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Gate tuned transition to the insulating phase of one-dimensional electrons in high magnetic fields
7.‘. “II
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Surface
Science 305 (1994) 629-632
Gate tuned transition to the insulating phase of one-dimensional electrons in high magnetic fields S.W. Hwang *, D.C. Tsui, M. Shayegan Depart~~t
of Electrical Engineering, Princeton ~~iL~ersi~,Princeton, NJ O&5&. USA
(Received 19 April 1993;accepted for publication 21 July 1993)
Abstract We report experimental results on charge transport in a low disorder and low density one-dimensional electron system. We find a gate tuned transition to an insulating phase at high magnetic fields and conductance oscillations as a function of density in this B-induced insulating phase. A carefu1 study of the temperature dependence and the current-voltage characteristics of these conductance oscillations is made and the results fit the recent theoretical model of one-dimensional Wigner solid.
Recently, Scott-Thomas et al. [l] found, in a narrow Si inversion layer, periodic conductance oscillations as a function of the density. They interpreted the oscillations as manifestation of a one-dimensional charge-density-wave (1D CDW) [2] pinned by impurities in the channel. The pinning energy is maximized whenever the density is commensurate with an integer number of electrons between two pinning centers, resulting in periodic oscillations of the conductance as the electron density is changed. Subsequently, the formation of a one-dimensional Wigner solid flD WS> in a low disorder and low density one-dimensional electron system (1D ES) was theoretically studied by Averin and Likharev [3], and by Glazman, Ruzin, and Shklovskii (GRS) 141. Both theoretical papers emphasize that in the low density limit, when lo,,, <(l/a,> (where n,, is the
* Corresponding
0039-~28/94/$07.~
author.
0 1994 Elsevier
SSDI 0039-6028(93)E0711-3
Science
1D ele$ron density and a, is the Bohr radius, N 100 A in GaAs), the electrons are expected to form a 1D WS pinned by weak potential fluctuations in the channel. In a more recent paper, Averin and Nazarov (AN) 151calculated the temperature (7”) dependence of the maxima and minima of such conductance oscillations, based on the charge density wave tunneling theory of Larkin and Lee [6]. In principle, the nioan < 1 regime can be reached by simply decreasing the channel width (IV>. In a usual split-gate channel fabricated on high density GaAs/Al,Ga,_,As 2D ES, the ~lioaa < 1 condition is expected to be obtained by biasing the gate strongly such that the channe1 is near pinch-off. In reality, however, such channels have strong disorder and it has not been possible to maintain an ideal, open channel near pinch-off. One way of overcoming this problem is to fabricate a split-gate channel on a low density, low disorder 2D ES. In this case, the n,,,ua < 1 limit can be reached even when W is not extremely
B.V. All rights reserved
630
S. W. Hwang et al. /Surface
small and the channel is still far away from pinchoff. In this paper, we report experimental results on charge transport in the low disorder and lowdensity (n,,a, < 1) regime where the formation of 1D WS is anticipated. The sample is a 1 p wide split-gate channel fabricated on a wafer of GaAs/Al,Ga, _,As heterostructure with an extremely low two-dimensional density, n2,, = 6.5 x 10”’ cm-’ and high mobility, I_L= 10h cm*/V . s. In this sample, W and n,, can be varied in a wide range by biasing the split-gate bias, I’,. The details of charge transport characteristics at B = 0 and at low B is reported elsewhere [7] and can be summarized as follows: (1) At B = 0, quantized conductance steps, at integer multiples of 2e*/h, are observed in our channel. The fabricated channel length is 1.2 pm at I(, = 0 and an even longer channel is expected for more negative I$. To our knowledge, the conductance quantization in such a long channel fabricated on such a low 2D electron density wafer has not been observed before [8]. Analysis of these quantized steps shows that we can change n,, in a wide reproducibly reaching the n,,,as < 1 range, regime (corresponding to the bias range, - 2.2 < I(, < - 1.45 V), and more importantly, that unprecedentedly smooth and weak potential fluctuations are obtained in the channel. (2) In the
Science 305 (1994) 629-632
presence of B, we observe the integer (IQHE) and fractional quantum Hall effect (FQHE) states (well defined l/3 state) in the channel. Analysis of the IQHE states shows that the channel has a V, dependence on the electron density predicted by the calculation of split-gate channel with no disorder [9]. For B higher than that for the l/3 FQHE state, the channel becomes an insulator. Fig. la shows the diagonal resistance across the channel, R,,, at I$ = - 1.855 V in the v < 1 regime, at T = 35, 112, 300 mK. Qualitatively, the T dependence of R, changes drastically in the range, 3.5 3.8 T, R,, increases as T decreases and the channel behaves as an insulator. This is especially apparent for the B > 4.0 T data, where the increase is rapid and R,, is approaching to cc as T + 0. This transition of the 1D channel into an insulating state can therefore be characterized by a B field, B,_,, which is - 3.8 T at V, = - 1.855 V.
IO41
,
0
0.1
8:3.0,316,3.7
0.3
c
TO&) Fig. 1. (a) The diagonal resistance, R,, at VP = - 1.855 V in the v < 1 regime, at 7‘= 35, 112, 300 mK. Notice a dramatic change in the T dependence of R,, in the range, 3.5
S. U! Hwang et al. /Surface
8
/
0
6
E+l cQz
I
x cool#l
2
0 cool#2 * cool#3
p
0 -2.5
I -2
-1.5
-1
-0.5
0
vg (V) Fig. 2. (B, VP) phase diagram showing the B,_, deduced from the data of 3 different cool-downs. Two straight lines are eye-guide lines.
We have investigated the T dependences of R, as a function of B and V,, and the B,-, deduced from the data taken from 3 cool-downs are summarized in Fig. 2. The vertical dashed line in the figure indicates the V, below which the channel is already insulating at B = 0 (the conductance of the channel, G < e2/h, VP< - 2.2 V), and the IQHE is not observed. The upper left part of the graph is the B field induced insulating phase where R, increases with decreasing T. The lower right part is the metallic phase where the QHE states are observed. The IQHE states are observed in the entire bias range on the right-hand side of the dashed line mentioned above and the FQHE states are observed at V, > - 1.9 V. B,-, clearly varies with V,, being smaller at larger -V,. In terms of the Landau level filling factor, the transition occurs in the range 0.28 < v < 0.33, when 2.8 X 1O’O cme2 < n2,, < 6.5 X 10’” cm-* (V, > - 1.9 V), and at larger filling factors (0.57 10’” cmd2 CV,<
Science 305 (I 994) 629-632
631
phase diagram, Fig. 2. We find that the conductance oscillations are observed in the insulating phase n,,,an < 1 regime (V, < - 1.45 V). Fig. 3 is a demonstration of the fact that conductance oscillations are observed only in the B-induced insulating phase, showing a set of G versus VBG data, taken at V, = -2.061 V, for several B’s for B < 2.7 T. As seen in Fig. 2, B,_, = 1.7 T at this V,. In the metallic regime (B = 0 and 1 T), there are no conductance oscillations. In particular, at B = 1 T, G shows a wide plateau of e2/h, suggesting the v = 1 IQHE state in the narrow channel. When the channel is in the insulating phase (B = 2.0, 2.5, and 2.7 T data), conductance oscillations appear. These oscillations become sharper and more developed as B increases. Fig. 3f shows the Fourier transform of the data taken at B = 2.7 T, clearly showing a peak corresponding to a dominant oscillation periodicity of 0.83 V. A careful study of the T dependences and the current-voltage (Z-V) characteristics of these oscillations was made and the results were analyzed in terms of the 1D WS model of GRS and AN. The most important prediction of the AN model
4.5
Cd) B = 2.5
T
4
2.5
L
1.1
0.2 (b) B = 1.0 T
(e) B = 2.7 T
Fig. 3. A set of G vs. VBG data taken at V, = - 2.061 V, for several B’s for B < 2.7 T.
632
S. W. Hwcmg et al. /Surface
is that both the conductance maxima (G,,,) and the conductance minima (G,,,) follow power law dependences on T, and as a result, both G,,, and G,i, increase with increasing T. Our data show wide Vsc ranges where both G,,,, and G,i, increase with T. We fit the data with the predicted power law and obtain parameters in reasonable agreement with estimation. We also observe a sharp threshold in the I-V taken from the oscillation data. When V is smaller than the threshold voltage, the differential conductance also fits the power law predicted by GRS, again with reasonable parameters [7]. In conclusion, we have studied our low density low disorder 1D ES at high B. We identify a transition into a B-field-induced insulating phase. More importantly, we observe conductance oscillations, predicted to occur in 1D WS, in this B-induced insulating phase in the n,oan < 1 low density limit. The analysis of the observed conductance oscillations shows that many of the quantitative and qualitative features of our data are consistent with the ID WS model. In particular, the T dependences and the I-V characteris-
Scirncr
305 (1994) 629-632
tics can be quantitatively explained model with reasonable parameters. This work is supported the ONR.
by the ID WS
in part by the NSF and
1. References III J.H.F. Scott-Thomas,
S.B. Field, M.A. Kastner. 11.1. Smith and D.A. Antoniadis, Phys. Rev. Lett. 62 (19X9) 583. l21 For a review, see. G. Gruner, Rev. Mod. Phys. 60 (1YXX)
I 120. l31 D.V. Averin and K.K. Likharev,
Proceedings of I YYI International Symposium on Nanostructures and Mesoscopic Systems (Santa Fe. NM, May 1991). [41 L.I. Glazman, I.M. Ruzin and B.I. Shklovskii. Phys. Rev. R 4s (1992) 8454. 151 D.V. Averin and Yu.V. Nazarov, Phys. Rev. B 47 (IY’U) 9944. 161A.I. Larkin and P.A. Lee. Phys. Rev. B 17 (lY7X) 1596. [71 SW. Hwang, D.C. Tsui and M. Shayegan, to be published. quantization in the 7-4 pm channel l81 The conductance fabricated on a 2.0X I()” cm-~’ density sample was reported by K. Ismail. S. Washburn and K.Y. Lee, Appl. Phys. Lett. 59 (1991) 1998. [‘)I L.1. Glazman and I.A. Larkin. Semicond. Sci. Technol. h (IVYI) 32.
........~“..~~.~.~.~::~;~~,::~,: :,.:”‘,,,~ ..‘,., ,,,;“::,: ,,,,;,,, ::~~ i.‘.‘.?:.:~:.:.~.:.:.::.:..>:;:.:.: .,\ y . .. . . .. . _. .,.,
‘..“ ‘.-.‘,T ....‘.‘.,. .,...:... ..:.::$::::;.:~~,:;,:::(.: . . . j;$:,:.y. i..
.i . . .
surface
.:...
;.~,:,x,:,
. . .. . . ,:,,,
science
.+,.:::.:;;i ...,. ...:.: .-.:.::...:~:~:~~~:~zi8x ‘_“‘,‘. :‘.‘.:.+.:i:;... .....:_>:.: :..: ~(:. ~.:,~.“‘... “‘i.~::.:::::‘:.:.:...:..:, ....:O’,i ,.,., ~;~ _..,:,:: ~,:,.:‘i...., ELSEVIER
Surface
Science 305 (1994) 629-632
Gate tuned transition to the insulating phase of one-dimensional electrons in high magnetic fields S.W. Hwang *, D.C. Tsui, M. Shayegan Depart~~t
of Electrical Engineering, Princeton ~~iL~ersi~,Princeton, NJ O&5&. USA
(Received 19 April 1993;accepted for publication 21 July 1993)
Abstract We report experimental results on charge transport in a low disorder and low density one-dimensional electron system. We find a gate tuned transition to an insulating phase at high magnetic fields and conductance oscillations as a function of density in this B-induced insulating phase. A carefu1 study of the temperature dependence and the current-voltage characteristics of these conductance oscillations is made and the results fit the recent theoretical model of one-dimensional Wigner solid.
Recently, Scott-Thomas et al. [l] found, in a narrow Si inversion layer, periodic conductance oscillations as a function of the density. They interpreted the oscillations as manifestation of a one-dimensional charge-density-wave (1D CDW) [2] pinned by impurities in the channel. The pinning energy is maximized whenever the density is commensurate with an integer number of electrons between two pinning centers, resulting in periodic oscillations of the conductance as the electron density is changed. Subsequently, the formation of a one-dimensional Wigner solid flD WS> in a low disorder and low density one-dimensional electron system (1D ES) was theoretically studied by Averin and Likharev [3], and by Glazman, Ruzin, and Shklovskii (GRS) 141. Both theoretical papers emphasize that in the low density limit, when lo,,, <(l/a,> (where n,, is the
* Corresponding
0039-~28/94/$07.~
author.
0 1994 Elsevier
SSDI 0039-6028(93)E0711-3
Science
1D ele$ron density and a, is the Bohr radius, N 100 A in GaAs), the electrons are expected to form a 1D WS pinned by weak potential fluctuations in the channel. In a more recent paper, Averin and Nazarov (AN) 151calculated the temperature (7”) dependence of the maxima and minima of such conductance oscillations, based on the charge density wave tunneling theory of Larkin and Lee [6]. In principle, the nioan < 1 regime can be reached by simply decreasing the channel width (IV>. In a usual split-gate channel fabricated on high density GaAs/Al,Ga,_,As 2D ES, the ~lioaa < 1 condition is expected to be obtained by biasing the gate strongly such that the channe1 is near pinch-off. In reality, however, such channels have strong disorder and it has not been possible to maintain an ideal, open channel near pinch-off. One way of overcoming this problem is to fabricate a split-gate channel on a low density, low disorder 2D ES. In this case, the n,,,ua < 1 limit can be reached even when W is not extremely
B.V. All rights reserved
630
S. W. Hwang et al. /Surface
small and the channel is still far away from pinchoff. In this paper, we report experimental results on charge transport in the low disorder and lowdensity (n,,a, < 1) regime where the formation of 1D WS is anticipated. The sample is a 1 p wide split-gate channel fabricated on a wafer of GaAs/Al,Ga, _,As heterostructure with an extremely low two-dimensional density, n2,, = 6.5 x 10”’ cm-’ and high mobility, I_L= 10h cm*/V . s. In this sample, W and n,, can be varied in a wide range by biasing the split-gate bias, I’,. The details of charge transport characteristics at B = 0 and at low B is reported elsewhere [7] and can be summarized as follows: (1) At B = 0, quantized conductance steps, at integer multiples of 2e*/h, are observed in our channel. The fabricated channel length is 1.2 pm at I(, = 0 and an even longer channel is expected for more negative I$. To our knowledge, the conductance quantization in such a long channel fabricated on such a low 2D electron density wafer has not been observed before [8]. Analysis of these quantized steps shows that we can change n,, in a wide reproducibly reaching the n,,,as < 1 range, regime (corresponding to the bias range, - 2.2 < I(, < - 1.45 V), and more importantly, that unprecedentedly smooth and weak potential fluctuations are obtained in the channel. (2) In the
Science 305 (1994) 629-632
presence of B, we observe the integer (IQHE) and fractional quantum Hall effect (FQHE) states (well defined l/3 state) in the channel. Analysis of the IQHE states shows that the channel has a V, dependence on the electron density predicted by the calculation of split-gate channel with no disorder [9]. For B higher than that for the l/3 FQHE state, the channel becomes an insulator. Fig. la shows the diagonal resistance across the channel, R,,, at I$ = - 1.855 V in the v < 1 regime, at T = 35, 112, 300 mK. Qualitatively, the T dependence of R, changes drastically in the range, 3.5 3.8 T, R,, increases as T decreases and the channel behaves as an insulator. This is especially apparent for the B > 4.0 T data, where the increase is rapid and R,, is approaching to cc as T + 0. This transition of the 1D channel into an insulating state can therefore be characterized by a B field, B,_,, which is - 3.8 T at V, = - 1.855 V.
IO41
,
0
0.1
8:3.0,316,3.7
0.3
c
TO&) Fig. 1. (a) The diagonal resistance, R,, at VP = - 1.855 V in the v < 1 regime, at 7‘= 35, 112, 300 mK. Notice a dramatic change in the T dependence of R,, in the range, 3.5
S. U! Hwang et al. /Surface
8
/
0
6
E+l cQz
I
x cool#l
2
0 cool#2 * cool#3
p
0 -2.5
I -2
-1.5
-1
-0.5
0
vg (V) Fig. 2. (B, VP) phase diagram showing the B,_, deduced from the data of 3 different cool-downs. Two straight lines are eye-guide lines.
We have investigated the T dependences of R, as a function of B and V,, and the B,-, deduced from the data taken from 3 cool-downs are summarized in Fig. 2. The vertical dashed line in the figure indicates the V, below which the channel is already insulating at B = 0 (the conductance of the channel, G < e2/h, VP< - 2.2 V), and the IQHE is not observed. The upper left part of the graph is the B field induced insulating phase where R, increases with decreasing T. The lower right part is the metallic phase where the QHE states are observed. The IQHE states are observed in the entire bias range on the right-hand side of the dashed line mentioned above and the FQHE states are observed at V, > - 1.9 V. B,-, clearly varies with V,, being smaller at larger -V,. In terms of the Landau level filling factor, the transition occurs in the range 0.28 < v < 0.33, when 2.8 X 1O’O cme2 < n2,, < 6.5 X 10’” cm-* (V, > - 1.9 V), and at larger filling factors (0.57 10’” cmd2 CV,<
Science 305 (I 994) 629-632
631
phase diagram, Fig. 2. We find that the conductance oscillations are observed in the insulating phase n,,,an < 1 regime (V, < - 1.45 V). Fig. 3 is a demonstration of the fact that conductance oscillations are observed only in the B-induced insulating phase, showing a set of G versus VBG data, taken at V, = -2.061 V, for several B’s for B < 2.7 T. As seen in Fig. 2, B,_, = 1.7 T at this V,. In the metallic regime (B = 0 and 1 T), there are no conductance oscillations. In particular, at B = 1 T, G shows a wide plateau of e2/h, suggesting the v = 1 IQHE state in the narrow channel. When the channel is in the insulating phase (B = 2.0, 2.5, and 2.7 T data), conductance oscillations appear. These oscillations become sharper and more developed as B increases. Fig. 3f shows the Fourier transform of the data taken at B = 2.7 T, clearly showing a peak corresponding to a dominant oscillation periodicity of 0.83 V. A careful study of the T dependences and the current-voltage (Z-V) characteristics of these oscillations was made and the results were analyzed in terms of the 1D WS model of GRS and AN. The most important prediction of the AN model
4.5
Cd) B = 2.5
T
4
2.5
L
1.1
0.2 (b) B = 1.0 T
(e) B = 2.7 T
Fig. 3. A set of G vs. VBG data taken at V, = - 2.061 V, for several B’s for B < 2.7 T.
632
S. W. Hwcmg et al. /Surface
is that both the conductance maxima (G,,,) and the conductance minima (G,,,) follow power law dependences on T, and as a result, both G,,, and G,i, increase with increasing T. Our data show wide Vsc ranges where both G,,,, and G,i, increase with T. We fit the data with the predicted power law and obtain parameters in reasonable agreement with estimation. We also observe a sharp threshold in the I-V taken from the oscillation data. When V is smaller than the threshold voltage, the differential conductance also fits the power law predicted by GRS, again with reasonable parameters [7]. In conclusion, we have studied our low density low disorder 1D ES at high B. We identify a transition into a B-field-induced insulating phase. More importantly, we observe conductance oscillations, predicted to occur in 1D WS, in this B-induced insulating phase in the n,oan < 1 low density limit. The analysis of the observed conductance oscillations shows that many of the quantitative and qualitative features of our data are consistent with the ID WS model. In particular, the T dependences and the I-V characteris-
Scirncr
305 (1994) 629-632
tics can be quantitatively explained model with reasonable parameters. This work is supported the ONR.
by the ID WS
in part by the NSF and
1. References III J.H.F. Scott-Thomas,
S.B. Field, M.A. Kastner. 11.1. Smith and D.A. Antoniadis, Phys. Rev. Lett. 62 (19X9) 583. l21 For a review, see. G. Gruner, Rev. Mod. Phys. 60 (1YXX)
I 120. l31 D.V. Averin and K.K. Likharev,
Proceedings of I YYI International Symposium on Nanostructures and Mesoscopic Systems (Santa Fe. NM, May 1991). [41 L.I. Glazman, I.M. Ruzin and B.I. Shklovskii. Phys. Rev. R 4s (1992) 8454. 151 D.V. Averin and Yu.V. Nazarov, Phys. Rev. B 47 (IY’U) 9944. 161A.I. Larkin and P.A. Lee. Phys. Rev. B 17 (lY7X) 1596. [71 SW. Hwang, D.C. Tsui and M. Shayegan, to be published. quantization in the 7-4 pm channel l81 The conductance fabricated on a 2.0X I()” cm-~’ density sample was reported by K. Ismail. S. Washburn and K.Y. Lee, Appl. Phys. Lett. 59 (1991) 1998. [‘)I L.1. Glazman and I.A. Larkin. Semicond. Sci. Technol. h (IVYI) 32.