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A two-dimensional array of tunnel junctions used for Coulomb blockade thermometry
Physica B 284}288 (2000) 1788}1789
A two-dimensional array of tunnel junctions used for Coulomb blockade thermometry Tobias Bergsten*, Tord Claeson, Per Delsing Chalmers University of Technology and Go( teborg University, Department of Microelectronics and Nanoscience, SE-412 96 Go( teborg, Sweden
Abstract We have measured current}voltage characteristics of a two-dimensional array of small tunnel junctions at temperatures from 1.4 to 4.2 K. This corresponds to thermal energies larger than the charging energy. We show that 2D-arrays can be used as primary thermometers in the same way as 1D-arrays, and have several advantages over 1D-arrays. 2000 Elsevier Science B.V. All rights reserved. Keywords: 2D array; Coulomb blockade; Primary thermometer
Coulomb blockade thermometry (CBT) is a relatively new primary thermometry method, which was "rst demonstrated by Pekola et al. [1]. It is based on the Coulomb blockade of tunnelling [2,3] in one-dimensional (1D) arrays of very small tunnel junctions at temperatures where k ¹'E "e/2C where E is the charging ! ! energy of the tunnel junctions and C is the junction capacitance. The Coulomb blockade gives rise to a bellshaped dip (Fig. 1) when the di!erential conductance, dI/d<, is plotted against the bias voltage, <. The halfwidth of the dip depends only on the temperature, ¹, and natural constants, which makes it useful for primary thermometry. Pekola et al. have used it successfully in the range from 20 mK to 30 K with an accuracy down to 0.5%. In this paper we demonstrate that a two-dimensional (2D) array of tunnel junctions can be used for CBT in the same way as the 1D-arrays demonstrated by the group of Pekola et al. [1,4]. The motivation for this approach was that 1D-arrays of tunnel junctions are dependent on every junction to function properly, like a chain is only as strong as its weakest link. A 2D-array on the other hand can be functional even if several junctions are damaged. Another motivation was that the noise and the measure-
* Corresponding author. E-mail address: [email protected] (T. Bergsten)
ment speed can be improved in the 2D-case, due to the lower resistance. The theoretical conductance versus voltage has been calculated for 1D-arrays of tunnel junctions [1]. To "rst order the half-width voltage < is directly proportional to the temperature e< "5.439Nk ¹, (1) where N is the number of junctions in series. At low temperature (k ¹+E ) higher-order corrections should ! be included [4], e< "5.439Nk ¹#0.7108NE . (2) ! The measurements were carried out in a pumped He cryostat at di!erent temperatures from 1.4 to 4.2 K. The sample was an array of 256;256 Al}AlO }Al tunnel V junctions with E /k "0.85 K. We measured I<-curves ! at 50 di!erent temperature points and di!erentiated them numerically. Using Eq. (2) we calculated the temperature and compared it with the temperature calculated from the equilibrium pressure in the He bath, using the de"nition of ITS-90 [5]. As can be seen in Fig. 2 the two temperatures are essentially the same. The standard deviation between the two temperature measurements is 0.3% in the temperature range from 2.0 to 4.2 K. At lower temperatures the deviation is larger because the temperature of the He bath is more sensitive to pressure #uctuations.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 9 7 5 - 0
T. Bergsten et al. / Physica B 284}288 (2000) 1788}1789
Fig. 1. The di!erential conductance as a function of bias voltage shows a bell-shaped dip around zero voltage. These measurements were made on a 256;256 junction 2D-array with a resistance of 17 k) and a capacitance of 2.2 fF.
1789
It can be shown [6] that the measurement uncertainty is generally lower in a 2D-array than in a 1D-array and that a 2D-array has a higher tolerance to non-uniformities in the tunnel junction properties. The conclusion is that a 2D-array of tunnel junctions can be used the same way as a 1D-array as a cryogenic thermometer, with certain advantages. The advantages are the robustness to damage, the lower measurement uncertainty, the higher tolerance to non-uniformity and the higher measurement speed due to lower resistance. This should make it more suitable for thermometry at temperatures above a few hundred mK. At lower temperatures the heating of the electrons requires cooling "ns to be attached to the array islands [7] and this is not practical in a 2D-array.
References [1] J.P. Pekola, K.P. Hirvi, J.P. Kauppinen, M.A. Paalanen, Phys. Rev. Lett. 73 (1994) 2903. [2] D.V. Averin, K.K. Likharev, in: B. Al'tshuler, P. Lee, R. Webb (Eds.), Mesoscopic Phenomena in Solids, ISBN 0444-88454-8, Elsevier, Amsterdam, 1991, p. 173. [3] H. Grabert, M. Devoret (Eds.), Single Charge Tunneling, Coulomb Blockade Phenomena in Nanostructures, ISBN 0-306-44229-9, Plenum Press, New York, 1992. [4] Sh. Farhangfar, K.P. Hirvi, J.P. Kauppinen, J.P. Pekola, J.J. Toppari, D.V. Averin, A.N. Korotkov, J. Low Temp. Phys. 108 (1997) 191. [5] H. Preston-Thomas, Metrologia 27 (1990) 3. [6] T. Bergsten, T. Claeson, P. Delsing, J. Appl. Phys. 86 (1999) 3844. [7] J.P. Kauppinen, J.P. Pekola, Phys. Rev. B 54 (1996) 8353. Fig. 2. The di!erence between the temperature calculated from the half-width of the Coulomb blockade in the I<-curve and the temperature calculated from the He equilibrium pressure. The inset shows the two temperature measurements directly. The notable deviation at the lowest temperature point is most likely due to a transition to the superconducting state in the aluminium.
A two-dimensional array of tunnel junctions used for Coulomb blockade thermometry Tobias Bergsten*, Tord Claeson, Per Delsing Chalmers University of Technology and Go( teborg University, Department of Microelectronics and Nanoscience, SE-412 96 Go( teborg, Sweden
Abstract We have measured current}voltage characteristics of a two-dimensional array of small tunnel junctions at temperatures from 1.4 to 4.2 K. This corresponds to thermal energies larger than the charging energy. We show that 2D-arrays can be used as primary thermometers in the same way as 1D-arrays, and have several advantages over 1D-arrays. 2000 Elsevier Science B.V. All rights reserved. Keywords: 2D array; Coulomb blockade; Primary thermometer
Coulomb blockade thermometry (CBT) is a relatively new primary thermometry method, which was "rst demonstrated by Pekola et al. [1]. It is based on the Coulomb blockade of tunnelling [2,3] in one-dimensional (1D) arrays of very small tunnel junctions at temperatures where k ¹'E "e/2C where E is the charging ! ! energy of the tunnel junctions and C is the junction capacitance. The Coulomb blockade gives rise to a bellshaped dip (Fig. 1) when the di!erential conductance, dI/d<, is plotted against the bias voltage, <. The halfwidth of the dip depends only on the temperature, ¹, and natural constants, which makes it useful for primary thermometry. Pekola et al. have used it successfully in the range from 20 mK to 30 K with an accuracy down to 0.5%. In this paper we demonstrate that a two-dimensional (2D) array of tunnel junctions can be used for CBT in the same way as the 1D-arrays demonstrated by the group of Pekola et al. [1,4]. The motivation for this approach was that 1D-arrays of tunnel junctions are dependent on every junction to function properly, like a chain is only as strong as its weakest link. A 2D-array on the other hand can be functional even if several junctions are damaged. Another motivation was that the noise and the measure-
* Corresponding author. E-mail address: [email protected] (T. Bergsten)
ment speed can be improved in the 2D-case, due to the lower resistance. The theoretical conductance versus voltage has been calculated for 1D-arrays of tunnel junctions [1]. To "rst order the half-width voltage < is directly proportional to the temperature e< "5.439Nk ¹, (1) where N is the number of junctions in series. At low temperature (k ¹+E ) higher-order corrections should ! be included [4], e< "5.439Nk ¹#0.7108NE . (2) ! The measurements were carried out in a pumped He cryostat at di!erent temperatures from 1.4 to 4.2 K. The sample was an array of 256;256 Al}AlO }Al tunnel V junctions with E /k "0.85 K. We measured I<-curves ! at 50 di!erent temperature points and di!erentiated them numerically. Using Eq. (2) we calculated the temperature and compared it with the temperature calculated from the equilibrium pressure in the He bath, using the de"nition of ITS-90 [5]. As can be seen in Fig. 2 the two temperatures are essentially the same. The standard deviation between the two temperature measurements is 0.3% in the temperature range from 2.0 to 4.2 K. At lower temperatures the deviation is larger because the temperature of the He bath is more sensitive to pressure #uctuations.
0921-4526/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 4 5 2 6 ( 9 9 ) 0 2 9 7 5 - 0
T. Bergsten et al. / Physica B 284}288 (2000) 1788}1789
Fig. 1. The di!erential conductance as a function of bias voltage shows a bell-shaped dip around zero voltage. These measurements were made on a 256;256 junction 2D-array with a resistance of 17 k) and a capacitance of 2.2 fF.
1789
It can be shown [6] that the measurement uncertainty is generally lower in a 2D-array than in a 1D-array and that a 2D-array has a higher tolerance to non-uniformities in the tunnel junction properties. The conclusion is that a 2D-array of tunnel junctions can be used the same way as a 1D-array as a cryogenic thermometer, with certain advantages. The advantages are the robustness to damage, the lower measurement uncertainty, the higher tolerance to non-uniformity and the higher measurement speed due to lower resistance. This should make it more suitable for thermometry at temperatures above a few hundred mK. At lower temperatures the heating of the electrons requires cooling "ns to be attached to the array islands [7] and this is not practical in a 2D-array.
References [1] J.P. Pekola, K.P. Hirvi, J.P. Kauppinen, M.A. Paalanen, Phys. Rev. Lett. 73 (1994) 2903. [2] D.V. Averin, K.K. Likharev, in: B. Al'tshuler, P. Lee, R. Webb (Eds.), Mesoscopic Phenomena in Solids, ISBN 0444-88454-8, Elsevier, Amsterdam, 1991, p. 173. [3] H. Grabert, M. Devoret (Eds.), Single Charge Tunneling, Coulomb Blockade Phenomena in Nanostructures, ISBN 0-306-44229-9, Plenum Press, New York, 1992. [4] Sh. Farhangfar, K.P. Hirvi, J.P. Kauppinen, J.P. Pekola, J.J. Toppari, D.V. Averin, A.N. Korotkov, J. Low Temp. Phys. 108 (1997) 191. [5] H. Preston-Thomas, Metrologia 27 (1990) 3. [6] T. Bergsten, T. Claeson, P. Delsing, J. Appl. Phys. 86 (1999) 3844. [7] J.P. Kauppinen, J.P. Pekola, Phys. Rev. B 54 (1996) 8353. Fig. 2. The di!erence between the temperature calculated from the half-width of the Coulomb blockade in the I<-curve and the temperature calculated from the He equilibrium pressure. The inset shows the two temperature measurements directly. The notable deviation at the lowest temperature point is most likely due to a transition to the superconducting state in the aluminium.