The electronic properties of graphene,” Rev. of Mod. Phys., 81, 109, 2009.
T = 10KB = 0
1 µm σ S ≈ 3.0 mS
nS ≈ 7.1×1012 cm-2
L ≈ 5000 nmtox = 300 nm
1) How close to the ballistic conductance?
2) What is the mfp?
7
analysis
Lundstrom ECE-656 F15
G2D =2q2
hλ EF( )L
M EF( ) = σ SWL
GB =2q2
hM EF( )
nS EF( ) = 1
πEF
!υF
⎛⎝⎜
⎞⎠⎟
2
→ EF = 0.3 eV
M (EF ) =W 2EF π!υF
For more about the conductance of graphene, see: “Low-bias transport in graphene,” by M.S. Lundstrom and D. Berdebes, NCN@Purdue 2009 Summer School, nanoHUB.org
λ 0.3 eV( ) ≈ 130 nm
σ S meas( ) = λ L( )σ S ball( )
σ S meas( ) ≈ σ S ball( ) 40
Mobility of graphene
Lundstrom ECE-656 F15 8
G2D =2q2
hλ EF( )L
M EF( ) = nS qµnWL
nS EF( ) = 1
πEF
!υF
⎛⎝⎜
⎞⎠⎟
2
M (E) =W 2EF π!υF
λ(E) = π2υFτ EF( )
µn =2qh
λ EF( )M EF( ) WnS
µn =qτ EF( )EF υF
2( )
Example: MFP in a MOSFET
Lundstrom ECE-656 F15 9
Question: Do we expect this device to be closer to ballistic or to diffusive?
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
RCH ≈ 215Ω-µm
µn ≈ 260 cm2 V-s
nS ≈ 6.7 ×1012 cm-2
L ≈ 60 nm
MFP from mobility
Lundstrom ECE-656 F15
RCH ≈ 215Ω-µm
µn ≈ 260 cm2 V-s
nS ≈ 6.7 ×1012 cm-2
L ≈ 60 nm Dn =
kBTq
µn
(near-equilibrium, MB statistics)
Dn =
υTλ0
2(near-equilibrium, MB statistics, constant mfp)
λ0 =
2 kBT q( )υT
µn
υT =
2kBTπm* = 1.2×107 cm/s
λ0 ≈ 11 nm << L
diffusive
Apparent mfp
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
2260 cm V-snµ ≈
13 -21 10 cmSn ≈ ×
60 nmL ≈
λ0 ≈11 nm
1λapp
= 1λ0
+ 1L
λapp = 9.3 nm
Apparent mobility
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
2260 cm V-snµ ≈
13 -21 10 cmSn ≈ ×
60 nmL ≈
λ0 ≈11 nm
λapp = 9.3 nm
Dn =
υTλ0
2 µn =
υTλ0
2 kBT q( ) µapp =
υTλapp
2 kBT q( ) = 215 cm2 V-s( )
Ballistic mobility
Lundstrom ECE-656 F15
µB ≈
υT L2
1kBT q
≈1400 cm2 V-s
µapp =
1µn
+ 1µB
⎛⎝⎜
⎞⎠⎟
−1
µapp =
υTλapp
2 kBT q( )
µapp =
υT
2 kBT q( )1
λapp
⎛
⎝⎜
⎞
⎠⎟
−1
µapp =
υT
2 kBT q( )1λ0
+ 1L
⎛⎝⎜
⎞⎠⎟
−1
1µapp
=2 kBT q( )υTλ0
+2 kBT q( )
υT L⎛
⎝⎜
⎞
⎠⎟
1µapp
= 1µn
+ 1µB
⎛⎝⎜
⎞⎠⎟
µB =
υT L2 kBT q( )
L = 40 nm HEMT
Lundstrom ECE-656 F15
0 200 nmλ ≈
210,000 cm V-snµ ≈
0
2T
nDυ λ=
7*
2 2.7 10 cm/sBT
k Tm
υπ
= = ×
( )*00.041m m=
D. H. Kim and J. A. del Alamo, "Lateral and Vertical Scaling of In0.7Ga0.3As HEMTs for Post-Si-CMOS Logic Applications," IEEE TED, 55, pp. 2546-2553, 2008.
λ0 >> L ballistic
Transverse modes
Lundstrom ECE-656 F15 15
Question: How many transverse modes are there in a 1 micron wide FET?
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
Modes in a 2D resistor
Lundstrom ECE-656 F15 16
x
yz W t
M EF( ) = gVW
2m* EF − EC( )π!
M EF( ) = gV
W kF
π
depends on bandstructure!
!2kF2
2m* = EF − EC( ) How is M related to the sheet carrier density?
Sheet carrier density at T = 0 K
Lundstrom ECE-656 F15 17
nS =πkF
2
2π( )2× 2→ kF = 2πnS
nS = gvπkF
2
2π( )2× 2→ kF = 2πnS gv
Transverse modes
Lundstrom ECE-656 F15 18
Question: How many transverse modes are there in a MOSFET?
(Courtesy, Shuji Ikeda, ATDF, Dec. 2007)
M 2D EF( ) =WgV
2m* EF − ε1( )π!
M 2D kF( ) = gV WλF 2( ) = gV
W kFπ
M 2D kF( ) =W 2gvnSπ
kF = 2πnS gv
Sheet carrier density at T = 0 K
Lundstrom ECE-656 F15 19
x
yz W t
M EF( ) = gV
W kF
π
M EF( ) =W
2gV nS
πdepends only on nS
nS = gV
kF2
2π
Example: How many modes in a MOSFET?
Lundstrom ECE-656 F15 20
x
yz W t
nS = 1013 cm-2
M EF( ) = 36
for a rough estimate, assume TL = 0 K
M EF( ) =W
2gV nS
π W = 2L = 100 nm
