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Low temperature characterization of effective mobility in uniaxially and biaxially strained nMOSFETs
Solid-State Electronics 50 (2006) 644–649 www.elsevier.com/locate/sse
Low temperature characterization of effective mobility in uniaxially and biaxially strained nMOSFETs F. Lime a, F. Andrieu b, J. Derix a, G. Ghibaudo a
a,*
, F. Boeuf c, T. Skotnicki
c
IMEP, ENSERG, Federation de Recherche FMNT-RA, BP 257, 38016 Grenoble Cedex, France b CEA-LETI, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France c STMicroelectronics, 850 rue Jean Monnet, 38960 Crolles Cedex, France Received 22 November 2005; received in revised form 8 March 2006; accepted 15 March 2006
The review of this paper was arranged by G. Ghibaudo and T. Skotnicki
Abstract In this paper, a detailed investigation of the mobility in two different strained-Si technologies has been conducted. The mobility gain due to strain is nearly lost in short channel devices. The short channel loss observed in both cases is attributed to a bigger contribution of Coulomb scattering and to a larger subband splitting due to pocket implants. Finally, the mobility gain reduction with temperature lowering has been quantitatively interpreted by a simple analytical model accounting for the fourfold valley depopulation and considering different scattering rates for the two first subbands. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Strained-Si technologies are of great interest since it allows enhanced carrier mobility and in turn increased drive current [1]. However, it is not clear whether the mobility gain of strained-Si nMOSFETs is due to a gain in conduction mass [2,3] or to a reduced phonons scattering rate [4]. In this paper, we present a detailed analysis of mobility limiting mechanisms between process induced (uniaxial) and substrate induced (biaxial) strained nMOSFETs. Owing to a simple two-band mobility model, we explain satisfactorily the mobility gain dependence with temperature and channel length. 2. Experimental details We have used tensely strained-Si nMOS devices with two different architectures and featuring poly gate material. *
Corresponding author. Tel.: +33 4 76 85 60 58; fax: +33 4 76 85 60 70. E-mail address: [email protected] (G. Ghibaudo).
0038-1101/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2006.03.036
Indeed, two different approaches were used to strain the channel [5]. The first one consists of an epitaxy of 15 nm Si on a relaxed Si0.8Ge0.2 substrate (sSi) to induce biaxial stress in the Si layer. The second one is a process induced type of stress, as it relies on the contact Etch Stop Layer (ESL) to induce uniaxial stress [6]. These two different strained-Si devices were compared with their own reference, a process matched for ESL and a threshold matched for sSi. The measurements were done with a HP 4155B semiconductor parameter analyzer at wafer level. Mobility was extracted from the drain current at low Vd (10–50 mV). It is well known that this extraction can be difficult in modern cutting edge devices due to the strong gate current and uncertainties in the effective gate length of short channel devices. That’s why we compensated for the gate current using the method given in [7,8], and, we extracted the effective gate length using capacitance measurements made on short channel devices [9]. Moreover, it should be noted that the method used for mobility extraction is independent of the source–drain series resistance [10].
F. Lime et al. / Solid-State Electronics 50 (2006) 644–649
energy, together with the out-of-plane twofold valleys. However, as our devices are strained along h1 1 0i, the band splitting in the uniaxially and biaxially strained devices is qualitatively the same (see Fig. 1). First, the channel length mobility dependence will be discussed. It is plotted in Fig. 2 for the two types of stress and their reference. The mobility of the sSi reference is lower than that of the ESL due to a higher channel doping. It is clear from Fig. 3 that the low field mobility gain is quite different between the two architectures. For the sSi case, the higher gain is obtained on the longer channels with a maximum of 50% and decreases with gate length. For ESL, the mobility gain is zero for the long channel case, but increases with gate length until it reaches a maximum of 50% at approximately 0.16 lm before to fall off. This latter behavior was expected as the strain in the ESL case is located near the source and drain and therefore becomes stronger with short channels. Furthermore, it is worth noting that, even if the mobility gain decreases below
3. Results and discussion The mobility gain of strained-Si nMOSFETs comes from the splitting of the conduction band. For tensile biaxial strain, the energy of the twofold degenerate valley is lower compared to the fourfold one. This results in an increase of the twofold valley occupancy, which has a lower conductivity mass, and decrease the phonon scattering rate between the twofold and the fourfold valleys (intervalley scattering). These two effects cumulate to give rise to a better mobility. This is true in bulk strained Si but was questioned by Takagi et al. [4] for 2DEG (i.e. inversion layer). Indeed Takagi argue that, for intravalley phonon scattering, the lower conductivity mass is compensated by the valley degeneracy and the inversion layer thickness. In that case, the better mobility of the twofold valley would then comes from a lower intervalley phonon scattering rate compared to the fourfold valley. In uniaxially strained devices along the h1 0 0i direction, two of the fourfold degenerate valleys should be lowered in
No strain
645
Biaxial
Uniaxial along <010>
<010>
Δ2in plane
Δ4in plane Ec
Ec
ΔEstrain
ΔEstrain
Ec
Δ2out of plane+ Δ2in plane
Δ2out of plane
(a) Uniaxial along <110>
<110>
Δ4in plane Ec
ΔE strain Δ2out of plane
(b) Fig. 1. Ellipsoids of constant energy and the associated conduction band split, corresponding to no strain, biaxial strain, uniaxial strain along h0 1 0i (a) and uniaxial strain along h1 1 0i (b) cases.
646
F. Lime et al. / Solid-State Electronics 50 (2006) 644–649 600
500
2
µ0 (cm /V/s)
400
300 sSi Ref sSi ESL Ref ESL
200
100 0.01
0.1
1
10
L (µm) Fig. 2. Low field mobility behavior with effective gate channel length for the strained nMOS and their respective reference.
60
In Fig. 4, is plotted the temperature dependence of the low field mobility l0 for short and long channel transistors of the two technologies. The mobility is enhanced when decreasing the temperature, until it eventually saturates when Coulomb contribution dominates. So the slower increase that we observe for short channel devices can be attributed to a higher Coulomb scattering contribution originating from halo and pockets. By comparing the sSi and ESL case, we can also see that sSi has a higher Coulomb contribution, confirming the higher doping. We can also see in Fig. 5 that the low field mobility gain of short channel devices is weak and nearly constant with temperature, meaning that the mobility is governed by a similar mechanism in the strained-Si devices and their references. The constant positive mobility gain of sSi short channel devices, however, is an artifact due to different implantation/diffusion conditions compared to the reference. To confirm this issue, we have extracted the mobility gain at high field (Qi = 106 C/cm2). In that case, the Coulomb scattering is strongly attenuated due to the screening of the inversion layer. As expected, in these
2000
40
1800
30 sSi ESL
20
2
10 0 -10 0.01
sSi
1600 µ 0 (cm /V/s)
Mobility gain (%)
50
1400 1200 L = 2 µm
1000 800
0.1
1
10
Gate length (µm)
400
Fig. 3. Low field mobility gain extracted from Fig. 1.
where sph, sc and ssr are the scattering rates of phonons, Coulomb and surface roughness contributions, and m* is the effective conductivity mass. i refer to the twofold or fourfold degenerate valleys. The total mobility is calculated by averaging over the population count of each valley. m* and sph are responsible for the better mobility of strainedSi devices. It should also be noted that the phonon contribution decreases when lowering the temperature.
200
0
50
100
150
200
250
300
T (K)
(a)
2000 1800
ESL
1600
2
µ 0 (cm /V/s)
0.1 lm in both cases, the ESL gain seems better between 50 nm and 0.3 lm gate lengths. In order to get more insights into the mobility behavior of these devices, we performed low temperature measurements. Indeed, there are three main scattering mechanisms that limit the mobility. The effective mobility can be written in the following form for the twofold or the fourfold degenerate valleys: 1 mi i 1 i 1 i 1 ðs ¼ Þ þ ðs Þ þ ðs Þ ð1Þ ph c sr lieff q
L = 0.1 µm
600
1400 1200 1000 L = 2 µm
800
L = 0.1 µm
600 400 200 (b)
0
50
100
150
200
250
300
T (K)
Fig. 4. Low field mobility evolution with temperature and channel length for the two technologies.
F. Lime et al. / Solid-State Electronics 50 (2006) 644–649
60
60
30 20
-2
µ @ Qi /q = 6.25x10 cm µ0
12
10 0 -10
sSi
-20 0.01
0.1 Effective gate length (µm)
(a)
50
Mobility gain at high field (%)
Mobility gain (%)
40
T = 300, 80, 10 K
12
50
647
40
sSi Lg = 2 µm
30 20 ESL Lg = 0.1µm
10 0 -10
1
-2
Qi /q = 6.25x10 cm
50
100
150
12
20 10 0 -10
-20 0.01 (b)
Mobility gain at low field (%)
30
ESL
0.1 Effective gate length (µm)
350
60
T = 300, 200, 80 K
Mobility gain (%)
40
300
-2
µ @ Qi /q = 6.25x10 cm µ0
50
250
T (K)
(a)
60
200
50 40 30 20
ESL Lg = 0.1 µm sSi Lg = 2 µm
10 0
1
Fig. 5. Evolution with channel length of the low field mobility gain due to the decrease in temperature for sSi and ESL technologies and their reference.
curves, the artifact is removed. For the long channel case, Fig. 5 shows a slightly increased mobility gain at high field, which can also be attributed to the reduced Coulomb scattering. In the ESL case, the Coulomb scattering is lower and there is little difference between the high field and low field curves. These features show that Coulomb scattering is a critical issue for short channel devices, which could impede the mobility gain expected from strain. The evolution of the mobility gain with temperature (also partially indicated in Fig. 5) is plotted in Fig. 6 for the two technologies at gate lengths where the strain is most effective i.e. short channel for ESL and long channel for sSi. It is interesting to note that the two technologies exhibit the same temperature dependence even though different channel lengths are considered (see Fig. 4). We can again see that the sSi mobility gain is higher at high field, in opposition to the ESL case. It should be highlighted that the mobility gain at high field is generally and inexplicably found higher than what accurate simulation and theory predict [11].
-10
0
50
100
150
200
250
300
350
T (K)
(b)
Fig. 6. Evolution with temperature of the mobility gain at high field and low field for sSi and ESL technologies, at a channel length where the stress is effective. Dash line is a simulation for a Si1xGex virtual substrate with x = 0.2 (D = 0.13 eV).
In order to investigate the mechanisms responsible for the mobility gain at low field, we performed very simple calculations based on Boltzman statistics, which is reasonable at low field (i.e. close to Vt). In fact, the mobility can be expressed by the Kubo–Greenwood formula as
l¼
P q i i mc
Z
1
of ðEÞ gðEÞ dE ðE Ei Þ si ðEÞ oE Ei Z 1 P f ðEÞ gðEÞ dE i
ð2Þ
Ei
where the summation is over all the subbands i. g(E) = md/ ph2 is the density of states and si(E), mic and Ei the scattering rate, conduction mass and the energy of the bottom of the subband i. f(E) is the Fermi distribution function. If we consider in Eq. (2) the first two subbands of the twofold and fourfold valleys with different scattering times s1 and s2, and we replace the Fermi distribution by the
648
F. Lime et al. / Solid-State Electronics 50 (2006) 644–649
Boltzman distribution function, then the low field effective mobility can be equated to
0.8 300K
q s1 q s2 E0 þ D 2 1 md nd 1 1 þ md nd 2 2 exp kT mc mc l0 ¼ E0 þ D m1d nd 1 þ m2d nd 2 exp kT ð3Þ
Mobility gain
0.6
0.4
0.2
where indices 1 (2) stands for the twofold (fourfold) degenerate valleys, nd1 and nd2 are degeneracy factors, D is the splitting between the two subbands resulting from the strain, and E0 is the subband splitting with no strain. The mobility gain is obtained by normalizing Eq. (3) when D = 0. Table 1 gives the values of the conduction, mc, and density of states, md, masses, and degeneracy factor, nd, used for the twofold and fourfold valleys. This simple model has been used to explain the temperature dependence of the low field mobility gain (see dashed line of Fig. 6). Indeed, a good agreement with data has been obtained after adjusting the scattering rate ratio, s2/ s1 = 0.4 (The subband splitting E0 has been evaluated by Poisson–Schrodinger simulations, E0 = 50 mV). The higher scattering rate in the fourfold valleys can be attributed to a larger intervalley phonon scattering as compared to the twofold valleys [4]. The monotonous decrease of the mobility gain as reducing the temperature can therefore be simply explained by the progressive de-activation of the fourfold subband contribution, canceling out any straininduced band splitting beneficial effect.
Table 1 Masses and degeneracy factors used in the model
Twofold sub. Fourfold sub.
md
mc
nd
0.19 0.42
0.19 0.32
2 4
100K 0
0
0.1
0.2 Ge content
0.3
0.4
Fig. 8. Mobility gain versus substrate Ge content (symbols from [12], solid line from Eq. 2, E0 = 40 meV, D = 0.67 Æ x).
In the case where the scattering rate ratio is taken equal to one (s2/s1 = 1), we found a maximum gain of 20% at 300 K (see Fig. 7), resulting only from the different conduction masses. This situation is encountered in short channel devices where Coulomb scattering dominates, yielding a significant mobility gain reduction as found for both technologies (Fig. 5). This reduction is also accentuated by the increasing doping level in short channels due to pocket implant, giving rise to enhanced subband splitting (larger E0 in Eq. 2) and, in turn, lower population in the fourfold valleys. The physical consistency of this mobility analysis has also been validated after changing the strain level in sSi through the substrate Ge content by comparison to literature data (see Fig. 8). To this end, the strain-induced band splitting D has been varied linearly with Ge content x as D = 0.67 Æ x (eV). The good agreement between model and experiment is worth to be noted. The 100 K curve underlines again the fact that the mobility gain is fully lost as the fourfold valleys are de-populated. 4. Conclusion
Mobility gain (%)
60
τ2/τ1=0.4
40
20
τ2/τ1=1 0
0
100
200
In this paper, we have investigated in details the mobility in two different strained-Si technologies. We found that the mobility gain of our devices is nearly lost on short channels. The short channel loss observed in both cases is attributed to a bigger contribution of Coulomb scattering and to a larger subband splitting due to pocket implants. Finally, the mobility gain reduction with temperature lowering has been quantitatively interpreted using a simple analytical model accounting for the fourfold valleys depopulation and considering different scattering rates for the two first subbands.
300
T (K) Fig. 7. Variation of mobility gain with temperature for two scattering time ratios corresponding to phonon (=0.4) and Coulomb (=1) scattering rates.
Acknowledgement This work has been partially supported by the European NANOCMOS integrated project.
F. Lime et al. / Solid-State Electronics 50 (2006) 644–649
References [1] Rim K, Welser J, Hoyt JL, Gibbons JF. IEDM Tech Dig 1995:517. [2] Sugii N, Washio K. Jpn J Appl Phys 2003;42:1924. [3] Haugerud BM, Bosworth LA, Belford RE. J Appl Phys 2004;95(5): 2792. [4] Takagi S, Hoyt JL, Welser JJ, Gibbons JF. J Appl Phys 1996;80(3):1567. [5] Boeuf F et al. SSDM 2004;3:16. [6] Thomson S et al. IEDM Tech Dig 2002:61–4.
649
[7] Lime F, Guiducci C, Clerc R, Ghibaudo G, Leroux C, Ernst T. SolidState Electron 2003;47:1147. [8] Romanjek K, Lime F, Ghibaudo G, Leroux C. Solid-State Electron 2003;47:1657. [9] Romanjek K et al. EDL 2004;25:583. [10] Cretu B, Boutchacha T, Ghibaudo G, Balestra F. Electron Lett 2001;37:717. [11] Fischetti MV, Gamiz F, Hansch W. J Appl Phys 2002;92(12): 7320. [12] Welser JJ et al. IEDM Tech Dig 1994:373–6.
Low temperature characterization of effective mobility in uniaxially and biaxially strained nMOSFETs F. Lime a, F. Andrieu b, J. Derix a, G. Ghibaudo a
a,*
, F. Boeuf c, T. Skotnicki
c
IMEP, ENSERG, Federation de Recherche FMNT-RA, BP 257, 38016 Grenoble Cedex, France b CEA-LETI, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France c STMicroelectronics, 850 rue Jean Monnet, 38960 Crolles Cedex, France Received 22 November 2005; received in revised form 8 March 2006; accepted 15 March 2006
The review of this paper was arranged by G. Ghibaudo and T. Skotnicki
Abstract In this paper, a detailed investigation of the mobility in two different strained-Si technologies has been conducted. The mobility gain due to strain is nearly lost in short channel devices. The short channel loss observed in both cases is attributed to a bigger contribution of Coulomb scattering and to a larger subband splitting due to pocket implants. Finally, the mobility gain reduction with temperature lowering has been quantitatively interpreted by a simple analytical model accounting for the fourfold valley depopulation and considering different scattering rates for the two first subbands. Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction Strained-Si technologies are of great interest since it allows enhanced carrier mobility and in turn increased drive current [1]. However, it is not clear whether the mobility gain of strained-Si nMOSFETs is due to a gain in conduction mass [2,3] or to a reduced phonons scattering rate [4]. In this paper, we present a detailed analysis of mobility limiting mechanisms between process induced (uniaxial) and substrate induced (biaxial) strained nMOSFETs. Owing to a simple two-band mobility model, we explain satisfactorily the mobility gain dependence with temperature and channel length. 2. Experimental details We have used tensely strained-Si nMOS devices with two different architectures and featuring poly gate material. *
Corresponding author. Tel.: +33 4 76 85 60 58; fax: +33 4 76 85 60 70. E-mail address: [email protected] (G. Ghibaudo).
0038-1101/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.sse.2006.03.036
Indeed, two different approaches were used to strain the channel [5]. The first one consists of an epitaxy of 15 nm Si on a relaxed Si0.8Ge0.2 substrate (sSi) to induce biaxial stress in the Si layer. The second one is a process induced type of stress, as it relies on the contact Etch Stop Layer (ESL) to induce uniaxial stress [6]. These two different strained-Si devices were compared with their own reference, a process matched for ESL and a threshold matched for sSi. The measurements were done with a HP 4155B semiconductor parameter analyzer at wafer level. Mobility was extracted from the drain current at low Vd (10–50 mV). It is well known that this extraction can be difficult in modern cutting edge devices due to the strong gate current and uncertainties in the effective gate length of short channel devices. That’s why we compensated for the gate current using the method given in [7,8], and, we extracted the effective gate length using capacitance measurements made on short channel devices [9]. Moreover, it should be noted that the method used for mobility extraction is independent of the source–drain series resistance [10].
F. Lime et al. / Solid-State Electronics 50 (2006) 644–649
energy, together with the out-of-plane twofold valleys. However, as our devices are strained along h1 1 0i, the band splitting in the uniaxially and biaxially strained devices is qualitatively the same (see Fig. 1). First, the channel length mobility dependence will be discussed. It is plotted in Fig. 2 for the two types of stress and their reference. The mobility of the sSi reference is lower than that of the ESL due to a higher channel doping. It is clear from Fig. 3 that the low field mobility gain is quite different between the two architectures. For the sSi case, the higher gain is obtained on the longer channels with a maximum of 50% and decreases with gate length. For ESL, the mobility gain is zero for the long channel case, but increases with gate length until it reaches a maximum of 50% at approximately 0.16 lm before to fall off. This latter behavior was expected as the strain in the ESL case is located near the source and drain and therefore becomes stronger with short channels. Furthermore, it is worth noting that, even if the mobility gain decreases below
3. Results and discussion The mobility gain of strained-Si nMOSFETs comes from the splitting of the conduction band. For tensile biaxial strain, the energy of the twofold degenerate valley is lower compared to the fourfold one. This results in an increase of the twofold valley occupancy, which has a lower conductivity mass, and decrease the phonon scattering rate between the twofold and the fourfold valleys (intervalley scattering). These two effects cumulate to give rise to a better mobility. This is true in bulk strained Si but was questioned by Takagi et al. [4] for 2DEG (i.e. inversion layer). Indeed Takagi argue that, for intravalley phonon scattering, the lower conductivity mass is compensated by the valley degeneracy and the inversion layer thickness. In that case, the better mobility of the twofold valley would then comes from a lower intervalley phonon scattering rate compared to the fourfold valley. In uniaxially strained devices along the h1 0 0i direction, two of the fourfold degenerate valleys should be lowered in
No strain
645
Biaxial
Uniaxial along <010>
<010>
Δ2in plane
Δ4in plane Ec
Ec
ΔEstrain
ΔEstrain
Ec
Δ2out of plane+ Δ2in plane
Δ2out of plane
(a) Uniaxial along <110>
<110>
Δ4in plane Ec
ΔE strain Δ2out of plane
(b) Fig. 1. Ellipsoids of constant energy and the associated conduction band split, corresponding to no strain, biaxial strain, uniaxial strain along h0 1 0i (a) and uniaxial strain along h1 1 0i (b) cases.
646
F. Lime et al. / Solid-State Electronics 50 (2006) 644–649 600
500
2
µ0 (cm /V/s)
400
300 sSi Ref sSi ESL Ref ESL
200
100 0.01
0.1
1
10
L (µm) Fig. 2. Low field mobility behavior with effective gate channel length for the strained nMOS and their respective reference.
60
In Fig. 4, is plotted the temperature dependence of the low field mobility l0 for short and long channel transistors of the two technologies. The mobility is enhanced when decreasing the temperature, until it eventually saturates when Coulomb contribution dominates. So the slower increase that we observe for short channel devices can be attributed to a higher Coulomb scattering contribution originating from halo and pockets. By comparing the sSi and ESL case, we can also see that sSi has a higher Coulomb contribution, confirming the higher doping. We can also see in Fig. 5 that the low field mobility gain of short channel devices is weak and nearly constant with temperature, meaning that the mobility is governed by a similar mechanism in the strained-Si devices and their references. The constant positive mobility gain of sSi short channel devices, however, is an artifact due to different implantation/diffusion conditions compared to the reference. To confirm this issue, we have extracted the mobility gain at high field (Qi = 106 C/cm2). In that case, the Coulomb scattering is strongly attenuated due to the screening of the inversion layer. As expected, in these
2000
40
1800
30 sSi ESL
20
2
10 0 -10 0.01
sSi
1600 µ 0 (cm /V/s)
Mobility gain (%)
50
1400 1200 L = 2 µm
1000 800
0.1
1
10
Gate length (µm)
400
Fig. 3. Low field mobility gain extracted from Fig. 1.
where sph, sc and ssr are the scattering rates of phonons, Coulomb and surface roughness contributions, and m* is the effective conductivity mass. i refer to the twofold or fourfold degenerate valleys. The total mobility is calculated by averaging over the population count of each valley. m* and sph are responsible for the better mobility of strainedSi devices. It should also be noted that the phonon contribution decreases when lowering the temperature.
200
0
50
100
150
200
250
300
T (K)
(a)
2000 1800
ESL
1600
2
µ 0 (cm /V/s)
0.1 lm in both cases, the ESL gain seems better between 50 nm and 0.3 lm gate lengths. In order to get more insights into the mobility behavior of these devices, we performed low temperature measurements. Indeed, there are three main scattering mechanisms that limit the mobility. The effective mobility can be written in the following form for the twofold or the fourfold degenerate valleys: 1 mi i 1 i 1 i 1 ðs ¼ Þ þ ðs Þ þ ðs Þ ð1Þ ph c sr lieff q
L = 0.1 µm
600
1400 1200 1000 L = 2 µm
800
L = 0.1 µm
600 400 200 (b)
0
50
100
150
200
250
300
T (K)
Fig. 4. Low field mobility evolution with temperature and channel length for the two technologies.
F. Lime et al. / Solid-State Electronics 50 (2006) 644–649
60
60
30 20
-2
µ @ Qi /q = 6.25x10 cm µ0
12
10 0 -10
sSi
-20 0.01
0.1 Effective gate length (µm)
(a)
50
Mobility gain at high field (%)
Mobility gain (%)
40
T = 300, 80, 10 K
12
50
647
40
sSi Lg = 2 µm
30 20 ESL Lg = 0.1µm
10 0 -10
1
-2
Qi /q = 6.25x10 cm
50
100
150
12
20 10 0 -10
-20 0.01 (b)
Mobility gain at low field (%)
30
ESL
0.1 Effective gate length (µm)
350
60
T = 300, 200, 80 K
Mobility gain (%)
40
300
-2
µ @ Qi /q = 6.25x10 cm µ0
50
250
T (K)
(a)
60
200
50 40 30 20
ESL Lg = 0.1 µm sSi Lg = 2 µm
10 0
1
Fig. 5. Evolution with channel length of the low field mobility gain due to the decrease in temperature for sSi and ESL technologies and their reference.
curves, the artifact is removed. For the long channel case, Fig. 5 shows a slightly increased mobility gain at high field, which can also be attributed to the reduced Coulomb scattering. In the ESL case, the Coulomb scattering is lower and there is little difference between the high field and low field curves. These features show that Coulomb scattering is a critical issue for short channel devices, which could impede the mobility gain expected from strain. The evolution of the mobility gain with temperature (also partially indicated in Fig. 5) is plotted in Fig. 6 for the two technologies at gate lengths where the strain is most effective i.e. short channel for ESL and long channel for sSi. It is interesting to note that the two technologies exhibit the same temperature dependence even though different channel lengths are considered (see Fig. 4). We can again see that the sSi mobility gain is higher at high field, in opposition to the ESL case. It should be highlighted that the mobility gain at high field is generally and inexplicably found higher than what accurate simulation and theory predict [11].
-10
0
50
100
150
200
250
300
350
T (K)
(b)
Fig. 6. Evolution with temperature of the mobility gain at high field and low field for sSi and ESL technologies, at a channel length where the stress is effective. Dash line is a simulation for a Si1xGex virtual substrate with x = 0.2 (D = 0.13 eV).
In order to investigate the mechanisms responsible for the mobility gain at low field, we performed very simple calculations based on Boltzman statistics, which is reasonable at low field (i.e. close to Vt). In fact, the mobility can be expressed by the Kubo–Greenwood formula as
l¼
P q i i mc
Z
1
of ðEÞ gðEÞ dE ðE Ei Þ si ðEÞ oE Ei Z 1 P f ðEÞ gðEÞ dE i
ð2Þ
Ei
where the summation is over all the subbands i. g(E) = md/ ph2 is the density of states and si(E), mic and Ei the scattering rate, conduction mass and the energy of the bottom of the subband i. f(E) is the Fermi distribution function. If we consider in Eq. (2) the first two subbands of the twofold and fourfold valleys with different scattering times s1 and s2, and we replace the Fermi distribution by the
648
F. Lime et al. / Solid-State Electronics 50 (2006) 644–649
Boltzman distribution function, then the low field effective mobility can be equated to
0.8 300K
q s1 q s2 E0 þ D 2 1 md nd 1 1 þ md nd 2 2 exp kT mc mc l0 ¼ E0 þ D m1d nd 1 þ m2d nd 2 exp kT ð3Þ
Mobility gain
0.6
0.4
0.2
where indices 1 (2) stands for the twofold (fourfold) degenerate valleys, nd1 and nd2 are degeneracy factors, D is the splitting between the two subbands resulting from the strain, and E0 is the subband splitting with no strain. The mobility gain is obtained by normalizing Eq. (3) when D = 0. Table 1 gives the values of the conduction, mc, and density of states, md, masses, and degeneracy factor, nd, used for the twofold and fourfold valleys. This simple model has been used to explain the temperature dependence of the low field mobility gain (see dashed line of Fig. 6). Indeed, a good agreement with data has been obtained after adjusting the scattering rate ratio, s2/ s1 = 0.4 (The subband splitting E0 has been evaluated by Poisson–Schrodinger simulations, E0 = 50 mV). The higher scattering rate in the fourfold valleys can be attributed to a larger intervalley phonon scattering as compared to the twofold valleys [4]. The monotonous decrease of the mobility gain as reducing the temperature can therefore be simply explained by the progressive de-activation of the fourfold subband contribution, canceling out any straininduced band splitting beneficial effect.
Table 1 Masses and degeneracy factors used in the model
Twofold sub. Fourfold sub.
md
mc
nd
0.19 0.42
0.19 0.32
2 4
100K 0
0
0.1
0.2 Ge content
0.3
0.4
Fig. 8. Mobility gain versus substrate Ge content (symbols from [12], solid line from Eq. 2, E0 = 40 meV, D = 0.67 Æ x).
In the case where the scattering rate ratio is taken equal to one (s2/s1 = 1), we found a maximum gain of 20% at 300 K (see Fig. 7), resulting only from the different conduction masses. This situation is encountered in short channel devices where Coulomb scattering dominates, yielding a significant mobility gain reduction as found for both technologies (Fig. 5). This reduction is also accentuated by the increasing doping level in short channels due to pocket implant, giving rise to enhanced subband splitting (larger E0 in Eq. 2) and, in turn, lower population in the fourfold valleys. The physical consistency of this mobility analysis has also been validated after changing the strain level in sSi through the substrate Ge content by comparison to literature data (see Fig. 8). To this end, the strain-induced band splitting D has been varied linearly with Ge content x as D = 0.67 Æ x (eV). The good agreement between model and experiment is worth to be noted. The 100 K curve underlines again the fact that the mobility gain is fully lost as the fourfold valleys are de-populated. 4. Conclusion
Mobility gain (%)
60
τ2/τ1=0.4
40
20
τ2/τ1=1 0
0
100
200
In this paper, we have investigated in details the mobility in two different strained-Si technologies. We found that the mobility gain of our devices is nearly lost on short channels. The short channel loss observed in both cases is attributed to a bigger contribution of Coulomb scattering and to a larger subband splitting due to pocket implants. Finally, the mobility gain reduction with temperature lowering has been quantitatively interpreted using a simple analytical model accounting for the fourfold valleys depopulation and considering different scattering rates for the two first subbands.
300
T (K) Fig. 7. Variation of mobility gain with temperature for two scattering time ratios corresponding to phonon (=0.4) and Coulomb (=1) scattering rates.
Acknowledgement This work has been partially supported by the European NANOCMOS integrated project.
F. Lime et al. / Solid-State Electronics 50 (2006) 644–649
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