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doi:10.1016/j.ph

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Physica E 37 (2007) 168–172

www.elsevier.com/locate/physe

A study of the interface roughness effect in Si nanowires using afull 3D NEGF approach

A. Martinez�, K. Kalna, J.R. Barker, A. Asenov

Device Modelling Group, Department of Electronics and Electrical Engineering, University of Glasgow, Glasgow G12 8LT, UK

Available online 30 August 2006

Abstract

The effect of interface roughness in ballistic Si nanowires is investigated using a full 3D non-equilibrium Green’s Functions formalism.

The current density, the electron density and the transmission function are calculated for nanowires with different interface roughness

configurations. Interface roughness is randomly generated using an exponential autocorrelation function. The interface roughness profile

in nanowires with 2 nm diameter and 6 nm length is reflected in the current density landscape showing macroscopic 3D patterns. These

macroscopic patterns affect the transmission probability causing resonances coming from the constrictions in the channel. The shape of

the electron density in cross-sections along the wire follows the distortion of electron transversal wave function.

r 2006 Elsevier B.V. All rights reserved.

PACS: 73.63.b; 73.23.b

Keywords: Non-equilibrium Green’s functions; Interface roughness; Nanowires

1. Introduction

The impact of the atomic scale roughness at the Si/SiO2

interface increases with scaling of CMOS transistors tonanometre dimensions. The particular profile of the Si/SiO2 interface roughness varies from device to device andproduces fluctuations in the current and threshold voltageeven in an ensemble of macroscopically identical devices[1]. At such small dimensions, the nanoscaled transistorshave to be simulated using fully quantum mechanicalapproach because of the increasing importance of quantumconfinement effect and tunnelling. When the electrontransport channel is confined to a region with fewnanometres cross-section the sub-band energy shifts inthe order of the thermal energy. This introduces steps in theelectron transmission probability.

Using non-equilibrium Green’s functions (NEGF) ap-proach solved in 3D real space we investigate the problemof the complex interface landscape and the confinement ofthe electron transport in nanowire transistors as illustratedin Fig. 1. The cross-section of the channel of the nanowire

e front matter r 2006 Elsevier B.V. All rights reserved.

yse.2006.07.007

ing author. Tel.: +1413304792.

ess: antonio@elec.gla.ac.uk (A. Martinez).

transistor changes along the channel length due to therough interface between the Si and the gate insulator asdepicted in Fig. 2. The cross-section variations producefluctuations in the confinement, which consequentlyintroduce fluctuations in the energy levels along thetransport direction. These fluctuations in the energy levelsact as quantum mechanical barriers and introduce scatter-ing of the electron moving along the channel.Many experimental and theoretical publications suggest

that the Si/SiO2 interface has an approximate correlationlength of 2 nm [2]. This correlation length is of the order ofthe electron thermal wavelength (�5 nm). The interactionbetween the electrons and the rough interface needs to becarried out non-perturbatively using a fully quantummechanical approach since multiple interferences of theelectron wave function between the four surfaces of ananowire are expected to occur.

2. NEGF approach

One of the well-established techniques for nanoCMOSdevice simulation is the NEGF approach to quantumtransport [3]. This approach considers the amplitude of

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Fig. 2. Upper and lower panels present the potential and the current

landscape, respectively, through the zy-plane for the randomly generated

interface roughness (Rough 1). The scale for the current density is in

arbitrary units.

Contact Source

Gate

Drain

Silicon

Dielectric

Metal

Fig. 1. Schematic view of the Si nanowire transistor with heavily doped

source/drain contacts. Only the active region between the source and drain

is considered in the NEGF simulations.

0 10 20 30 40 50 60

0 10 20 30 40 50 60y (Angstroms)

20

15

10

5

0

20

15

10

5

0

x (A

ngst

rom

s)x

(Ang

stro

ms)

x 10-6

3

2

1

Fig. 3. The same as Fig. 1 but in the xy-plane.

0 10 20 30 40 50 60

0 10 20 30 40 50 60y (Angstroms)

20

15

10

5

0

20

15

10

5

0

x (A

ngst

rom

s)z

(Ang

stro

ms)

Fig. 4. The rough interface for the Rough 2 device in the plane x ¼ 10 A

upper panel and z ¼ 10 A lower panel.

A. Martinez et al. / Physica E 37 (2007) 168–172 169

probability of an electron (with an energy E) to go from apoint x in the real space to a point y, i.e., the propagatorGr(x, y, E). In the absence of scattering, the correlationmatrix F(x, y, E) can be computed as

F ¼ GrOGa,

where Ga represents the Hermitian conjugate of Gr. O is thestatistically weighted self-energy proportional to theFermi–Dirac function and the inverse of the electronlifetime in the device (i.e., the lead-device coupling energy).The diagonal elements of F are proportional to the electrondensity and the off-diagonal elements can be used tocalculate the current density. The electrons are injectedfrom the source and drain following the Fermi–Diracdistribution. The transmission probability between theterminal p and q in this approach can be calculated using

the following expression [4]:

TpqðEÞ ¼ Trace Sap � Sr

p

� �Gr Sa

p � Srp

� �Ga

n o,

where Sr and Sa are the retarded and advance self-energiesof the terminals. This expression is depends on the carrierenergy.2D NEGF simulations produce reasonably qualitative

results [1], but are restricted only to 1D representation ofinterface roughness that inherently has a 2D character.This results in an overestimation of the correspondingfluctuations in the device characteristics, especially in verysmall devices. Realistic treatment of the interface rough-

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0 10 20 30 40 50 60

0 10 20 30 40 50 60y (Angstroms)

20

15

10

5

0

20

15

10

5

0

x (A

ngst

rom

s)z

(Ang

stro

ms)

Fig. 5. The rough interface for the Rough 3 device in the plane x ¼ 10 A

upper panel and z ¼ 10 A lower panel.

0 10 20 30 40 50 60

0 10 20 30 40 50 60y (Angstroms)

20

15

10

5

0

20

15

10

5

0

x (A

ngst

rom

s)z

(Ang

stro

ms)

Fig. 6. The rough interface for the Rough 4 device in the plane x ¼ 10 A

upper panel and z ¼ 10 A lower panel.

0 10 20 30 40 50 60

0 10 20 30 40 50 60y (Angstroms)

20

15

10

5

0

20

15

10

5

0

x (A

ngst

rom

s)z

(Ang

stro

ms)

Fig. 7. The rough interface for the Rough 5 device in the plane x ¼ 10 A

upper panel and z ¼ 10 A lower panel.

20

20

15

15

10

105

5 2015105

2015105 2015105

20

15

10

5

20

15

10

5

20

15

10

5

x (Angstroms)

z (A

ngst

rom

s)z

(Ang

stro

ms)

x (Angstroms)

(a)

(c)

(b)

(d)

Fig. 8. The rough interface in the cross-sections for the Rough 5 device

along the channel at y ¼ 10 (a), 30 (b), 40 (c) and 50 A (d).

A. Martinez et al. / Physica E 37 (2007) 168–172170

ness problem requires 3D real space NEGF transportsimulations which are computationally very expensive andmemory consuming because of the need for a fine energystep integration and an inversion of huge sparse matrices.Fortunately, faster processors and larger memories arebecoming available thus allowing the 3D NEGF techniqueto become a realistic choice for the simulation of ballisticnanotransistors illustrated in Fig. 1.

3. Model and simulations

We present a preliminary study of the effect of interfaceroughness in Si nanowire transistors using a full 3D NEGFsimulator, with parallelization in the energy loop. TheHamiltonian used is the effective-mass Hamiltonian.Interface roughness is generated in 2D using an exponen-tial autocorrelation function [5]. The model essentially

introduces random steps in the interface on the scale of thelattice constant, The average length of these steps aredetermined by the autocorrelation function and is takenequal to 2 nm. Therefore, our model of rough interface isassociated with spatial fluctuations of the dielectricconstant around the ideal (plane) interface. A recursivealgorithm has been incorporated in order to calculate thedensity of states [6]. This allows us to calculate efficientlythe diagonal terms of Gr and F. The calculation of the firstoff-diagonal terms of F is needed in order to compute thecurrent density while self-energies are calculated followingRef. [7].

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20

20

15

10

10

2010 2010

2010

5

20

15

10

5

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10

5

20

15

10

5

x 1019

x 1018

x 1018

x 1018

2

4

6

5

10

15

2

4

6

3

2

1

x (Angstroms)

z (A

ngst

rom

s)z

(Ang

stro

ms)

x (Angstroms)

(a)

(c)

(b)

(d)

Fig. 9. The electron density in the cross-sections for the Rough 5 device along the channel at y ¼ 10 (a), 30 (b), 40 (c) and 50 A (d). The scale of density is

in cm�3.

A. Martinez et al. / Physica E 37 (2007) 168–172 171

A slab of 2nm� 2nm� 6nm undoped Si representing theactive region of a nanowire transistor has been used in thesimulations. A spacing of 1 A is used for generating theuniform device mesh. The energy mesh contains approxi-mately 600 points. We have computed the current density andthe transmission coefficient for five randomly generatedinterfaces with interface steps of 3 A which is approximatelyhalf of the Si lattice constant. All the nanowires considered inthis work are oriented in the 100 direction.

Fig. 2 shows in the upper/lower panel the potential/current density landscapes for a plane parallel to thetransport direction (y-axis) in a slab with one particularinterface roughness configuration. The extra confinementcreated by the roughness at the right side of the channelsqueezes the current into the middle of the slab, and shiftsup the transversal energy levels.

Fig. 3 shows the potential and electron density for aplane parallel to the transport direction but perpendicularto the plane of Fig. 2. In this plane, the electrons are lessconfined compared to the plane in Fig. 2 and are pushed tothe bottom from the middle of the channel as seen in thecurrent density profile. Figs. 4–7 show the surface rough-ness in the x ¼ 10 and z ¼ 10 planes for the cases labelledRough 2, Rough 3, Rough 4, and Rough 5, respectively.The cross-sections at y ¼ 10, 30, 40 and 50 A illustratingthe rough interface are shown in Fig. 8 for the deviceRough 5 and the corresponding densities in Fig. 9.

The density profile is deformed by different irregularcross-sections. This reflects the change of the transversalwave function due to the particular shape of each cross-section. Note that the density in the cross-sections aty ¼ 30 and 40 A are different but the interface profile inboth is the same. However, the interface profiles in theneighbouring cross-sections around this cross-section arenot the same. The Si cross-section for y440 A becomessmaller, affecting the transversal wave function not only inthose points but also in the neighbouring ones. Thesechanges in the transversal wave function and in the electrondensity demonstrate the need for a 3D real space treatmentof the electron transport in such small structures sincedifferent transversal states are mixing along the devicechannel.Fig. 10 shows the transmission probability as a function

of the electron energy for four cases of randomly generatedinterface roughnesses. The transmission for the smoothcase is a step function, counting the transversal states. Twoimportant observations can be made from Fig. 5. Firstly,there are resonances in the transmission in the nanowiretransistors with the rough interface. Secondly, there is asmoothing of the transmission in nanowires with the roughinterface compared to the nanowire with the flat interface.The transmissions in the cases Rough 1 and Rough 2 showresonances, which arise because the rough interfaceproduces constrictions in the cross-section, which behaves

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0.2 0.4 0.6 0.8 1

smooth

Rough 1

Rough 2

Rough 3

Rough 4

Rough 5

Energy (eV)

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Tran

smis

sion

Fig. 10. Transmission probability for nanowires with randomly generated

interface roughness. The transmission for a smooth interface is also shown

for a comparison.

A. Martinez et al. / Physica E 37 (2007) 168–172172

as a double barrier. In the cases Rough 3 and Rough 4, theparticular configurations of the random interface rough-ness do not have such constriction regions. As a result, noresonances in the transmission are observed. Note that theinterface roughness affects the transversal energy levels indifferent ways. For example, the case Rough 1 around0.65 eV has a larger transmission compared to the caseRough 3. However, the situation reverses above 0.9 eV.The additional step which is observed in the transmissionfunction for the cases Rough 1, Rough 2 and Rough 4indicates the removal of the degeneracy of the secondtransversal electron state.

4. Conclusion

We have simulated the current density, the electrondensity and the transmission probability for Si nanowireswith randomly generated interface roughness using a full3D NEGF formalism. The current density exhibits acomplex 3D character, meandering through the particularfeatures of the interface. The transversal electron wavefunction is distorted by the interface, which justifies theneed for a full 3D treatment of the electron transport. Thetransmission probability for the rough nanowires displaysresonances as a consequence of the body thicknessfluctuation. In some cases, the body thickness fluctuationsare sufficient to break the degeneration of the transversalelectron states.

References

[1] A. Martinez, A. Svizhenko, M.P. Anantram, J.R. Barker, A.R. Brown,

A. Asenov, IEDM Tech. Dig. 627 (2005).

[2] A. Asenov, S. Kaya, J.H. Davies, IEEE Trans. Electron Dev. 49 (2002)

112.

[3] L.V. Keldysh, Soviet Phys. JEPT 20 (1965) 1018;

R. Lake, G. Klimeck, R.C. Bowen, D. Jovanovich, J. App. Phys. 81

(1997) 7845.

[4] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge

University Press, Cambridge, 2003, p. 321.

[5] S.M. Goodnick, D.K. Ferry, C.W. Wilmsen, Z. Lilienthal, D. Fathy,

O.L. Krivanek, Phys. Rev. B 32 (1985) 8171.

[6] A. Svizhenkov, M.P. Anantram, T.R. Govindan, B. Siegel, J. App.

Phys. 91 (2002) 2343.

[7] R. Venugopal, Z. Ren, S. Datta, M. Lundstrom, J. App. Phys. 92

(2002) 3730.