Transport in nanowire MOSFETs: influence of the band-structure M. Bescond IMEP – CNRS – INPG (MINATEC), Grenoble, France Collaborations: N. Cavassilas, K. Nehari, M. Lannoo L2MP – CNRS, Marseille, France A. Martinez, A. Asenov University of Glasgow, United Kingdom SINANO Workshop, Montreux 22 nd of September

Motivation: improve the device performances Gate-all-around MOSFET: materials and orientations Ballistic transport within the Green’s functions Tight-binding description of nanowires Conclusion Outline 2

Towards the nanoscale MOSFET’s  Scaling of the transistors:  New device architectures  New materials and orientations  Improve carrier mobility  Gate-all-around MOSFET 1 : Increasing the number of gates offers a better control of the potential Ge, GaAs can have a higher mobility than silicon (depends on channel orientation). Effective masses in the confined directions determine the lowest band. Effective mass along the transport determines the tunnelling current.  Improve potential control 1 M. Bescond et al., IEDM Tech. Digest, p. 617 (2004). 3

3D Emerging architectures 3D Emerging architectures 3D simulations: The gate-all-around MOSFET

Gate-All-Around (GAA) MOSFETs T Si =W Si =4nm T OX =1nm Source and drain regions: N-doping of cm -3. Dimensions: L=9 nm, W Si =4 nm, and T Si =4 nm, T OX =1 nm. Intrinsic channel. 5

3D Mode-Space Approach*  3D Problem = N  1D Problems  Saving of the computational cost!!!!  Hypothesis:  n,i is constant along the transport axis. * J. Wang et al., J. Appl. Phys. 96, 2192 (2004).  The 3D Schr ö dinger = 2D (confinement) + 1D ( transport) 2D (confinement) 1D (transport) i th eigenstate of the n th atomic plan 6

Different Materials and Crystallographic Orientations

Different Materials and Orientations Ellipsoid coordinate system ( k L, k T1, k T2 ) + Device coordinate system (X, Y, Z) +    Rotation Matrices Effective Mass Tensor (EMT) 8

Theoretical Aspects* 3D Schrödinger equation: Potential energy H 3D : 3D device Hamiltonian Coupling * F. Stern et al., Phys. Rev. 163, 816 (1967). 9

Theoretical Aspects* The transport direction X is decoupled from the cross- section in the 3D Schrödinger equation: Where E’ is given by: m trans is the mass along the transport direction: Coupling M. Bescond et al., Proc. ULIS Workshop, Grenoble, p.73, April 20 th -21 st M. Bescond et al. JAP, submitted,

3D Mode-Space Approach  Resolution of the 2D Schrödinger equation in the cross-section: m YY, m ZZ, m YZ.  Resolution of the 1D Schr ö dinger equation along the transport axis: m trans.  The 3D Schrödinger = 2D (confinement) + 1D ( transport) 2D (confinement) 1D (transport) i th eigenstate of the n th atomic plane 11

Semiconductor conduction band  (spherical): m l =m t  diagonal EMT Three types of conduction band minima:  (ellipsoidal): m l  m t  non diagonal EMT  (ellipsoidal): m l  m t  non diagonal EMT E  E  E Δ    Electron Energy  -valleys  -valleys 12

Results: effective masses Wafer orientation: 13

Material: Ge m YY =0.2*m 0 m ZZ =0.95*m 0 m trans =0.2*m 0 m YY =0.117*m 0 m ZZ =0.117*m 0 m YZ -1 =±1/(0.25*m 0 ) m trans =0.6*m 0  4-valleys 1 st 2 nd  -valleys Z 6 nm  Non-diagonal terms in the effective mass tensor couple the transverse directions in the  -valleys Free electron mass Square cross-section: 4  4 nm, oriented wire 14

Material: Ge Square cross-section: T  T=5  5 nm, oriented wire  Total current is mainly defined by the electronic transport through the  -valleys (  bulk)  Tunneling component negligible due to the value of m trans in the  -valleys (0.6*m 0 ) 15

Square cross-section: 4  4 nm, oriented wire Material: Ge  The  4 become the energetically lowest valleys due to the transverse confinement  4-valleys: m YY =0.2*m 0, m ZZ =0.95*m 0  -valleys: m YY =0.117*m 0, m ZZ =0.117*m 0 16

Material: Ge*  4-valleys: m trans =0.2*m 0 versus  -valleys: m trans =0.6*m 0  The total current increases by decreasing the cross-section! * M. Bescond et al., IEDM Tech. Digest, p. 533 (2005). 17

3D Emerging architectures Influence of the Band structure: Silicon

Why? Scaling the transistor size  devices = nanostructures  Electrical properties depend on: Band-bap. Curvature of the bandstructure: effective masses.  Atomistic simulations are needed 1,2.  Aim of this work: describe the bandstructure properties of Si and Ge nanowires. 1 J. Wang et al. IEDM Tech. Dig., p. 537 (2005). 2 K. Nehari et al. Solid-State Electron. 50, 716 (2006). 19

Tight-Binding method Band structure calculation Concept: Develop the wave function of the system into a set of atomic orbitals. sp 3 tight-binding model: 4 orbitals/atom: 1 s + 3 p Interactions with the third neighbors. Three center integrals. Spin-orbit coupling. 1 st (4) 2 nd (12) 3 rd (12) Diamond structure: Reference 20

Tight-Binding method Band structure calculation E SS (000) eVE SS (111) eV E xx (000) eVE sx (111) eV E xx (111) eVE xy (111) eV E ss (220) eVE ss (311) eV E sx (220) eVE sx (311) eV E sx (022) eVE sx (113) eV E xx (220) eVE xx (311) eV E xx (022) eVE xx (113) eV E xy (220) eVE xy (311) eV E xy (022) eVE xy (113) eV  20 different coupling terms for Ge:* *Y.M. Niquet et al., Appl. Phys. Lett. 77, 1182 (2000).  Coupling terms between atomic orbitals are adjusted to give the correct band structure: semi-empirical method. * Y.M. Niquet et al. Phys. Rev. B, 62 (8): , (2000). 21

The dimensions of the Si atomic cluster under the gate electrode is [T Si x(W=T Si )xL G ]. Silicon Hydrogen Schematic view of a Si nanowire MOSFET with a surrounding gate electrode. Electron transport is assumed to be one-dimensional in the x-direction. Simulated device Si Nanowire Gate-All-Around transistor 22

Energy dispersion relations  In the bulk: The minimum of the conduction band is the DELTA valleys defined by six degenerated anisotropic bands.  -valleys  Constant energy surfaces are six ellipsoids 23

Energy dispersion relations Energy dispersion relations for the Silicon conduction band calculated with sp 3 tight-binding model. The wires are infinite in the [100] x-direction.  Direct bandgap semiconductor  The minimum of  2 valleys are zone folded, and their positions are in k 0 =+/  Splitting between  4 subbands T=1.36 nm T=2.72 nmT=5.15 nm 24

Conduction band edge and effective masses Bandgap increases when the dimensions of cross section decrease m* increases when the dimensions of cross section decrease : 25

Results Current-Voltage Caracteristics  No influence on I off, due to the reduction of cross section dimension which induces a better electrostatic control  Overestimation of Ion (detailled on next slide) I D (V G ) characteristics in linear/logarithmic scales for three nanowire MOSFET’s (L G =9nm, V D =0.7V) with different square sections nm 1.9 nm 2.98 nm 26

K. Nehari et al., Solid-State Electronics, 50, 716 (2006). K. Nehari et al., APL, submitted, Results Overestimation on ON-Current Overestimation of the I on current delivered by a L G =9nm nanowire MOSFET as a function of the wire width when using the bulk effective-masses instead of the TB E(k)-based values. When the transverse dimensions decrease, the effective masses increase and the carrier velocity decreases. 27

3D Emerging architectures Influence of the Band structure: Germanium

Three types of conduction band minima:  -valleys  -valleys Conduction band minima L point: four degenerated valleys (ellipsoidal).  point: single valley (spherical).  directions: six equivalent minima (ellipsoidal). 29

Indirect band-gap. The minimum of CB obtained in k X =  /a corresponding to the 4  bulk valleys. Second minimum of CB in k X =0, corresponding to the single  bulk valley (75% of s orbitals). T=5.65 nm Dispersion relations* 4  bulk valleys  2 bulk valleys Single  bulk valley  4 bulk valleys Ge * M. Bescond et al. J. Comp. Electron., accepted (2006). 30

The four bands at k X =  /a are strongly shifted. The minimum of the CB moves to k X =0. The associated state is 50% s (  character) and 50% p (  and  character)  Quantum confinement induces a mix between all the bulk valleys.  These effects can not be reproduced by the effective mass approximation (EMA). T=1.13 nm Dispersion relations Ge 31

Effective masses:  point Ge Significant increase compared to bulk value (0.04  m 0 ): From  m 0 at T=5.65nm to 0.29  m 0 at T=1.13nm  increase of 70% and 600% respectively.  Other illustration of the mixed valleys discussed earlier in very small nanowires. (1/m*)=(4  ²/h²)  (  ²E/  k²) 32

Effective masses: k X =  /a Small thickness: the four subbands are clearly separated and gives very different effective masses. Larger cross-sections (D>4nm): the effective masses of the four subbands are closer, and an unique effective mass can be calculated: around 0.7  m 0 (effective mass: m trans =0.6  m 0 for T=5nm) The minimum is not obtained exactly at k X =  /a: Ge 33

Band-gap: Ge vs Si For both materials: the band gap increases by decreasing the thickness T (EMA). E G of Ge increases more rapidly than the one of Si: Si and Ge nanowires have very close band gaps.  Beneficial impact for Ge nano-devices on the leakage current (reduction of band-to- band tunneling). Ge 34

Effective masses: Valence Band Strong variations with the cross-section: from  m 0 to  m 0 (  70% higher than the mass for the bulk heavy hole). 35

Conclusion Study of transport in MOSFET nanowire using the NEGF. Effective Mass Approximation: different materials and orientations (T>4-5nm). Thinner wire: bandstructure calculations using a sp 3 tight- binding model. Evolution of the band-gap and effective masses. Direct band-gap for Si and indirect for Ge except for very small thicknesses (« mixed » state appears at k X =0). Bang-gap of Ge nanowire very rapidly increases with the confinement: band-to-band tunneling should be attenuated. Ge is much more sensitive then Si to the quantum confinement  necessity to use an atomistic description + Full 3D* * A. Martinez, J.R. Barker, A. Asenov, A. Svizhenko, M.P. Anantram, M. Bescond, J. Comp. Electron., accepted (2006) * A. Martinez, J.R. Barker, A. Svizenkho, M.P. Anantram, M. Bescond, A. Asenov, SISPAD, to be published (2006) 36

 1D case: Concept of conduction channel and quantum of conductance Current density from Left to right: Total current density: Quantum of conductance: ++ Rq: If bosonic particles: Due to the Fermi-Dirac distribution (1 e - /state) which limits the electron injection in the active region  Resistance of the reservoirs  Description of ballisticity: the Landauer’s approach extra

Resistance of the reservoirs  Resistance of the reservoirs: the Fermi-Dirac distribution limit the electron quantity injected in a subband (D 0 =2e 2 /h). extra

Towards the nanoscale MOSFET’s M 42M M transistors /chip 10 µm 1 µm0.1 µm 10 nm Mean free path in perfect semiconductors  ballistic transport De Broglie length in semiconductors  quantum effects Channel length of ultimate R&D MOSFETs in 2006 extra

Semi-empirical methods  Effective Mass Approximation (EMA): E(k) k 0 Parabolic approximation of an homogeneous material Parabolic approximation of a finished system of atoms  (Infinite system at the equilibrium) Near a band extremum the band structure is approximated by an parabolic function: extra

New electrostatic potential New electron density 1D density (Green) Poisson Electrostatic potential Current  Simulation Code  Potential energy profile (valley (010)) Numerical Aspects 2D Schrödinger Resolution 3D density (Green) Self- consistent coupling  The transverse confinement involves a discretisation of the energies which are distributed in subbands (Neumann) Extra