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Two sub-band conductivity of Si quantum well
ARTICLE IN PRESS
Physica E 32 (2006) 281–284 www.elsevier.com/locate/physe
Two sub-band conductivity of Si quantum well Mika Prunnila, Jouni Ahopelto VTT Information Technology, P.O. Box 1208, FIN-02044 VTT, Espoo, Finland Available online 3 February 2006
Abstract We report on two sub-band/bi-layer transport in double gate SiO2–Si–SiO2 quantum well with 14 nm thick Si layer at 270 mK. At symmetric well potential the experimental sub-band spacing changes monotonically from 2.3 to 0.3 meV when the total electron density is adjusted by gate voltages between 0:7 1016 –3:0 1016 m2 . The conductivity is mapped in large gate bias window and it shows strong non-monotonic features. At symmetric well potential and high density these features are addressed to sub-band wave function delocalization in the quantization direction and to different disorder of the top and bottom interfaces of the Si well. In the gate bias regimes close to second sub-band/bi-layer threshold the non-monotonic behavior is interpreted to arise from scattering from the other electron sub-system with localized or low mobility states. r 2006 Elsevier B.V. All rights reserved. PACS: 73.40.c; 72.20.i Keywords: Two-dimensional electron gas; Localization; Resonant coupling; Bi-layer; Silicon
1. Introduction Development of silicon-on-insulator technology has enabled fabrication of silicon heterostructure devices where thin single crystalline Si film is sandwiched between amorphous SiO2 layers. This kind of SiO2–Si–SiO2 quantum well provides a unique material system where electron density can be tuned in a broad range due to high potential barriers formed by the SiO2 layers. Previous work on SiO2–Si–SiO2 quantum wells has mainly focused on single and bi-layer magneto transport [1–3]. In this work we focus on the issue how the two sub-band or bi-layer transport affects the low temperature conductivity of double gate SiO2–Si–SiO2 quantum well with 14 nm thick Si layer. 2. Experimental The samples were fabricated on commercially available (1 0 0) silicon-on-insulator wafer as described in detail in Corresponding author. Tel.: +358 20 722 6668; fax: +358 20 722 7012.
E-mail address: mika.prunnila@vtt.fi (M. Prunnila). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.12.093
Ref. [3]. The cross-sectional sample structure consist of nþ Si top gate, top gate oxide (OX), Si well, back gate oxide (BOX) and nþ Si back gate. The top gate is polycrystalline while the back gate is crystalline silicon. All results reported here were obtained from a 100 mm wide Hall bar sample with 400 mm voltage probe distance. The layer thickness for the Si well, top gate oxide and back gate oxide were tW ¼ 14 nm, tOX ¼ 40 nm, and tBOX ¼ 83 nm, respectively. Fig. 1 shows the schematic device cross-section together with self-consistent wave functions and effective potential (output of nextnano3 simulator, see http:// www.nextnano.de) at two total electron density n values. In the experiments the sample was mounted to a sample holder of a He-3 cryostat, which was at base temperature (270 mK). The electron density was determined from the Shubnikov–de Haas (SdH) oscillations of the diagonal resistivity rxx utilizing the standard methods: in the single sub-band gate bias windows rxx was measured as a function of the gate voltages at constant magnetic field B and n was determined from the minimum positions of rxx . In the presence of two sub-bands rxx was recorded as a function of B. Then a Fourier transform was performed numerically to rxx ð1=BÞ and a peak position multiplied by e=h
ARTICLE IN PRESS M. Prunnila, J. Ahopelto / Physica E 32 (2006) 281–284
ψB
EF
n+Si
∆BAB
ψAB
tBOX
10 mev (b)
(c)
Fig. 1. (a) Schematic cross-section of the sample and gate biasing in the experiments. The dimensions are tw ¼ 14 nm, tOX ¼ 40 nm and tBOX ¼ 83 nm. (b) and (c) Self-consistently calculated Hartree bonding and anti-bonding electron wave functions together with the effective potential V eff in the 14 nm thick Si well at balanced gate bias. The wave functions are offsetted so that CB;AB ¼ 0 is equal to the corresponding eigen energy E B;AB . (b): n ¼ 1:0 1016 m2 . (c): n ¼ 2:8 1016 m2 .
3.0
VBG = VTGtBOX/tTOX 2.0
2.5 2.0
1.5
n = nAB + nB
1.5 1.0
∆BAB (meV)
Veff
tw tOX
(a)
VBG Si
SiO2
VTG
Electron density (1016m-2)
282
1.0 nB 0.5 0.0 0.5
0.5
nAB 1.0
1.5
2.0 VTG (V)
2.5
3.0
(e fundamental charge, h Plank’s constant) in the spectrum gave the sheet density per degeneracy of a sub-band.
Fig. 2. Left axis: total electron density (n) and sub-band densities ðnB ; nAB Þ as a function of top gate voltage along the balanced gate line of a 14 nm thick Si well. The densities are determined from SdH oscillations at 270 mK. Right axis: sub-band energy spacing calculated from nB and nAB assuming ideal 2D density of states.
3. Results and discussion
we have neglected the exchange and correlation effects in the calculations.
3.1. Sub-band densities at balanced gate bias
3.2. Conductivity and mobility
Left vertical axis of Fig. 2 shows the different electron densities as a function of top gate voltage V TG along the balanced (or symmetric) gate bias line where the back gate voltage V BG ¼ V TG tBOX =tOX . This choice of gate biases produce symmetric potential in the Si well. The total density n is obtained by summing the bonding sub-band nB and anti-bonding sub-band densities nAB . When V TG 0:8–1:0 VnAB cannot be detected directly and it is obtained from nAB ¼ n nB , where n is linearly interpolated. On the balanced gate line the threshold for nAB is at V TG 0:8 V and n ¼ nB 0:7 1016 m2 . Note that the SdH method does not reveal the degeneracies of the system. We have made an assumption that both sub-bands arise from the high perpendicular mass valleys (non-primed valleys), which gives the total degeneracy of four after including the spin degeneracy. Our a priori assumption is validated by noting that V TG vs. n ¼ nB þ nAB slope corresponds accurately to the total gate capacitance. The light perpendicular mass valleys (non-primed valleys) have a total degeneracy of eight and this degeneracy would lead to incorrect total density. Right vertical axis of Fig. 2 shows the bonding antibonding energy gap DBAB which is obtained from nB and nAB by assuming ideal 2D density of states together with bulk effective mass m ¼ mt ¼ 0:19m0 parallel to the quantum well plane. The increased gate drive and n leads to expected reduction of DBAB due to the potential barrier formation in the middle of the Si well as demonstrated by the self-consistent calculations in Figs. 1(b) and (c). The calculated DBAB is larger than the experimental one by 0:72 meV (on average) in the density range of Fig. 2 (not shown). This discrepancy can be explained by noting that
Fig. 3(a) shows (zero magnetic field) conductivity s ¼ 1=rxx measured as a function of top and back gate voltages [4]. The scales of the voltage axes in Fig. 3(a) are chosen in such a manner that if we move perpendicularly to the balanced bias line the electron density stays (roughly) constant, i.e., we move along n ¼ const: line and merely shift the position of the electron gas inside the Si slab. In the gate bias regions V TG t0 or V BG t0 only single subband is occupied and s behaves monotonically, which is expected on the basis of simple Coulomb—surface roughness scattering picture and is consistent with mobility measurements of sub 10 nm thick Si well [3]. The overall asymmetry of the conductivity with respect to the symmetric bias line arises from the different quality of the top (Si-OX) and back (Si-BOX) interfaces of the Si well. The back interface has substantially larger disorder in comparison to the top interface as can be observed from Figs. 3(b) and (c), which show the back interface and top interface mobilities m ¼ s=en, i.e., the mobilities along the axes of Fig. 3(a). Note that the n-axis of Figs. 3(b) and (c), together with n plotted in Fig. 2, also give an idea of the magnitude of electron density in an arbitrary ðV TG ; V BG Þ point in Fig. 3(a). The observed disorder difference is mainly due to larger surface roughness of the Si-BOX interface [3] consistent with the fact that the top–back mobility ratio actually increases as a function of carrier density. At low density surface roughness plays a minor role and the top and back interface mobilities almost coincide, which is also partly due to electron wave function spreading throughout the Si well. The presence of the two Si–SiO2 interfaces also reduce the peak mobility and shift it towards higher electron densities [5].
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M. Prunnila, J. Ahopelto / Physica E 32 (2006) 281–284
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VTG (V)
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Fig. 3. (a) Contour plot of conductivity s measured at 270 mK. The contour spacing is 104 1=O. The dashed line is the balanced gate line V BG ¼ V TG tBOX =tOX . (b) Back interface mobility as a function of V BG (right vertical axis) and n (left vertical axis). (c) Top interface mobility as a function of V TG and n.
VTG = 3.24V 3.0
4.0
2.74 2.49 5 +
-
3.0
- 4 + 3 2 ∆z
2.0
|za|
ψB ψAB
distance (nm)
σ (10-3/Ω)
In the bias regime V TG;BG \0 the conductivity behaves strongly non-monotonically. We can observe, e.g., that at low density sðV TG ; V BG Þjn¼const: has a local minimum roughly at symmetric bias [feature (f1)]. This behavior has a striking similarity to the substrate bias experiments of bulk Si inversion layers, where it was found that a positive substrate bias tends to reduce the mobility [6]. The substrate bias mobility reduction has been addressed to scattering from singly populated localized band tail states of higher sub-bands [7] (whose population increases with substrate bias in bulk devices). Note that the local minimum is particularly clear around the nAB threshold (V TG 0:8 V, symmetric bias) and the sub-band spacing is at minimum at symmetric bias (when n ¼ const:). Therefore, straightforward explanation of the reduction in sjn¼const: towards the symmetric bias at low densities is the increased scattering of electrons in the lowest subband from localized/low mobility electrons in the upper sub-bands. Second strong feature in Fig. 3(a) is the local maximum and the succeeding minimum in sðV BG;TG ; V TG;BG ¼ const:Þ close to V BG;TG 0 at high densities [feature (f 2)]. This feature is illustrated more clearly in Fig. 4 which shows sðV BG ; V TG ¼ const:Þ. Note that if we would plot
1 0
-2
0
2
4
6
8
VBG (V) Fig. 4. Conductivity as a function of back gate voltage. # indicate V BG ¼ V TG tBOX =tOX . Inset: deviation (position uncertainty) Dz and absolute value of average position za of the sub-band wave functions. The zero is in the middle of the well and þ= indicate the sign of za . The calculation corresponds to the V TG ¼ 3:0 V curve.
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M. Prunnila, J. Ahopelto / Physica E 32 (2006) 281–284
sðV TG ; V BG ¼ const:Þ we would obtain a similar curve (only the magnitude of s would be different due to different quality of the Si-OX and Si-BOX interfaces). By comparing the high magnetic field rxx data of Ref. [3] and Fig. 3(a) we note that the local minima in sðV BG;TG ; V TG;BG ¼ const:Þ above V BG;TG ¼ 0 occurs at the position which is the threshold where the high-field rxx starts to show signatures of bi-layer transport. Thus, the observed drop in sðV BG;TG ; V TG;BG ¼ const:Þ is addressed to bi-layer threshold. Identical behavior has indeed been observed in low density bi-layers and it has been addressed to interlayer scattering [8]. Thus, the features (f1) and (f 2) in s have essentially the same origin: population in second (low density/low mobility) sub-system inside the Si well is increased and this increases the scattering in the dominating (higher density/higher mobility) sub-system. We have separated these features here, because the bi-layer picture is favorable at high densities (see below). Note that as the gate biasing in the experiments changes the surface potential of the Si-oxide interfaces with respect to the Fermi level in a broad range charging of traps in the vicinity of the interfaces may affect the observed phenomena. If we increase the electron density by gate bias far beyond the threshold of the second sub-band yet another feature in s appears around the symmetric bias above V TG 2 V. This is indicated by the down arrows in Fig. 4. The origin of this feature is related to the resonance effect observed in bi-layer systems (often referred as resistance resonance) [9–11]. Far away from the balanced gate bias the barrier in the middle of the Si well localizes the wave functions of the different layers or sub-bands to the different sides of the well. At balanced gate bias the both wave functions delocalize across the well. This localization–delocalization is illustrated in Fig. 4 (inset) where we show the average position za ¼ hzi ¼ hCjzjCi ðC ¼ CB;AB Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and deviation (position uncertainty) Dz ¼ hz2 i hzi2 of the sub-bands. At balanced bias Dz has a maximum and za ¼ 0 (middle of the well), which results in mobility reduction of the sub-band/layer that otherwise is localized in the vicinity of the less disordered interface while the mobility of the other sub-band stays practically unaffected
[10]. This leads to the local minimum in s at V BG ¼ V TG tBOX =tOX at high electron densities. 4. Summary The conductivity of the 14 nm thick Si quantum well showed strongly non-monotonic double gate bias dependency. At symmetric well potential and high density these were addressed to sub-band wave function delocalization in the quantization direction and to different disorder of the top and bottom interfaces of the Si well. In the gate bias regimes close to second sub-band threshold at symmetric bias and bi-layer threshold the non-monotonic behavior was interpreted to arise from scattering from localized (or low mobility) states of the second sub-band and interlayer scattering, respectively. Acknowledgements M. Markkanen is thanked for assistance in the sample fabrication. Academy of Finland is acknowledged for financial support through project # 205467. References [1] K. Takashina, A. Fujiwara, S. Horiguchi, Y. Takahashi, Y. Hirayama, Phys. Rev. B 69 (2004) 161304(R). [2] M. Prunnila, J. Ahopelto, H. Sakaki, Phys. Status. Solidi. (a) 202 (2005) 970. [3] M. Prunnila, J. Ahopelto, F. Gamiz, Solid State Electron. 49 (2005) 1516. [4] The obscured behavior of the s contours in the vicinity of the threshold and below the symmetric bias line is an experimental artefact explained in Ref. [3]. [5] M. Prunnila, J. Ahopelto, F. Gamiz, Appl. Phys. Lett. 84 (2004) 2298. [6] A.B. Fowler, Phys. Rev. Lett. 34 (1975) 15. [7] X.G. Feng, D. Popovic, S. Washburn, Phys. Rev. Lett. 83 (1999) 368. [8] Y. Katayama, D.C. Tsui, H.C. Manoharan, S. Parihar, M. Shayegan, Phys. Rev. B 52 (1995) 14817. [9] A. Palevski, F. Beltram, F. Capasso, L. Pfeiffer, K.W. West, Phys. Rev. Lett. 65 (1990) 1929. [10] Y. Ohno, M. Tsuchiya, H. Sakaki, Appl. Phys. Lett. 62 (1993) 1952. [11] Y. Berk, A. Kamenev, A. Palevski, L.N. Pfeiffer, K.W. West, Phys. Rev. B. 50 (1994) 15420.
Physica E 32 (2006) 281–284 www.elsevier.com/locate/physe
Two sub-band conductivity of Si quantum well Mika Prunnila, Jouni Ahopelto VTT Information Technology, P.O. Box 1208, FIN-02044 VTT, Espoo, Finland Available online 3 February 2006
Abstract We report on two sub-band/bi-layer transport in double gate SiO2–Si–SiO2 quantum well with 14 nm thick Si layer at 270 mK. At symmetric well potential the experimental sub-band spacing changes monotonically from 2.3 to 0.3 meV when the total electron density is adjusted by gate voltages between 0:7 1016 –3:0 1016 m2 . The conductivity is mapped in large gate bias window and it shows strong non-monotonic features. At symmetric well potential and high density these features are addressed to sub-band wave function delocalization in the quantization direction and to different disorder of the top and bottom interfaces of the Si well. In the gate bias regimes close to second sub-band/bi-layer threshold the non-monotonic behavior is interpreted to arise from scattering from the other electron sub-system with localized or low mobility states. r 2006 Elsevier B.V. All rights reserved. PACS: 73.40.c; 72.20.i Keywords: Two-dimensional electron gas; Localization; Resonant coupling; Bi-layer; Silicon
1. Introduction Development of silicon-on-insulator technology has enabled fabrication of silicon heterostructure devices where thin single crystalline Si film is sandwiched between amorphous SiO2 layers. This kind of SiO2–Si–SiO2 quantum well provides a unique material system where electron density can be tuned in a broad range due to high potential barriers formed by the SiO2 layers. Previous work on SiO2–Si–SiO2 quantum wells has mainly focused on single and bi-layer magneto transport [1–3]. In this work we focus on the issue how the two sub-band or bi-layer transport affects the low temperature conductivity of double gate SiO2–Si–SiO2 quantum well with 14 nm thick Si layer. 2. Experimental The samples were fabricated on commercially available (1 0 0) silicon-on-insulator wafer as described in detail in Corresponding author. Tel.: +358 20 722 6668; fax: +358 20 722 7012.
E-mail address: mika.prunnila@vtt.fi (M. Prunnila). 1386-9477/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2005.12.093
Ref. [3]. The cross-sectional sample structure consist of nþ Si top gate, top gate oxide (OX), Si well, back gate oxide (BOX) and nþ Si back gate. The top gate is polycrystalline while the back gate is crystalline silicon. All results reported here were obtained from a 100 mm wide Hall bar sample with 400 mm voltage probe distance. The layer thickness for the Si well, top gate oxide and back gate oxide were tW ¼ 14 nm, tOX ¼ 40 nm, and tBOX ¼ 83 nm, respectively. Fig. 1 shows the schematic device cross-section together with self-consistent wave functions and effective potential (output of nextnano3 simulator, see http:// www.nextnano.de) at two total electron density n values. In the experiments the sample was mounted to a sample holder of a He-3 cryostat, which was at base temperature (270 mK). The electron density was determined from the Shubnikov–de Haas (SdH) oscillations of the diagonal resistivity rxx utilizing the standard methods: in the single sub-band gate bias windows rxx was measured as a function of the gate voltages at constant magnetic field B and n was determined from the minimum positions of rxx . In the presence of two sub-bands rxx was recorded as a function of B. Then a Fourier transform was performed numerically to rxx ð1=BÞ and a peak position multiplied by e=h
ARTICLE IN PRESS M. Prunnila, J. Ahopelto / Physica E 32 (2006) 281–284
ψB
EF
n+Si
∆BAB
ψAB
tBOX
10 mev (b)
(c)
Fig. 1. (a) Schematic cross-section of the sample and gate biasing in the experiments. The dimensions are tw ¼ 14 nm, tOX ¼ 40 nm and tBOX ¼ 83 nm. (b) and (c) Self-consistently calculated Hartree bonding and anti-bonding electron wave functions together with the effective potential V eff in the 14 nm thick Si well at balanced gate bias. The wave functions are offsetted so that CB;AB ¼ 0 is equal to the corresponding eigen energy E B;AB . (b): n ¼ 1:0 1016 m2 . (c): n ¼ 2:8 1016 m2 .
3.0
VBG = VTGtBOX/tTOX 2.0
2.5 2.0
1.5
n = nAB + nB
1.5 1.0
∆BAB (meV)
Veff
tw tOX
(a)
VBG Si
SiO2
VTG
Electron density (1016m-2)
282
1.0 nB 0.5 0.0 0.5
0.5
nAB 1.0
1.5
2.0 VTG (V)
2.5
3.0
(e fundamental charge, h Plank’s constant) in the spectrum gave the sheet density per degeneracy of a sub-band.
Fig. 2. Left axis: total electron density (n) and sub-band densities ðnB ; nAB Þ as a function of top gate voltage along the balanced gate line of a 14 nm thick Si well. The densities are determined from SdH oscillations at 270 mK. Right axis: sub-band energy spacing calculated from nB and nAB assuming ideal 2D density of states.
3. Results and discussion
we have neglected the exchange and correlation effects in the calculations.
3.1. Sub-band densities at balanced gate bias
3.2. Conductivity and mobility
Left vertical axis of Fig. 2 shows the different electron densities as a function of top gate voltage V TG along the balanced (or symmetric) gate bias line where the back gate voltage V BG ¼ V TG tBOX =tOX . This choice of gate biases produce symmetric potential in the Si well. The total density n is obtained by summing the bonding sub-band nB and anti-bonding sub-band densities nAB . When V TG 0:8–1:0 VnAB cannot be detected directly and it is obtained from nAB ¼ n nB , where n is linearly interpolated. On the balanced gate line the threshold for nAB is at V TG 0:8 V and n ¼ nB 0:7 1016 m2 . Note that the SdH method does not reveal the degeneracies of the system. We have made an assumption that both sub-bands arise from the high perpendicular mass valleys (non-primed valleys), which gives the total degeneracy of four after including the spin degeneracy. Our a priori assumption is validated by noting that V TG vs. n ¼ nB þ nAB slope corresponds accurately to the total gate capacitance. The light perpendicular mass valleys (non-primed valleys) have a total degeneracy of eight and this degeneracy would lead to incorrect total density. Right vertical axis of Fig. 2 shows the bonding antibonding energy gap DBAB which is obtained from nB and nAB by assuming ideal 2D density of states together with bulk effective mass m ¼ mt ¼ 0:19m0 parallel to the quantum well plane. The increased gate drive and n leads to expected reduction of DBAB due to the potential barrier formation in the middle of the Si well as demonstrated by the self-consistent calculations in Figs. 1(b) and (c). The calculated DBAB is larger than the experimental one by 0:72 meV (on average) in the density range of Fig. 2 (not shown). This discrepancy can be explained by noting that
Fig. 3(a) shows (zero magnetic field) conductivity s ¼ 1=rxx measured as a function of top and back gate voltages [4]. The scales of the voltage axes in Fig. 3(a) are chosen in such a manner that if we move perpendicularly to the balanced bias line the electron density stays (roughly) constant, i.e., we move along n ¼ const: line and merely shift the position of the electron gas inside the Si slab. In the gate bias regions V TG t0 or V BG t0 only single subband is occupied and s behaves monotonically, which is expected on the basis of simple Coulomb—surface roughness scattering picture and is consistent with mobility measurements of sub 10 nm thick Si well [3]. The overall asymmetry of the conductivity with respect to the symmetric bias line arises from the different quality of the top (Si-OX) and back (Si-BOX) interfaces of the Si well. The back interface has substantially larger disorder in comparison to the top interface as can be observed from Figs. 3(b) and (c), which show the back interface and top interface mobilities m ¼ s=en, i.e., the mobilities along the axes of Fig. 3(a). Note that the n-axis of Figs. 3(b) and (c), together with n plotted in Fig. 2, also give an idea of the magnitude of electron density in an arbitrary ðV TG ; V BG Þ point in Fig. 3(a). The observed disorder difference is mainly due to larger surface roughness of the Si-BOX interface [3] consistent with the fact that the top–back mobility ratio actually increases as a function of carrier density. At low density surface roughness plays a minor role and the top and back interface mobilities almost coincide, which is also partly due to electron wave function spreading throughout the Si well. The presence of the two Si–SiO2 interfaces also reduce the peak mobility and shift it towards higher electron densities [5].
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M. Prunnila, J. Ahopelto / Physica E 32 (2006) 281–284
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0.5
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0.5 1.0 n (1016m-2)
Fig. 3. (a) Contour plot of conductivity s measured at 270 mK. The contour spacing is 104 1=O. The dashed line is the balanced gate line V BG ¼ V TG tBOX =tOX . (b) Back interface mobility as a function of V BG (right vertical axis) and n (left vertical axis). (c) Top interface mobility as a function of V TG and n.
VTG = 3.24V 3.0
4.0
2.74 2.49 5 +
-
3.0
- 4 + 3 2 ∆z
2.0
|za|
ψB ψAB
distance (nm)
σ (10-3/Ω)
In the bias regime V TG;BG \0 the conductivity behaves strongly non-monotonically. We can observe, e.g., that at low density sðV TG ; V BG Þjn¼const: has a local minimum roughly at symmetric bias [feature (f1)]. This behavior has a striking similarity to the substrate bias experiments of bulk Si inversion layers, where it was found that a positive substrate bias tends to reduce the mobility [6]. The substrate bias mobility reduction has been addressed to scattering from singly populated localized band tail states of higher sub-bands [7] (whose population increases with substrate bias in bulk devices). Note that the local minimum is particularly clear around the nAB threshold (V TG 0:8 V, symmetric bias) and the sub-band spacing is at minimum at symmetric bias (when n ¼ const:). Therefore, straightforward explanation of the reduction in sjn¼const: towards the symmetric bias at low densities is the increased scattering of electrons in the lowest subband from localized/low mobility electrons in the upper sub-bands. Second strong feature in Fig. 3(a) is the local maximum and the succeeding minimum in sðV BG;TG ; V TG;BG ¼ const:Þ close to V BG;TG 0 at high densities [feature (f 2)]. This feature is illustrated more clearly in Fig. 4 which shows sðV BG ; V TG ¼ const:Þ. Note that if we would plot
1 0
-2
0
2
4
6
8
VBG (V) Fig. 4. Conductivity as a function of back gate voltage. # indicate V BG ¼ V TG tBOX =tOX . Inset: deviation (position uncertainty) Dz and absolute value of average position za of the sub-band wave functions. The zero is in the middle of the well and þ= indicate the sign of za . The calculation corresponds to the V TG ¼ 3:0 V curve.
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sðV TG ; V BG ¼ const:Þ we would obtain a similar curve (only the magnitude of s would be different due to different quality of the Si-OX and Si-BOX interfaces). By comparing the high magnetic field rxx data of Ref. [3] and Fig. 3(a) we note that the local minima in sðV BG;TG ; V TG;BG ¼ const:Þ above V BG;TG ¼ 0 occurs at the position which is the threshold where the high-field rxx starts to show signatures of bi-layer transport. Thus, the observed drop in sðV BG;TG ; V TG;BG ¼ const:Þ is addressed to bi-layer threshold. Identical behavior has indeed been observed in low density bi-layers and it has been addressed to interlayer scattering [8]. Thus, the features (f1) and (f 2) in s have essentially the same origin: population in second (low density/low mobility) sub-system inside the Si well is increased and this increases the scattering in the dominating (higher density/higher mobility) sub-system. We have separated these features here, because the bi-layer picture is favorable at high densities (see below). Note that as the gate biasing in the experiments changes the surface potential of the Si-oxide interfaces with respect to the Fermi level in a broad range charging of traps in the vicinity of the interfaces may affect the observed phenomena. If we increase the electron density by gate bias far beyond the threshold of the second sub-band yet another feature in s appears around the symmetric bias above V TG 2 V. This is indicated by the down arrows in Fig. 4. The origin of this feature is related to the resonance effect observed in bi-layer systems (often referred as resistance resonance) [9–11]. Far away from the balanced gate bias the barrier in the middle of the Si well localizes the wave functions of the different layers or sub-bands to the different sides of the well. At balanced gate bias the both wave functions delocalize across the well. This localization–delocalization is illustrated in Fig. 4 (inset) where we show the average position za ¼ hzi ¼ hCjzjCi ðC ¼ CB;AB Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and deviation (position uncertainty) Dz ¼ hz2 i hzi2 of the sub-bands. At balanced bias Dz has a maximum and za ¼ 0 (middle of the well), which results in mobility reduction of the sub-band/layer that otherwise is localized in the vicinity of the less disordered interface while the mobility of the other sub-band stays practically unaffected
[10]. This leads to the local minimum in s at V BG ¼ V TG tBOX =tOX at high electron densities. 4. Summary The conductivity of the 14 nm thick Si quantum well showed strongly non-monotonic double gate bias dependency. At symmetric well potential and high density these were addressed to sub-band wave function delocalization in the quantization direction and to different disorder of the top and bottom interfaces of the Si well. In the gate bias regimes close to second sub-band threshold at symmetric bias and bi-layer threshold the non-monotonic behavior was interpreted to arise from scattering from localized (or low mobility) states of the second sub-band and interlayer scattering, respectively. Acknowledgements M. Markkanen is thanked for assistance in the sample fabrication. Academy of Finland is acknowledged for financial support through project # 205467. References [1] K. Takashina, A. Fujiwara, S. Horiguchi, Y. Takahashi, Y. Hirayama, Phys. Rev. B 69 (2004) 161304(R). [2] M. Prunnila, J. Ahopelto, H. Sakaki, Phys. Status. Solidi. (a) 202 (2005) 970. [3] M. Prunnila, J. Ahopelto, F. Gamiz, Solid State Electron. 49 (2005) 1516. [4] The obscured behavior of the s contours in the vicinity of the threshold and below the symmetric bias line is an experimental artefact explained in Ref. [3]. [5] M. Prunnila, J. Ahopelto, F. Gamiz, Appl. Phys. Lett. 84 (2004) 2298. [6] A.B. Fowler, Phys. Rev. Lett. 34 (1975) 15. [7] X.G. Feng, D. Popovic, S. Washburn, Phys. Rev. Lett. 83 (1999) 368. [8] Y. Katayama, D.C. Tsui, H.C. Manoharan, S. Parihar, M. Shayegan, Phys. Rev. B 52 (1995) 14817. [9] A. Palevski, F. Beltram, F. Capasso, L. Pfeiffer, K.W. West, Phys. Rev. Lett. 65 (1990) 1929. [10] Y. Ohno, M. Tsuchiya, H. Sakaki, Appl. Phys. Lett. 62 (1993) 1952. [11] Y. Berk, A. Kamenev, A. Palevski, L.N. Pfeiffer, K.W. West, Phys. Rev. B. 50 (1994) 15420.