SiGe modulation doped heterostructure

Thin Solid Films 517 (2009) 6105–6108

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Thin Solid Films j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / t s f

The effect of hot phonons on the hole drift velocity in a p-type Si/SiGe modulation doped heterostructure Ghassem Ansaripour ⁎ Department of Physics, Yazd University, P.O. Box 89195-741 Yazd, Iran

a r t i c l e

i n f o

Article history: Received 19 August 2008 Received in revised form 20 April 2009 Accepted 23 April 2009 Available online 3 May 2009 PACS: 72.20.Dp 73.40.Kp

a b s t r a c t In this article the effect of hot phonons on the drift velocity of holes in the high-field transport regime of a Si/ SiGe modulation doped heterostructure is presented. A theoretical model including the hot phonon production effect is implemented to compare the experimental results of hole drift velocity at high fields. At liquid helium temperature, our experimental results show that at an electric field of 1000 V/cm the hole drift velocity saturates at around vd = 7 × 105 cm/s which is in good agreement with the theoretical calculations based on the above model. The reduction of hole drift velocity at high fields is due to increasing momentum relaxation rate which is a result of the enhanced production of non-drifting longitudinal optical phonons. © 2009 Elsevier B.V. All rights reserved.

Keywords: Hole drift velocity SiGe Hot phonons

1. Introduction The advantages of the Si1-xGex/Si strained layer heterojunction system to improve the performance of traditional Si metal oxide semiconductor field effect transistor (MOSFET) devices are well recognized [1]. This potential for transport enhancement is especially exciting for p-MOSFETs, which suffer from almost three times lower transconductance compared to their n-channel counterparts at equal gate dimensions, a result of relatively poor hole mobility in Si [2–5]. The increase in low-field mobility and modified valence band structure in compressively strained Si1-xGex provides the momentum for this interest. Although significant research has been done in optimizing the Ge content and layer structure for strained Si1-xGex p-MOSFETs [6–8], the assessment of the velocity-field characteristics in strained channels has still not been adequately addressed. An accurate knowledge of high-field transport in Si1-xGex p-MOSFETs is necessary for evaluating its true potential for complementary metal oxide semiconductor (CMOS) applications and to understand the impact of non-equilibrium transport in deep-submicron devices [9]. 2. Experiment The sample used in this work is a modulation doped SiGe/Si heterostructure grown by molecular beam epitaxy, processed into

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dumbbell shape with the channel length (l = 630 µm) and width (w = 130 µm). It was grown on (100) Si at 650 °C, consisting of a 300 nm Si buffer followed by a 30 nm SiGe channel, an undoped Si spacer and a 50 nm B doped cap layer. Details of the layer structure are given elsewhere [10]. The carrier sheet density was obtained from Hall bar measurements. The two dimensional hole gas (2DHG) sample was measured to have Hall mobility and carrier sheet density of 9300 cm2 V− 1 s− 1 and 1.13 × 1011 cm− 2 at 4.2 K. The schematic structure of the sample used in this work is shown in the inset of Fig. 1(a). Aluminum was evaporated and annealed to provide Ohmic contacts. Pulsed current–voltage (I–V) measurements were performed at liquid helium temperature. In the measurements the width of the voltage pulses was in the range 1–5 µs which was applied along the length of the sample to avoid Joule heating. A Tektronix 2467B oscilloscope was used to measure both the applied voltage and current through the sample. All experimental results have been carried out at the Physics Department of Warwick University. The average drift velocity was deduced from the output voltage using vd = I / ewnh with, w being the width of the sample and nh the low-field hole sheet density. In Fig. 1(a) and (b), the open symbols show the measured drift mobility and velocity versus electric field at liquid helium temperature respectively. It is seen that the drift velocity saturates around v d = 6.9 × 10 5 cm/s at an electric field of 1.08 × 103 V/cm for the carrier sheet density of 1.13 × 1011 cm− 2. It is clear that the peak velocities predicted by theory and the measurements in the transient mode are much higher than the highest drift velocity attainable in the steady state [11,12]. The reduced drift velocity in the steady state may be related to production of warm

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G. Ansaripour / Thin Solid Films 517 (2009) 6105–6108 Table 1 Material constants of SiGe used in the calculation. m⁎ = 0.3 m0 [5,19] ħω = 40 meV L = 40 Å q0 = 5.6 × 108 m− 1 [14] nh = 1.13 × 1011 cm− 2 τp = 9 ps [15–17] τ0 = 4 fs [18]

Hole effective mass Optical phonon energy Width of quantum well Hole wave vector Hole sheet density (2DHG) Phonon lifetime Time constant for hole–phonon interaction

It can be shown that the energy relaxation rate is given by [14] eFvd =

  ħω EL 1 = 2 ½ðnðωÞ + 1Þ expð−ħω = kB TC Þ − nðωÞ τ0 ħω

ð3Þ

where EL = ħ2π2 / 2m⁎L2 is the energy shift of the lowest subband in the triangle well, F is the electric field, e is electronic charge, vd is hole drift velocity, and m⁎ is the effective mass of holes in SiGe. In what follows an effective mass of m⁎ = 0.3 m0 is assumed [5,19]. It is readily shown that momentum relaxation via the interaction with interface modes is affected solely by the momentum-conservation approximation (MCA). Thus vd is given by [14] vd =

Fig. 1. (a) Measured drift mobility and (b) velocity as a function of electric field for the 2DHG SiGe sample. The line represents the calculated drift velocity for τp = 9 ps and nh = 1.13 × 1011 cm− 2. The inset illustrates schematically the sample structure.

eFτ0 kB Tc −1 ½ðnðωÞ + 1Þ expð −ħω =kB Tc Þ : m⁎ ðħωEL Þ1 = 2

Substituting Eq. (4) in Eq. (3) gives an expression relating hole temperature to electric field: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ħωm⁎EL f½ðnðωÞ + 1Þ expð − ħω =kB Tc Þ F= kB Tc eτ 0 1=2

×½ðnðωÞ + 1Þ expð −ħω =kB Tc Þ−nðωÞg (non-equilibrium) phonons at high fields and the surface charge effects [13]. However the phonon drift would reduce substantially due to scattering of elastic phonons by imperfections during their lifetime.

ð4Þ

:

ð5Þ

Therefore Eqs. (4) and (5) can be used to deduce the drift velocity versus electric field curves. The material parameters used in the calculation are given in Table 1.

3. Model transport and results The theoretical model used in this work was developed for GaAs/ AlGaAs by Ridley [14], but adopted for our Si/SiGe modulation doped sample. In the non-degenerate case, the effective relaxation time which takes into account all the hot phonon effects, τ⁎ is given by [14]

τ⁎ = τ 0 +

τp 1=2 ðπkB Tc =ħωÞ ðn = Nc Þð1 − expð−ħω = kB Tc ÞÞ 2q0 L

ð1Þ

where τp is the optical phonon lifetime which is assumed to be 9 ps [15–17], τ0 is the time constant for hole–optical phonon interaction and is taken to be 4 fs [18], n is the hole sheet density per well width (or bulk carrier density), Tc is the hole temperature, ħω is the optical phonon energy, kB is the Boltzmann constant, Nc is the effective density of states, L is the well width and is the phase-matched wave vector (q0 = (2m⁎ħω)1/2 / ħ) [14]. The phonon occupancy n(ω) is given by
 n0 ðωÞ + τp = 2τ0 q0 L ðnkB Tc =ħωÞ1 = 2 ðn = Nc Þ expð−ħω = kB Tc Þ
 nðωÞ = 1 + τp = 2τ0 q0 L ðπkB Tc =ħωÞ1 = 2 ðn = Nc Þð1 − expð−ħω = kB Tc ÞÞ

ð2Þ

where n0(ω) is the occupancy at thermodynamic equilibrium and determined by the hole temperature, for Tc = TL(n0(ω) = 1 / ((exp (ħω / kBTC) − 1))) in which TL is the lattice temperature.

Fig. 2. (a) Variation of the effective energy relaxation time, τ⁎, and (b) calculated average phonon number per mode, n(ω), with inverse carrier temperature for several hole sheet densities.

G. Ansaripour / Thin Solid Films 517 (2009) 6105–6108

In Fig. 1(b) the line represents the calculated drift velocity as a function of electric field for carrier sheet density 1.13 × 1011 cm− 2 to fit the measured drift velocities. The fit is in good agreement with the experimental data, although it tends to slightly overestimate. At low fields (particularly around the knee point of the curve) the agreement is not quite as good as the high-field region, but still reasonable. The discrepancy between the calculated and measured results is discussed later. Fig. 2(a) shows the variation of the effective energy relaxation time with inverse carrier temperature for different carrier sheet densities at TL = 4.2 K calculated from Eq. (1). It is seen that for a given carrier sheet density the effective energy relaxation time decreases for increasing carrier temperature, while increasing the carrier density leads to higher energy relaxation time. In Fig. 2(b) we show the calculated average phonon number per mode against reciprocal carrier temperature at TL = 4.2 K. As expected, by increasing the carrier sheet density and hole temperature the average phonon number increases.

4. Discussion The drift mobility shown in Fig. 1(a) is reduced by increasing the electric field. This is explained in view of Fig. 2(b). By increasing the electric field the carrier temperature is increased and so more phonons are created. More phonons lead to more scattering and hence the mobility is reduced. From the calculated line in Fig. 1(b) a saturation velocity of ~ 7.8 × 105 cm/s at ~1.04 × 103 V/cm is obtained, whilst the experimental results as stated before show ~ 7 × 105 cm/s at ~ 1.1 × 103 V/cm. The higher calculated drift velocity (line) compared with the measured data (open symbols) over the whole velocity-field characteristics is explained as follows: Firstly, over the whole electric field range when using vd = I / ewnh to calculate the drift velocity from the experimental I–V data, it is assumed that the carrier sheet density nh is constant. However at high fields the carriers might be excited into the immobile regions where they become non-conducting. At high electron temperatures electrons in the extended states can real-space transfer to localized states giving rise to negative differential resistance (NDR) [20]. In fact this effect has been seen in a low mobility GaAs/AlGaAs structure [12,14]. In the modulation doped sample used in this work, the holes in the high mobility SiGe well are separated from their parent acceptors in the Si (doped) layer. By increasing the field the hot holes might begin to transfer in real space from low mass high mobility, SiGe layer, to a high mass low mobility, Si (doped) layer. This would cause a reduction in the carrier sheet density and because vd is inversely proportional to nh the carrier drift velocity is increased. This effect has also been observed in strained Si/SiGe heterostructures [21,22]. Secondly, during the pulse duration, the temperature of the sample may substantially rise. This is because it is not possible to eliminate Joule heating fully, even though the I–V characteristics are measured by a short pulse width. Therefore, heating causes more scattering, hence a drop in carrier mobility leads to a reduction in the measured drift velocity. It is evident from some experimental investigations as well as theoretical work that hot optical phonons can be produced by hot carriers of sufficiently high density resulting in a drop in the rate of energy relaxation [23–27]. In addition to decreasing the energy relaxation rate the production of hot phonons changes the mobility. A larger increase of carrier temperature with electric field enhances the mobility at low fields if this is influenced by ionized impurity scattering, but the main effect at high fields is a reduction of mobility caused by the enhancement of phonon scattering. However at high fields where scattering by the latter mechanism is unimportant the main effects would be the increased carrier temperature for a given field and the drop in drift velocity [14].

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A number of remarks may be made about the experimental and calculated results. In Fig. 1(b), at low fields, the measured drift velocity begins to rise at a rate less than the present theory can account for, this is well documented and is due to charged impurity scattering [28,29] and remote ionized impurity scattering [30] which correspond to the trapped impurities and relatively thin Si spacer layer (20 nm) of the sample respectively. However, beyond this regime, at high operating electric fields, the mobility of the system further decreases owing to energy dependent scattering between the 2DHG and the optical phonons. The carriers initially gain energy from the field at a higher rate, until balanced by the rate of loss of energy to the lattice by collisions, so that in steady state the mean energy of the carrier gas is increased. This causes an increase in the carrier temperature and a reduction in the 2DHG mobility. In Fig. 2(b), for a given field, by increasing the carrier sheet density from 1013 m− 2 to 1016 m− 2 the average phonon number per mode, n(ω) increases and hence more phonons means more scattering and the mobility is reduced provided the phonons are non-drifting in k-space. Also, assuming the non-degenerate regime the magnitude of the hot phonon effects would be proportional to phonon number and phonon lifetime. It is worth noting that all the hot phonon effects appear in the effective time constant, τ⁎ which is given by Eq. (1). This equation predicts that τ⁎ (hence, the hot phonon effects) is proportional to the product of bulk carrier density n (or phonon number) and phonon lifetime τp. 5. Conclusions In conclusion, pulsed field measurements were performed on a pseudomorphic remote modulation doped Si/SiGe heterostructure to study high-field hole transport and velocity-field characteristics in this system at liquid helium temperature. The Hall technique was used to measure the carrier sheet density and Hall mobility with the drift velocity of holes deduced using pulsed I–V measurements. The measured data show that at low fields the drift velocity of holes increases monotonically with electric field, but at high fields it saturates. To analyze and explain the measured results a theoretical model based on the effects of hot phonons has been used. The theoretical calculations suggest that (i) when comparing with the high-field data the production of high density non-drifting (hot) phonons which cause an increase in the effective energy relaxation time might play a significant role in determining the hot carrier transport, (ii) the drift mobility is reduced as a result of carrier scattering by the high density phonons and (iii) the modeling should be improved to take into account the aforementioned low-field scattering mechanisms. We found that an important parameter in the calculation is τ0, which is the time constant for the carrier–phonon interaction. This time constant is not known for Si and Ge. A value for τ0 of 4 fs was found appropriate with the experimental velocity-field data which is small compared to the energy relaxation time of holes in Si, i.e. ~30 fs [18]. This seems to be consistent with [30] in which a smaller time constant for germanium in comparison to silicon has been suggested. Moreover, notice that in deriving Eq. (4) the carrier–phonon interaction has been assumed to be the unscreened Fröhlich interaction [14]. In the present SiGe system, screening is more effective than in AlGaAs because the effective mass is much larger [5,19]. However, contrary to popular belief [31] screening is very effective at low temperatures and should be included. This is already clear from the work on low temperature transport in InGaAs [32,33]. The increased scattering rates reduce the drift velocity and this has immediate consequences for the speed of field effect devices. Firstly, the hot phonon reduction in drift velocity causes harm to the speed of many devices, unless velocity overshoot occurs [4] which is a result of times too short for phonon production to affect the motion of

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electrons. Secondly, many devices work at field strengths and carrier densities for which hot phonon effects are bound to occur, with (nh) inevitably present in the measured system making it more difficult to obtain high speed devices and observe two-stream instability in coupled quantum systems [14,34]. Acknowledgements The author would like to thank the following past and present members of the University of Warwick Physics Department P.J. Phillips and C.P. Parry for layer growth, T.E. Whall and E.H.C. Parker for valuable discussions and G. Braithwaite for processing and characterization of the sample used in this study. References [1] F. Schäffler, Semicond. Sci. Technol. 12 (1997) 1515. [2] J.O. Weidner, K.R. Hofmann, F. Hofmann, L. Risch, J. Mater. Sci.: Mater. Electron. 6 (1995) 325. [3] R.J.P. Lander, M.J. Kearney, A.L. Horrell, E.H.C. Parker, P.J. Phillips, T.E. Wall, Semicond. Sci. Technol. 12 (1997) 1064. [4] S. Kaya, Y.-P. Zhao, J.R. Watling, A. Asenov, J.R. Barker, G. Ansaripour, G. Braithwaite, T.E. Whall, E.H.C. Parker, Semicond. Sci. Technol. 5 (2000) 573. [5] G. Ansaripour, G. Braithwaite, M. Myronov, O.A. Mironov, E.H.C. Parker, T.E. Whall, Appl. Phys. Lett. 76 (2000) 1140. [6] S. Verdonckt-Vandebroek, E.F. Crabbe, B.F. Meyerson, D.I. Harame, P.J. Restle, J.M.C. Stork, A.C. Megdanis, C.L. Stanis, A.A. Bright, G.M.W. Kroesen, A.C. Warren, IEEE Electron Device Lett. 12 (1991) 447. [7] D.K. Nayak, K. Goto, A. Yutani, J. Murota, Y. Shiraki, IEEE Trans. Electron Devices 43 (1996) 1709. [8] R. Oberhuber, G. Zandler, P. Vogl, Phys. Rev. B 58 (1998) 9941.

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