Thin Solid Films 517 (2008) 412–415

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SO-limited mobility in a germanium inversion channel with non-ideal metal gate

Raheel Shah a,⁎, M.M. De Souza b

a Emerging Technologies Research Centre, De Montfort University, LE1 9BH, UKb EEE Department, University of Sheffield, Mappin Street, Sheffield S1 3JD, UK

⁎ Corresponding author.E-mail addresses: raheel@dmu.ac.uk (R. Shah), m.de

(M.M. De Souza).

0040-6090/$ – see front matter © 2008 Elsevier B.V. Adoi:10.1016/j.tsf.2008.08.089

a b s t r a c t

a r t i c l e i n f o

Available online 22 August 2008

Keywords:

Germanium is an attractivelectron mobility and quadthe International Technolog

GermaniumMOSFETMobilitySO phononMetal gate

y Roadmap for Semiconductors (ITRS), high-κ insulators and metal gates will berequired in conjunction with Ge technology. Key issues which will have to be addressed in achieving Getechnology are: trap free insulators, assessment of appropriate crystallographic orientations and the selectionof gate metals for the best mobility. In this work mobilities are evaluated for Ge-nMOSFET with two metalgates (Al and TiN) and high-κ (HfO2) insulator. Scattering with bulk phonons, surface roughness and high-κphonons are taken into account. It is predicted that Al as the gate material on Ge {100} substrate performs50% better than Ge {111} orientation at a sheet concentration of 1×1013 cm−2. Surface roughness is likely to

e candidate for ultra fast CMOS technology due to its potential for doublingrupling hole mobility in comparison to silicon. To maintain the requirements of

be the most damaging mobility degradation mechanism at high fields for Ge {111}.© 2008 Elsevier B.V. All rights reserved.

1. Introduction

Silicon based device scaling has almost reached its limits leading toa search for alternativematerials which can reliably be introduced intomanufacturing. Of these silicon–germanium is already in use instrained silicon technologies, whereas germanium is being activelyexplored for replacing silicon on account of its potential for highmobility [1,2]. Unlike silicon, however few studies are available onpredicted mobility of Ge-nMOSFETs [3,4]. This study is focused on theeffect of metal gates on Ge-nMOSFETs, since the effect of high-κdielectric in conjunction with non-metallic gates has already beenreported in [5]. It was reported in [5] that high-κ dielectrics have amore damaging effect on Ge based devices as compared to Si basedMOSFETs. This observation is confirmed in this study as well but forlow-medium sheet densities (Ns). For high Ns similar to Si-MOSFETsthe limiting factor is the scattering with the rough interface of thechannel. This observation is based on the fact that the degree ofroughness for Ge/insulator interface is assumed to be similar as thatfor Si/SiO2. In experiment, one can expect it to be worst, due to thedifficulties of obtaining a good insulator on Ge.

2. Model description

The models for surface-optical (SO) phonons in high-κ dielectricMOSFETs have been reported in [6,7], for poly-silicon and ideal metal,respectively. In this work, the gate metal is assumed as non-ideal (i.e.metal that can hold electrostatic potential for a finite thickness) and its

souza@sheffield.ac.uk

ll rights reserved.

impact on high-κ insulator is evaluated for the Ge substrate with twodifferent principal surface orientations i.e. {111} and {100}.

The normal component of electrostatic potential, ϕz, arising fromthe hybrid polaron modes (phonons–plasmons) is assumed to be [8]:

/z ω;Qð Þ ¼ae

zδ zb0

be−Qz þ ceQz 0Vzbtde−Qz zzt

ð1Þ

where δ is the Thomas–Fermi screening length for which a real metalcan hold the potential. The direction normal to the two interfaces isdenoted by z, t is the oxide thickness, Q is the transferred wave vectorand ω is the angular frequency with which the polarons oscillate. Thedispersion relation of the coupled modes is found to be of the sameform as that for the poly-Si gate [6] with the exception that the gatedielectric function is modulated by the term Qδ, mathematically:

e2i ωð Þ þ ei ωð Þ eg ωð ÞQδ

þ es ω;Qð Þ� �

coth Qtð Þ þ eg ωð ÞQδ

es ω;Qð Þ ¼ 0 ð2Þ

The dielectric functions in the insulator (εi), substrate (εs) and gate(εg) regions are given in the long wave length limit by the expressions:

ei ωð Þ ¼ e∞iω2−ω2

LO1

� �ω2−ω2

LO2

� �ω2−ω2

TO1

� �ω2−ω2

TO2

� � ð3aÞ

es ω;Qð Þ ¼ e∞s 1−ω2

ps Qð Þω2

!ð3bÞ

eg ωð Þ ¼ e∞g 1−ω2

pg

ω2

!ð3cÞ

Fig. 1. Solution of the dispersion relation for Si and Ge substrates.

413R. Shah, M.M. De Souza / Thin Solid Films 517 (2008) 412–415

The subscripts LO and TO stand for the Longitudinal and TransverseOptical modes of the insulator phonons, respectively, while thesuperscript ∞ over the dielectric constants ε• represent the highfrequency optical dielectric constants. The plasma frequency ωps forthe 2DEG in the substrate is given by:

ωps Qð Þ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffie2

e∞sQ ∑

i

Ni

mi

sð4Þ

where e is the charge on an electron, Ni is the population of electronsper unit area in the ith subband and mi is the effective electron massin the same subband. The gate plasma frequency ωpg is:

ωpg ¼ffiffiffiffiffiffiffiffiffiffiffie2Ng

e∞gmg

sð5Þ

where Ng and mg are respectively, the electron density and theeffective mass in the gate material. The choice of the gate materialaffects the plasma frequency via the three parameters i.e. Ng, mg

and εg∞. The electron density Ng is computed using the simpleexpression:

Ng ¼ 6:022� 1023 ZρA

ð6Þ

where ρ is volume density, A is the mass number of the material and Zdenotes the number of conduction electrons per atom (e/at.) [9].Experimentally it is observed that the “effective” number density ofelectrons (Ng′) participating in the optical transitions is different fromthe theoretical prediction of Eq. (6). Numerically it is given forintraband transitions (in International System of Units) as:

NVg ¼ m0e0ω2pg

e2SIð Þ ð7Þ

m0 and ε0 being the free electron mass and the permittivity of freespace, respectively. The ratio of theoretical electron density and theeffective electron density gives the optical mass mg (sometimesreferred to as the conductivity mass [10,11]) appearing in Eq. (5) i.e.[12].

mg

m0¼ Ng

NVgð8Þ

In case the optical dielectric constant εg∞ is not known experimen-tally then using Eqs. (7), (8) and (5) the high frequency dielectricconstant can be calculated. The plasma frequency is usually extractedfrom the real part of the frequency (ω) dependent dielectric functionat ω=0.

The modulation of the gate dielectric function by the factor Qδdescribed in Eq. (2) also affects the total dielectric function of the MOSstructure (cf. Eq. (33) of [6]).

3. Model application

The main purpose of this study is to evaluate the effect of metalgate on Ge based MOSFETs with high-κ insulators. It is thus

Table 1Effective masses for b111N valleys, parameters for phonon scattering [15] and surfaceroughness [18]

Phonon mode or energy Coupling constant

ml=1.588 m0 Intravalley: Acoustic 11.0 eVmt=0.08152 m0 Intravalley: Optical: 37.054 meV 5.5×108 eV cm−1

SR: Δ=0.48 nm Intervalley (LA, LO): 27.575 meV 3.0×108 eV cm−1

SR: Λ=1.3 nm Intervalley (TA): 10.3407 meV 0.2×108 eV cm−1

appropriate to recall the band structure of germanium. The lowestconduction band of Ge has 4 absolute minima along the “Λ” directionat the “L” point of the Brillouin zone, 6 relative minima along the “Δ”direction and another relative minimum at the “Γ” point. The “L-valleys” are 0.18 eV below the Δ valleys and 0.14 eV below the Γ valley[13]. It is thus argued that for small longitudinal effective field, onlythe “L-valleys” are occupied [14]. In this context it is reasonable toaccount only for L-valleys (along the b111N direction) in the transportcalculations for low source to drain bias. The longitudinal andtransverse effective masses which characterise the conduction valleysare given in [15] and are used in this study. For completeness theeffective masses are repeated in Table 1.

The quantization of electron energy in the inversion layer isevaluated in the triangular well approximation [16]. In this approx-imation the electrostatic potential is given by:

V zð Þ ¼ eEeffz ð9Þ

where Eeff is the transverse effective field in the substrate. Thequantized energies (Eigen values) in the subbands are evaluatedthrough the relation:

Eij ¼J2

2mzj

!1=33πeEeff

2i−14

� �� �2=3 j ¼ 1;2i ¼ 1;2;3; N

ð10Þ

The index j stands for the two ladders i.e. valleys with differentdegeneracy factors, while i runs for the number of finite subbands in aconduction valley. Here mz denotes the effective mass normal to theinsulator/substrate interface. The wave functions in terms of Eigenvalues are then given as:

ψij zð Þ ¼ ΝAi2mzjeEeff

J2

� �1=3

z−EijeEeff

� �" #ð11Þ

where N is the normalization factor and Ai is the airy function given as(among the various forms):

Ai xð Þ ¼ 1π

ffiffiffix3

rK1=3

23x3=2

� �ð12Þ

K(x) is the modified Bessel function of the second kind. In the nextstep the subband population is calculated via:

Nij ¼njmdjkBT

πJ2 ln 1þ eEF−EijkBT

� �ð13Þ

Fig. 4. Effective mobilities for Ge substrates. Bulk phonon, surface roughness and high-κphonon scattering included.

Fig. 2. Scattering strength of gate plasmons as a function of scattered wave vector.

414 R. Shah, M.M. De Souza / Thin Solid Films 517 (2008) 412–415

The density of state mass is denoted by md and nj is the valleydegeneracy of the jth ladder. EF is the Fermi energy of the system andis calculated by numerically solving the equation:

Ns ¼ ∑j∑iNij ¼ ∑

j∑i

njmdjkBT

πJ2 ln 1þ eEF−EijkBT

� �ð14Þ

The non-ideal metal gate/high-κ model is applied to germaniumsubstrate with two different metal gates (TiN and Al) and with HfO2 asthe high-κ insulator. Simulations are performed for the two surfaceorientations ({111} and {100}) of the Ge substrate. Comparison is alsomade for the (100) silicon substrate, since it is the most favourableorientation for silicon based devices.

4. Results

Fig. 1 shows a comparison of the phonon dispersions for (100)silicon and (111) germanium as a function of transferred wave vectorQ. For illustration, titanium nitride (TiN) is selected as the metal gate.Solid lines correspond to Si (100) substrate and broken lines to Ge(111) substrate. The optical parameters for TiN are selected from[10,17], i.e. Ng=1.87×1022 (cm−3), mg=1.4 m0 and εg=2.599ε0. Themost significant difference arises in the substrate mode (the lowestmode) is due to the different density of state masses appearing in theevaluation of the substrate plasma frequency ωp,s. The remainingmodes in increasing order are LO1, LO2 and the gate plasmon mode,

Fig. 3. Optical phonon scattering strength for Si and Ge.

which remain the same for both materials, being insulator-like andgate-like mode respectively.

Fig. 2 illustrates the scattering strength of gate plasmons (ΛSP in thenotation of Ref. [6]). The differences for the two materials arise fromthe definition of the total effective dielectric function appearing in ΛSP,which is sensitive to the differences in the dielectric functions in eachregion of the stack. Fig. 3 highlights the scattering strength, ΛSO, of theSO phonons, for the two materials. The same difference in substrateplasma frequency is propagated in this scattering strength as well. Asin Fig. 1, solid lines correspond to Si (100) and broken lines to Ge (111)substrate in both Figs. 2 and 3.

Finally in Fig. 4 the effective mobility for a range of sheet densitiesis given. Mobilities for the two different germanium substrateorientations are plotted for TiN and Al metal gates. The optical datafor Al are taken from [12] i.e. Ng=1.57×1023 (cm−3), mg=1.15 m0 andεg=1.05ε0. Here in Fig. 4, solid lines correspond to Ge (100) and brokenlines to Ge (111) substrate, while red lines represent Al-gate and blacklines are for TiN-gate. In the effective mobility, bulk phonon andsurface roughness scatterings along with the high-κ phonons areincluded, see Table 1 (there Δ and Λ are the parameters for theexponential autocovariance function, not to be confused with thedirectional lines of the Brillouin zone).

5. Conclusion

The main results of this study are given in Fig. 4. The simulationsreveal that phonon limited mobility is higher for the Ge {111}orientation while surface roughness limited mobility is better alongthe Ge {100} surface. This situation ultimately leads to the cross overof effective mobilities between Ge {111} and {100} surface orientationwith the evolving Ns.

Simulations also predict higher mobilities for Ge {111} surfaces inthe low-medium sheet concentration regime but poor mobilities formedium-high Ns values as compared to the Ge {100} orientation. Aland TiN show similar behaviour for medium-high Ns range but for lowsheet densities Al shows strong nonlinearity due to screening andanti-screening effects.

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