Journal of Computational Electronics 2: 443–448, 2003c© 2003 Kluwer Academic Publishers. Manufactured in The Netherlands.

Self-Consistent Subband Structure and Mobility of Two Dimensional Holesin Strained SiGe MOSFETs

SANTHOSH KRISHNAN AND DRAGICA VASILESKADepartment of Electrical Engineering, Arizona State University, Tempe, AZ, 85287-5706 USA

skrish2@imap2.asu.eduvasilesk@asu.edu

Abstract. The motivation to research strained SiGe layers on relaxed silicon is to enhance the very low holemobility observed in conventional Si MOSFETs. This is mainly due to large hole effective masses and strongsurface roughness scattering. In this work, the buried SiGe quantum well, chosen as the active channel for carriertransport, enhances mobility in two ways. First, the carriers are confined in the SiGe well and are removed from theSi-SiO2 interface, so interface roughness is not a big factor. Secondly, strained SiGe has a lower hole effective massthan unstrained Si and the mobility is expected to increase as the Ge concentration increases. We present the resultsof a self-consistent subband structure and low field mobility calculations for holes in a Si-SiGe heterostructure FET.Our results are in agreement with the experimental work by Garone and coworkers, indicating the validity of ourmodel.

1. Introduction

Mobility in the inversion layers of a MOSFET is a veryimportant physical quantity since it limits the drain cur-rent. It is well known that the electron and hole mobil-ities in an inversion layer on a (100) substrate followuniversal curves at room temperature independent ofsubstrate doping or substrate bias when plotted as afunction of the effective electric field, Eeff,

Eeff = q

εsc(Ndepl + ηNs). (1)

In Eq. (1), q is the elementary charge, εsc is the per-mittivity of the semiconductor, Ndepl is the depletioncharge density and Ns is the inversion layer density.The quantity η is a key parameter in defining the uni-versal mobility relationship and its value is 1/2 for elec-trons and 1/3 for holes. Experimental results of elec-tron and hole mobilities [1] reveal that the universalbehavior holds over a wide range of substrate impu-rity concentrations and temperatures. The differencein the universal slope for holes at low temperaturesindicates that surface roughness has a different Eeff de-

pendency for electrons and holes. Furthermore, holemobility is about a factor of three less than electronmobility and is a limiting factor to the speed of CMOSdevices.

A different device structure proposed by severalgroups [2–4], involves moving the holes away fromthe Si-SiO2 interface by confining them in a Si1−x Gex

quantum well. Pseudomorphic strain in the activeSi1−x Gex layer has the advantages of lifting the de-generacy of the heavy and light hole bands, therebyincreasing the hole density in the well (which has beenconfirmed by Hall effect and capacitance—voltagemeasurements) and lowering the effective masses ofboth bands, particularly the heavy hole band by alarge extent. Therefore, increasing the Ge concentra-tion would increase the strain and thus enhance thehole mobility.

2. Theoretical Model

The simulated structure has a 10 nm thick gate and2 nm of gate oxide followed by a 2 nm thick Si caplayer. Underneath lies the strained SiGe quantum well,

444 Krishnan

Figure 1. Strained SiGe p type MOSFET. The quantization of holestates is calculated along the dashed line AB.

the thickness of which is 5 nm. The Si depletion layerand substrate then make up the rest of the device. The Sicap, SiGe well and Si substrate are all uniformly dopedto 1017 cm−3. This structure is schematically shown inFig. 1.

In calculating the inversion layer hole mobility, itis necessary to know the subband structure. The holesubband structure is calculated self-consistently alongthe device depth by solving the 1D Poisson and the 1DSchrodinger equations. The valence band energy levelspitting between the heavy and the light hole bands iscalculated following the approach of Kahan et al. [5].This band splitting results in a greater hole density inthe SiGe well relative to the Si cap region. In particular,the heavy hole band is much closer to the Fermi energyand so the contribution to the sheet hole density fromthe SiGe well is much higher.

The transport in the strained 2D SiGe quantum wellis modeled using the ensemble Monte Carlo (EMC)technique and the holes are initially distributed betweenthe different subbands in a way which is consistent withthe quantum simulation results. The holes are then as-signed energies such that their average energy equalsthe thermal energy for 2D carriers. A parabolic bandmodel is used since this is a low field calculation. Thisallows us to use a simple effective mass for the mobil-ity calculations. The scattering mechanisms included inthe transport model are phonon scattering (acoustic andoptical), and non phonon mechanisms like alloy scat-tering, surface roughness scattering and Coulomb scat-

tering, including screening within the random phaseapproximation (RPA).

2.1. Phonon Scattering

The elastic and equipartition approximation is used tocalculate the acoustic phonon scattering rate, which isgiven by,

�acn (k‖) = Dacm ∗ kB T

ρv2s h3

∑m

Wnm θ (Ek‖ + εn − εm)

(2)Wnm =

∫dzψ2

m(z)ψ2n (z)

where the summation is over the final subbands, n is theinitial subband and m is the final subband. Consideringthe optical phonons, the scattering rate is given by,

�opn (k‖) = m ∗ D2

opn0

2ρh2ω0

∑m

Wnm θ

× (Ek‖ + εn − εm ± hω0), (3)

where the positive and negative signs indicate phononabsorption and emission respectively and Wnm is theoverlap factor of the carrier wavefunctions along thedirection of confinement. In Eqs. (2) and (3), the Heavi-side step function in energy, θ (E), has a simple physicalexplanation. The carriers are scattered to the final sub-band only when they have acquired enough energy fromthe electric field to make the intersubband transition.

2.2. Non Phonon Scattering—Alloy, SurfaceRoughness and Coulomb Scattering

Disorders in alloys are a source of scattering for carri-ers and thus limit the mobility enhancement expectedfrom strain in the active layer. Disorders such as atomicordering and phase separation change the crystal po-tential that a carrier experiences as it moves through thecrystal. Virtual crystal approximation is used in defin-ing the average potential the carrier experiences in analloy of type A1−x Bx as,

Utot = UA(1 − x) + UB x . (4)

The difference between the real potential and theassumed virtual crystal potential is represented withineach unit cell by a highly localized potential whichfor A type atoms is x�U , and for B type atoms is

Self-Consistent Subband Structure and Mobility of Two Dimensional Holes 445

(1 − x)�U . The quantities, UA, UB and �U are notwell defined and it is still debated whether �U shouldbe defined as the difference in the electron affinities,or the difference in the bandgap of the two species, orthe difference in the band offsets. The quantity �Uis called alloy potential and is usually adjusted to fitexperimental data. The scattering rate for alloy disorderscattering is calculated to be,

�alloyn (k‖) = m∗

4πh2 �U 2x(1 − x)a3

×∑

m

θ (Ek‖ + εn − εm)Wnm. (5)

Scattering due to surface roughness at the semicon-ductor oxide interface, is included according to a modelproposed by Goodnick et al. [6]. This model character-izes the fluctuations in the interface by two parameters,the root mean square (rms) height � and the roughnesscorrelation length L , which is roughly the distance be-tween the fluctuations. The model proposed assumesan exponential form (instead of a Gaussian form) forthe correlation function, which results in a better fitto experimental data of the surface fluctuations. Thesurface roughness scattering rate is calculated as,

�surfn (k‖) = e2m∗

4πh3 F2nm�2L2

∑m

θ (Ek‖ + εn − εm)

×∫

ϕ

1

1 + L2k2‖ sin2

(ϕ

2

)dϕ. (6)

In Eq. (6), Fnm = ∫dzψm(z)εz(z)ψn(z) and εz(z) is

the electric field along the depth. It is to be noted thatthe integration over the angle ϕ appears in the scatteringrate calculation indicating that surface roughness is anon isotropic scattering process.

Coulomb scattering due to a charge center in thedepletion region is modeled using the method of imagecharges to determine the potential at a point as,

ϕdepl(r, z > 0)

= e

4πεsc

[1√

(r − ri )2 + (z − zi )2

+ εsc − εox

εsc + εox

1√(r − ri )2 + (z + zi )2

](7)

After calculating the matrix elements for the transi-tion probability, the scattering rate due to all the chargesin the depletion region is obtained by integrating alongthe device depth direction, giving rise to the following

expression for the scattering rate,

�depln (k‖) = e4m ∗ Ndepl

16π2κ2h2Wdepl

∑m

θ (Ek‖ + εn − εm)

×∫ dϕ

ϕ

Snm(q(k‖, ϕ)). (8)

The explicit form of the symbols in Eq. (8) is as follows,

Snm(q) = 1

q2A2

nm(q)

Wdepl∫zi =0

dzi O2nm(q, zi ) (9)

Anm(q) =∫ ∞

z=0dz ψm(z)eqzψn(z) (10)

Onm(q, zi ) =[

c1eqzi + c2e−qzi + c1

(e−qzi

a+nm(q, zi )

Anm(q)

− eqzia−

nm(q, zi )

Anm(q)

)](11)

a±nm(q, z) =

∫ z

z′=0dz′ ψm(z′)e±qz′

ψn(z′) (12)

c1 = 0.5

(1 + εox

εsc

); c2 = 0.5

(1 − εox

εsc

)

(13)

2.3. Screening

An electron in a system of mobile carriers at low den-sities experiences the full effect of the perturbing po-tential. At high carrier densities however, the electrondoes not feel the full effect of the potential because ofscreening. The inverse of screening wave vector rep-resents a characteristic length in real space over whichthe effect of the potential dies out. The screening wavevector qs(q) is generally taken to be independent of q.For example in a 2DEG the Debye-Huckel screeningwave vector is given by,

qDHs � e2

2εsc

∂ns

∂ E f= e2

2εsc

ns

kB T(14)

But in the extreme quantum limit, qs is a function ofq (decreasing rapidly at short wavelengths or large q),so that the effect of screening is less pronounced thanin Eq. (14) and is given by

qns (q)=− e2

2εscgv

m∗πh2

×∫ 1

0

dy

1 + exp[

µ−εn

kB T + εn4kB T (1 − y2)

] . (15)

446 Krishnan

This screening wave vector is used in calculatingthe screened matrix elements for Coulomb scattering.The screened matrix elements are much smaller thanthe bare matrix element (especially at low q), implyinga smaller transition rate and an increase in mobility.

Figure 2. Energy band profile along device depth.

Figure 3. Strain splitting of the valence band, normalized probability densities and eigenenergies.

3. Simulation Results

Figure 2 shows the energy band profile along the devicedepth. The zero of the energy axis is the Fermi energy inthe semiconductor. The inset shows the valence band

Self-Consistent Subband Structure and Mobility of Two Dimensional Holes 447

splitting due to pseudomorphic strain induced in theSiGe quantum well. The applied gate bias is −2.0 V.It is also seen that the heavy hole band in the quantumwell is much closer to the Fermi energy and this resultsin greater hole charge density in this band. As the bias

Figure 4. Comparison of classical (unstrained SiGe) and self-consistent quantum (strained SiGe) hole concentrations.

Figure 5. Hole effective mobility for different doping densities and different Ge concentrations.

increases, the band bending becomes more pronouncedand so holes spill over into the Si cap region.

Figure 3 summarizes the results of the subband struc-ture calculation. The negative potential energies −Evhh

and −Evlh show the confinement in the SiGe quantum

448 Krishnan

well. The normalized probability densities correspond-ing to four subbands—three in heavy hole and one inlight hole band are shown. Also shown is the positionof the eigenenergies with respect to the Fermi level.

In Fig. 4, the self consistent hole concentration(strained SiGe) is compared with the classical (relaxedSiGe) hole concentration. There are two things thatare obvious from the results shown. First, the quan-tum hole concentration is displaced from the interfaceand its maximum occurs at 15 A from the Si cap-SiGeinterface. Secondly strain in the SiGe layer has liftedthe degeneracy of the valence band resulting in a muchhigher hole concentration in the well.

Figure 5 shows the effective mobility of holes incomparison with experimental results. It is seen thatthe results agree closely with the experimental resultsof Garone et al. [7]. Also note that the simulationsand the experimental results have a characteristic slopein agreement with that of the universal hole mobilitycurve. The improvement is about a factor of 50%, inagreement with Garone’s results.

4. Conclusion

It is clearly seen that strain improves the mobility instrained SiGe but one has to contend with alloy scat-tering which limits the expected mobility increase. It isto be noted that the mobility does not increase mono-tonically with the Ge concentration. This is due to thefact that the heavy hole mass becomes more or lessconstant after about 40% Ge [8], leading to saturation

of the mobility enhancement. Experimental work doneby Collaert et al. [9] indicates that strained SiGe de-vices exhibit excellent device characteristics down to50 nm. Thus, strained SiGe devices offer an attrac-tive yet, relatively inexpensive alternative to Si devicesin terms of improved device performance, use in longwavelength optoelectronic applications and most im-portantly, speed of operation.

Acknowledgments

The authors would like to thank the National ScienceFoundation for the financial support to this project.

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