WO2002014815A1

WO2002014815A1 - Coulomb blockade thermometer

Info

Publication number: WO2002014815A1
Application number: PCT/SE2001/001779
Authority: WO
Inventors: Tobias Bergsten; Per Delsing
Original assignee: Chalmers Technology Licensing Ab
Priority date: 2000-08-18
Filing date: 2001-08-20
Publication date: 2002-02-21
Also published as: SE0002950D0; SE0002950L; JP2004506886A; AU2001282771A1; SE515920C2

Abstract

The present invention relates to a method for measuring a temperature in a Coulomb blockade thermometer, consisting of tunnel junctions and means for measuring its current-voltage characteristics. The method comprises the step of measuring the third derivative of the current-voltage characteristics, d3I/dV3, which derivate shows a zero crossing at the voltage which scales linearly depending on the temperature so that said voltage forms a primary thermometer. The invention also relates to a Coulomb blockade thermometer.

Description

COULOMB BLOCKADE THERMOMETER

TECHNICAL FIELD

The present invention relates to a method for measuring a temperature in a Coulomb blockade thermometer, consisting of tunnel junctions and means for measuring its current- voltage characteristics.

BACKGROUND OF THE INVENTION

Coulomb Blockade Thermometry (CBT) [1, 2, 3] is a primary thermometry method that is suitable for cryogenic temperatures in the range 20mk - 30K. It is based on the properties of the Coulomb blockade [4, 5] in one- or two-dimensional arrays of tunnel junctions at temperatures where the charging energy E_c< kβT. Here Ec= e²/2C_eff, where C_eff is the effective capacitance of the tunnel junctions [3]. In the CBT method, the first derivative of the current- voltage characteristics (the TV-curve) is measured and from the properties of this curve the temperature can be extracted, using natural constants and calculable prefactor. The major advantage of the CBT method is the simple electrical measuring and that it does not influence the insensitivity to magnetic field [6].

In US 5,947,601 a CBT including a sensor component and devices for measuring its voltage-current dependence is described. The sensor includes a chain of several nanoscale tunnel junctions and connection electrodes for connecting the measuring devices to the end of the chain. The temperature is measured on the basis of characteristic quantities of the descriptor G7Gχ of the voltage-current dependence. This CBT works completely different than that one according to the present invention, which provides a much faster measurement.

BREIF DESCRIPTION OF THE FIGURES Figure 1 shows the third derivative of the IV-curve, d³I/dV³, as a function of the voltage, measured at three different temperatures.

Figure 2 shows the temperature extracted from the zero crossing drawn towards the temperature extracted from the steam pressure in a ⁴He-bath. Figure 3 shows a time lapse for the temperature extracted from a voltage V₀ at a zero crossing.

Figure 4 shows the feedback circuit used to demonstrate our temperature measuring method.

Figure 5 shows an example of a measuring circuit according to the invention.

Figure 6 shows three different examples of tunnel junction arrays.

DESCRIPTION OF THE INVENTION

The third derivative of the current- voltage characteristics, d/³/dF³'in two-dimensional arrays of small tunnel junctions HAS been measured by means of a cover-amplifier. It has been shown that this derivative is zero at a voltage that is linearly depending on the temperature and is only due to the temperature and some natural constants. This voltage therefore forms a primary thermometer. Here a method for measuring is described that generates a voltage at the zero crossing directly by means of a feedback circuit. This method only requires measuring of one voltage, which makes it substantially faster than the original method of Coulomb Blockade Thermometry (CBT).

Hereby a method is described that gives a much faster measurement. The same type of arrays of tunnel junctions can be used and instead the third derivative of the IV-curve is measured. The third derivative has a zero crossing at a voltage which (to the first order of Ec/kβT) is proportional to the temperature. Hereby only one point for measuring is needed, the voltage at the zero crossing, for measuring the temperature. Using the original method the whole curve di/dFis needed as a function of V. There is also a secondary method for measuring where one only measures the conductance at 0V. However it requires calibration of the resistance and capacitance of the tunnel junctions.

If the expression for dZ/dF [1] is derivatived twice with reference to Fthe third derivative can be written d VdF³, to the first order of Ec/k_BT.

Here N and M are the number of tunnel junctions in series and in parallel (in case of a two-dimensional array) respectively, Rj is the tunnelling resistance of one junction at voltages well above the Coulomb blockade and g (x) is defined in the paper by Pekola et al [1] and can be written

, (x 12)coh (x 11)- ^r'^~ 2sinli²( 2) (2)

Then the function g '(x) becomes

(( /2)coth( /2)-lX3coh²( /2)-2) ^{8 )~} 2sinh²(x/2) ⁰⁾

The equation (1) is valid within Ec « k_BT and R_τ » RR = h/e² « 25.8 kΩ. Lower temperatures and lower resistances give deviations that can be calculated theoretically [2,7]. Hereby consideration has been made only to those effects that derive from low temperatures. The deviations, due to low resistance, has been disregarded, but approximately are the errors less than 1% in the accounted measurements.

The voltage V₀ at the zero crossing of Eq. (1) can be calculated numerically and the result is

eV₀=±2.144Nk_BT. (4)

At low temperatures (where kβT approaches Ec) higher order corrections should be included [2,8]. This gives a correctionterm to V₀ in Eq. (1), which is independent to the temperature:

eV₀ = +2.U4Nk_BT - 0.465NE_C . (5) The object of measurement according to the invention was a two-dimensional array of 256 x 256 tunnel junctions and each junction had an effective capacitance of 2.2 fF and a tunnelling resistance of 17 kΩ . The array was fabricated using standard shadow evaporation [9,10] of aluminium and in situ oxidation. The measurements were carried out by applying a DC voltage and an additional AC excitation (123 Hz) to the sample, in series with a resistor R with a resistance of 20 kΩ. The voltage over the resistor was measured with a Stanford SRS830 lock-in amplifier, which was locked to the third harmonic (369 Hz) of the excitation AC voltage. This signal δV_3ω is proportional to the third derivative of the IV-curve. The excitation voltage δV_ω over the sample was measured with another lock-in amplifier, which was locked to the basic frequency (123 Hz) and the DC voltage was measured with a voltmeter. Additional low-, high- and band-pass filtering were used to improve the measurement.

A pumped ⁴He cryostat equipped with a vacuum regulator was used, which kept the bath at a constant pressure, and thereby constant temperature, during the measurements.

To find the connection between the third derivative and δV_3ω a Taylor development was accomplished of the TV-curve to the third order:

dX 24 δ V _. _ ά V* (6)

R_b δv-

where δV_3ω and δV_ω are voltage amplitudes (not rms values).

This method requires relatively large excitation amplitude, comparable with to V₀, which introduces errors due to higher derivatives. If including fifth and seventh derivatives in the Taylor development this can be written

I±*__. - _L₊ *__ L₊-__£1 _m

R_b δV dV³ 16 dV^s 640 άV ^'

If the size of the excitation δV_ω is known the errors that are generated due to higher derivatives can be calculated and the result can be corrected. On the other hand the higher derivatives are rather small right at the zero crossing, which means that a relatively large δV_ω can be used without initiating too large errors.

Figure 1 shows a measurement of d³1/dV³ at three different temperatures. The shape of the curve follows the expected function g"(x), and the voltage at the zero crossing follows Eq. (5) within a few percent. Figure 2 shows the temperature calculated from the zero crossing plotted against the temperature calculated from the ⁴He vapour pressure [11] from 1.6 K to 4.2 K.

Figure 3 shows the feedback circuit that was used to demonstrate our temperature measuring method. The lock-in amplifier generates a sine wave that is added to the DC voltage produced by the PID circuit. The resulting voltage signal is applied to the sensor in series with the resistor, and the lock-in amplifier senses the voltage over the resistor. The lock-in amplifier is set up to detect the third harmonic of the generated sine wave, to effectively measure the third derivative of the TV-curve. The DC output voltage is proportional to the amplitude of the detected signal and serves as an input error signal to the PID circuit. With proper feedback parameters the voltage over the sensor will be stabilised at V₀, as defined in Eq. (5).

To test the principal advantage of this measuring method compared to the one used by Pekola, a feedback circuit was arranged, which is illustrated in Figure 4. A DC voltage from the lock-in amplifier was used, which was proportional to the amplitude for the δV_3ω signal, as an error signal to a PID regulator. A PID regulator (Proportional Integrating and Deriving) is a general feedback circuit with three adjustable parameters, which can be used with a wide range of applications.

The output of the regulator was added to the excitation from the lock-in amplifier and was applied over the array in series with a resistor. The voltage over the resistor was measured with the lock-in amplifier, which was set to extract the third harmonic of the excitation frequency. Then the PID parameters were adjusted to proper values so the DC voltage over the array was stabilised at the voltage V₀, as defined by Eq. (5). Figure 4 shows a time trace of the temperature, calculated from the zero-crossing voltage Vo [Eq. (5)] while the temperature was varied in four steps. The dashed horizontal lines represent the temperature calculated from the vapour pressure of the ⁴He bath [11]. The agreement is reasonable, except at the lowest temperature where higher order effects and higher derivatives affect the measurement.

As a demonstration that the voltage follows the temperature as expected, a time trace of the voltage V₀ was taken while the temperature was adjusted in steps, by changing the bath pressure, and the result is the graph in Figure 3, where the voltage V₀ is converted to temperature according to Eq. (5). The temperature steps are evident in the figure, and aggress well with the temperature-calculated form the ⁴He vapour pressure, except for the lowest step. The disagreement at this step is probably due to higher derivatives and higher order corrections to Eq. (5) [2]. At the beginning of the fourth step, at 600 s in Figure 4, the P gain of the feedback was too large and the signal started to oscillate, but after reducing the gain (at around 680 s in the graph) the signal become stable again. Note that the relatively slow time response is not due to the thermometer or the measurement, but rather due to the time it takes to pump down the pressure in the ⁴He bath.

While Figure 4 shows that this method is working, the precision is not very impressive, with fluctuations up to 10%. There are several ways to improve this situation. The use of a lock-in amplifier picks up the very weak third harmonic signal below the main excitation signal and the noise, but by notching out the basic frequency before the input, the dynamic range of the amplifier can be increased, and get a cleaner signal. The PID parameters can also be better optimised to the measurement, and continuously adjusted.

Looking at Eq. (1), it is obvious that we can make another improvement by increasing M, i.e. the number of parallel junctions in the array. The signal amplitude increases linearly with M. This speaks in favour of using 2D arrays with this measuring method. Note that no advantage is gained by decreasing N the number of junctions in series, because to avoid higher order derivatives to affect the measurement, the excitation amplitude δV_ω must be decreased by the same amount. Even though it was not done in this experiment, it would be natural to let δV_ω become proportional to the temperature, to compensate for the strong signal dependence on temperature (~ ). In conclusion, the third derivative of the IV-curve of a two-dimensional array tunnel junctions was measured. By means of experiments as well as theoretically it is shown, that the zero crossing of this curve scales linearly with the temperature, in the first order, and provides a primary temperature measurement. A feedback circuit is also demonstrated that can be used to create a fast primary thermometer. This feedback circuit is possible because only one single point of measurement is needed to give a voltage value, which is proportional to the temperature.

Figure 5 shows a detailed example of a measuring circuit. A DC voltage and an AC voltage at a frequency ω are added and applied to the tunnel junction network in series with a resistor. The current through the resistor at the frequency 3_ω is measured with a lock-in amplifier or an equivalent circuit. The amplitude of the 3_ω component, represented by a DC voltage, is fed to a feedback circuit, e.g. a PID regulator circuit, which adjusts the DC voltage applied to the junctions network until the 3_ω component of the current is zero. The DC voltage over the network is then proportional to the temperature, and with proper scaling it can be fed to a display, which shows the temperature. Some optional components, represented by dashed boxes in the figure, can be used to improve the measurement. The 3_ω component of the current can be singled out by using a notch and/or a band pass filter, providing a higher signal-to-noise ratio. The amplitude of the AC voltage at frequency ω can be scaled with the temperature to give a constant AC to DC voltage ratio. It is very important that the ω signal does not contain any 3ω components, so a notch and/or a band pass filter can ensure that there is no 3ω signal in the AC voltage. Furthermore in the lock-in circuit a phase tuning may be used to optimize the signal detection.

Figure 6 shows three examples of connections for measuring the voltage Vo and the third derivative of the current I. V_outis the voltage connected to a lock-in amplifier or the like for measuring the signal at three times the frequency for V_AC- The top two circuits of the figure use an operational amplifier with feedback while the third one uses an instrumentation amplifier. The output voltage is then measured by the lock-in circuit to extract the 3ω signal in the current.

REFERENCES

[1] J. P. Pekola, K. P. Hirvi, J. P. Kauppinen and M. A. Paalanen, Phys. Rev. Lett. 73, 2903 (1994). [2] Sh. Farhangfar, K. P. Hirvi, J. P. Kauppinen, J. P. Pekola, J. J. Toppari, D. V. Averin and A. N. Korotkov, J. Low Temp. Phys. 108, 191 (1997).

[3] T. Bergsten, T. Claeson and P. Delsing, J. Appl. Phys. 86, 3844 (1999).

[4] D. V. Averin and K. K. Likharev, i Mesoscopic phenomena in solids, red. B. Al'tshuler, P. Lee and R. Webb, ISBN 0-444-88454-8 (Elsevier, Amsterdam, 1991), s. 173.

[5] Single charge tunneling, Coulomb blockade phenomena in nanostructures, red. H. Grabert and M. Devoret, ISBN 0-306-44229-9 (Plenum, New York, 1992).

[6] J. P. Pekola, J. J. Toppari, J. P. Kauppinen, K. M. Kinnunen och A. J. Manninen, J. Appl. Phys 83, 5582 (1998). [7] Sh. Farhangfar, R. S. Poikolainen, J. P. Pekola, D. S. Golubev och A. D. Zaikin, cond-mat/0005283.

[8] J. Kinaret (enskilda diskussioner).

[9] Jurgen Niemeyer, PTB-Mitteilungen 84, 251 (1974).

[10] G. J. Dolan, Appl. Phys. Lett., 31, 337 (1977). [11] H. Preston-Thomas, Metrologia 27, 3 (1990).

Claims

1. A method for measuring a temperature in a Coulomb Blockade Thermometry, consisting of tunnel junctions and means for measuring its current- voltage characteristics, characterized in, that the method comprises the step of measuring a third derivative of the current- voltage characteristics, d³I/dV³, which derivate shows a zero-crossing at a voltage which linearly depends on the temperature so that the said voltage constitute a primary thermometer.

2. Method according to claim 1, characterized in, that said voltage is generated at the zero crossing by means of a feedback circuit.

3. Method according to claim 1 or 2, characterized in, that said tunnel junctions are arranged in a two-dimensional array.

4. Method according to any of claim 1-3, characterized in, that the voltage is measured by means of a lock-in amplifier.

5. Method according to claim 1, characterized in, that said third derivative is

where N and M are the number of tunnel junctions in series and in parallel, respectively, in case of a two-dimensional array, R_τ is the tunnelling resistance of one junction and g(x) is a function defined as ₍ , (x /2)co h (x /2)- l ^g^^X)~ 2 smh ² (x /2)

6. A Coulomb Blockade Thermometry (CBT), consisting of a sensor component and devices for measuring its voltage-current characteristics, characterized in, that said device for measuring its voltage-current characteristics comprises a feedback circuit, an amplifier that generates a sine wave, which is added to a DC voltage from the feedback circuit, when a resulting voltage signal is applied to the sensor in series with the resistor, and the voltage over the resistor is sensed by the amplifier, and the amplifier is set up to detect the third harmonic of the generated sine wave.

7. The CBT according to claim 6, characterized in, that a DC output voltage from the amplifier is proportional to the amplitude of the detected signal and serves as an error signal to the feedback circuit.

8. The CBT according to claim 6 or 7, characterized in, that the amplifier is a lock-in amplifier.

9. The CBT according to any of claim 6-8, characterized in, that the feedback circuit is a PID regulator.

10. The CBT according to any of claim 6-9, characterized in, that the tunnel junctions are arranged in a two-dimensional array.

11. A Coulomb Blockade Thermometry (CBT), comprising a tunnel junction network and devices for measuring its voltage-current characteristics, characterized in, that the said devices comprises an adding means in which a DC voltage and an AC voltage at a first frequency (ω) are added and applied to the tunnel junction network in series with a resistor, a lock-in amplifier for measuring the current through the resistor at a second frequency (3ω), the amplitude of a 3ω component, represented by a DC voltage, is fed to a feedback circuit, said feedback circuit, which adjusts the DC voltage applied to the junctions network until the 3ω component of the current is zero, the DC voltage over the network is then proportional to the temperature, and can with proper scaling be fed to a display which shows the temperature.

12. The CBT according to claim 11 , characterized in, that the 3ω component of the current can be singled out by using notch and/or bandpass filter, providing a higher signal-to-noise ratio.

13. The CBT according to claim 11 or 12, characterized in, that the amplitude of the AC voltage can be scaled at the first frequency with the temperature to provide a constant AC to DC voltage ratio.

14. The CBT according to claim 11 , characterized in, that a phase tuning is used in the lock-in circuit to optimise the signal detection.