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Tunneling time and offset charging in small tunnel junctions
Physica B 165&166 (1990) 973-974 North· Holland
TUNNELING TIME AND OFFSET CHARGING IN SMALL TUNNEL JUNCTIONS LJ. GEERLIGS, V.F. ANDEREGG and J.E. MOOD Department of Applied Physics, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands Measurements of single electron charging effects in serially coupled tunnel junctions are presented. The results show that the tunneling time is much longer than the barrier traversal time. We find evidence for the presence of offset charging of the junctions. 1. INTRODUCTION Coulomb blockade of single electron tunneling has been observed in a large number of experiments in the last few years (1). Planar tunnel junctions, produced with nanolithographic methods, and especially a simple double junction, have received relatively little attention (2). In this paper we describe results for arrays of 2, 3 and 5 normal metal tunnel junctions. We find a lower limit for the tunneling time that is much longer than the traversal time of the barrier (3). The experiments show that random offset charging of the junctions is usually present. Both results are important for possible new experiments (4) and applications of these junctions in practical devices (1,5).
2. lHEORY
The tunneling rate across a junction with resistance R t is determined by the energy change ~ during the tunneling process (1): r=~E[exp(~E/kBT)-I]·1/(e2RV. Under the assumption of a short tunneling time (3,6) ~ follows from the change in capacitive energy of only the junction (a 'local rule' (7»: l1E=-e(Q-eI2)/C. where Q is the charge before tunneling on the junction with capacitance C. Recent experiments (8,9) showed the absence of Coulomb blockade in a single junction of small area. This suggested that during tunneling the electron probes the environment on a large scale. Then ~E is equal to the change in (Gibbs) free energy of the total system of capacitors and voltage sources, ~E=~(LQi2/2Ci - LQtiVj). The summations are over all capacitors and voltage sources in the system, Qti is the charge transferred through the voltage source Vi. This 'global rule' is valid if the tunneling time is longer than the response time of the voltage sources. A voltage source can be a large stray capacitor close to and shunting the junctions, which is present in many experiments. ~E can be conveniently expressed (10) as ~E=-e(Q-QC>/C, where the critical charge for tunneling Qc=eI2(1 +CdC) only depends on junction capacitance and the equivalent capacitance Ce of the circuit shunting the junction. For local rule Ce=O and Qc=cl2. For a linear array of n equal junctions, the threshold voltage for conduction (the Coulomb gap) is ne/2C for local rule and (n-l)e/2C for global rule.The Coulomb gap for local rule is equal to the asymptote of the I-V curve at large voltage (3,11). Serially coupled junctions are sensitive to externally applied electric fields. With a gate voltage Vg a 'gate charge' C g V 8. can be induced on the metal island (capacitance CgJ between junctions, resulting in a change of 0921.4526/90/$03.50
© 1990 -
the charges on the junctions (2). This change is periodic in the gate charge with period e. For a gate charge e/2 and zero bias voltage the junction charges are e/4. For a double junction this is equal to Qc under global rule (Ce=C), therefore the Coulomb gap is completely suppressed. Under local rule a voltage threshold el2C for tunneling remains. 3. EXPERIMENTS Fig. 1 shows current voltage (I-V) characteristics for two aluminum-aluminum oxide-aluminum junctions in series. A magnetic field of 2 Tesla was used to bring the junctions in the normal (non-superconducting) state. The I-V curves are consistent with global rule. The voltage offset of the large scale asymptote of the I-V curve (dotted line) should be equal to e/C. The Coulomb gap,
0.2
<' oS 0.5
v
o ~~:::::"'_--,-L----,-_-::-,::--------.:l o
0.1
0.2
V [mV] FIGURE 1 I-V curves for a double junction at 10 mK. The gate voltage was used to maximize (solid curve) and minimize (dashed curve) the Coulomb gap. Dotted line: asymptote of the I-V curve, crosses: fit to experiments as described in the text for kBTlEc=O.05. The inset shows the I-V curve with maximum threshold voltage for a linear array of 5 junctions, together with asymptote.
Elsevier Science Publishers B.V. (North-Holland)
L.J. Geerligs, V.F. Anderegg, J.E. Mooij
974
under global rule equal to e/2C, is indeed approximately half this value (solid curve). The value of e/C",120 IJ.V (C=1.3 fF) is consistent with the junction area of about 0.015 IJ.m 2. The dashed curve shows that by applying a gate voltage the Coulomb gap can be completely suppressed. Also shown in Fig. 1 are calculated I-V curves (crosses) for one junction having C=O.7 fF and the other having C=l.l fF (SEM photographs showed that the junction areas differed by a factor 1.5). In our circuits the voltage source, taken to be a stray capacitance of a few fF, is at a distance of at least 10-20 IJ.ffi from the junctions. With an estimated inductance of the onchip leads of I pH/IJ.ffi, the response time of the voltage bias is at least 10- 13 S. This is therefore a lower limit for the effective tunneling time at low voltages. In previous experiments on two junctions in a scanning tunneling microscope configuration (6) the relevant stray capacitance was so close to the junctions that the authors combined a tunneling time of 10- 15 S (the expected traversal time through the barrier of a tunneling electron (3» with global rule. In our much larger devices, if this timescale were really relevant local rules would apply. In Fig. 2 the current versus gate voltage (1-Vg) is shown for several arrays. At fixed Vg the slope iNg/dl can be used to calculate the charge noise l1n on the central island from the observed current noise in as qn=inCgdVg/dI. Preliminary measurements for a double junction (curve a) yielded 1.5·IO-4e between 2 and 200 Hz, which is one of the lowest values reported so far, but still an order of magnitude higher than expected (1). Curve b shows an example of telegraph noise that we observed in several devices. Effectively two curves were measured, with jumps between the two. Obviously at least one of the two curves corresponds to offset charging of the junctions. Generally offset charging of junctions (polarization of the island between junctions with a. non-integer charge) results from trapped charges, e.g. near the junctions. For curve b apparently a charge could jump between two positions. For more than 2 junctions (c-g), we observe fine structure similar to results for arrays of superconducting junctions frustrated by a magnetic field (12). If we define a frustration CgVgte, then the number of current dips in one period is equal to the number of islands between the junctions, the equivalent of the width in unit cells of an array of superconducting junctions. Curves d end e show beating, probably due to differences in electrode self capacitances. The difference of the curves for the various 5 junction arrays is another indication for offset charging of the junctions. Also the observation that the threshold voltage for more than 2 junctions is systematically less than the expected (n-l)e/2C can thus be understood (inset of Fig. I).
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4. CONCLUSIONS We determined a lower limit of order 10- 13 S for the tunneling time in small planar tunnel junctions. The possibility of offset charges in these systems should be seriously considered. ACKNOWLEDGEMENTS We thank M. Devoret, D. Esteve, U. Geigenmiiller, K Likharev, M. Peters and G. Schon for valuable
-2
-1
2
FIGURE 2 Current versus gate voltage Vg for several arrays with junction resistance between 50 and 100 ill. For each curve the bias voltage is fixed. (a) and (b) 2 junctions. (c) 3 junctions. (d)-(g) 5 junctions. (d) and (e) are I-Vg curves of the same device, at Vg differing by 10 periods. discussions. This work was supported by the Dutch Foundation for Fundamental research on Matter (FOM) and L. Kouwenhoven. REFERENCES (1) See the review by D.V. Averin and KK Likharev, Single electronics, in: Quantum effects in small disordered systems, eds B.L. Al'tshuler, P.A. Lee, and R.A. Webb (Elsevier, Amsterdam) to be published. (2) T.A. Fulton and G.J. Dolan, Phys. Rev. Lett. 59, (1987) 109 have reported experiments on a double planar junction. (3) M. Biittiker, and R. Landauer, ffiM J. Res. Dev. 30 (1986) 451. (4) J.E. Mooij et al., Charge-anticharge unbinding in arrays of small tunnel junction!Y, preprint submitted to Phys. Rev. Lett.. (5) V.F. Anderegg et al., Frequency-determined current in a turnstile device for single electrons, this volume. (6) R. Wilkins et al., Phys. Rev. Lett. 63 (1989) 801. (7) U. Geigenmiiller and G. SchOn, Europhys. Lett. 10 (1989) 765. (8) L.J. Geerligs et ai., Europhys, Lett. 10 (1989) 79. P. Delsing et ai., Phys. Rev. Lett. 63 (1989) ll80. (9) (10) D. Esteve, private communication. (ll) Yu. V. Nazarov, JETP Lett. 49 (1989) 126. (12) D.J. van Harlingen et ai., Physica B152 (1988) 134.
TUNNELING TIME AND OFFSET CHARGING IN SMALL TUNNEL JUNCTIONS LJ. GEERLIGS, V.F. ANDEREGG and J.E. MOOD Department of Applied Physics, Delft University of Technology, P.O. Box 5046, 2600 GA Delft, The Netherlands Measurements of single electron charging effects in serially coupled tunnel junctions are presented. The results show that the tunneling time is much longer than the barrier traversal time. We find evidence for the presence of offset charging of the junctions. 1. INTRODUCTION Coulomb blockade of single electron tunneling has been observed in a large number of experiments in the last few years (1). Planar tunnel junctions, produced with nanolithographic methods, and especially a simple double junction, have received relatively little attention (2). In this paper we describe results for arrays of 2, 3 and 5 normal metal tunnel junctions. We find a lower limit for the tunneling time that is much longer than the traversal time of the barrier (3). The experiments show that random offset charging of the junctions is usually present. Both results are important for possible new experiments (4) and applications of these junctions in practical devices (1,5).
2. lHEORY
The tunneling rate across a junction with resistance R t is determined by the energy change ~ during the tunneling process (1): r=~E[exp(~E/kBT)-I]·1/(e2RV. Under the assumption of a short tunneling time (3,6) ~ follows from the change in capacitive energy of only the junction (a 'local rule' (7»: l1E=-e(Q-eI2)/C. where Q is the charge before tunneling on the junction with capacitance C. Recent experiments (8,9) showed the absence of Coulomb blockade in a single junction of small area. This suggested that during tunneling the electron probes the environment on a large scale. Then ~E is equal to the change in (Gibbs) free energy of the total system of capacitors and voltage sources, ~E=~(LQi2/2Ci - LQtiVj). The summations are over all capacitors and voltage sources in the system, Qti is the charge transferred through the voltage source Vi. This 'global rule' is valid if the tunneling time is longer than the response time of the voltage sources. A voltage source can be a large stray capacitor close to and shunting the junctions, which is present in many experiments. ~E can be conveniently expressed (10) as ~E=-e(Q-QC>/C, where the critical charge for tunneling Qc=eI2(1 +CdC) only depends on junction capacitance and the equivalent capacitance Ce of the circuit shunting the junction. For local rule Ce=O and Qc=cl2. For a linear array of n equal junctions, the threshold voltage for conduction (the Coulomb gap) is ne/2C for local rule and (n-l)e/2C for global rule.The Coulomb gap for local rule is equal to the asymptote of the I-V curve at large voltage (3,11). Serially coupled junctions are sensitive to externally applied electric fields. With a gate voltage Vg a 'gate charge' C g V 8. can be induced on the metal island (capacitance CgJ between junctions, resulting in a change of 0921.4526/90/$03.50
© 1990 -
the charges on the junctions (2). This change is periodic in the gate charge with period e. For a gate charge e/2 and zero bias voltage the junction charges are e/4. For a double junction this is equal to Qc under global rule (Ce=C), therefore the Coulomb gap is completely suppressed. Under local rule a voltage threshold el2C for tunneling remains. 3. EXPERIMENTS Fig. 1 shows current voltage (I-V) characteristics for two aluminum-aluminum oxide-aluminum junctions in series. A magnetic field of 2 Tesla was used to bring the junctions in the normal (non-superconducting) state. The I-V curves are consistent with global rule. The voltage offset of the large scale asymptote of the I-V curve (dotted line) should be equal to e/C. The Coulomb gap,
0.2
<' oS 0.5
v
o ~~:::::"'_--,-L----,-_-::-,::--------.:l o
0.1
0.2
V [mV] FIGURE 1 I-V curves for a double junction at 10 mK. The gate voltage was used to maximize (solid curve) and minimize (dashed curve) the Coulomb gap. Dotted line: asymptote of the I-V curve, crosses: fit to experiments as described in the text for kBTlEc=O.05. The inset shows the I-V curve with maximum threshold voltage for a linear array of 5 junctions, together with asymptote.
Elsevier Science Publishers B.V. (North-Holland)
L.J. Geerligs, V.F. Anderegg, J.E. Mooij
974
under global rule equal to e/2C, is indeed approximately half this value (solid curve). The value of e/C",120 IJ.V (C=1.3 fF) is consistent with the junction area of about 0.015 IJ.m 2. The dashed curve shows that by applying a gate voltage the Coulomb gap can be completely suppressed. Also shown in Fig. 1 are calculated I-V curves (crosses) for one junction having C=O.7 fF and the other having C=l.l fF (SEM photographs showed that the junction areas differed by a factor 1.5). In our circuits the voltage source, taken to be a stray capacitance of a few fF, is at a distance of at least 10-20 IJ.ffi from the junctions. With an estimated inductance of the onchip leads of I pH/IJ.ffi, the response time of the voltage bias is at least 10- 13 S. This is therefore a lower limit for the effective tunneling time at low voltages. In previous experiments on two junctions in a scanning tunneling microscope configuration (6) the relevant stray capacitance was so close to the junctions that the authors combined a tunneling time of 10- 15 S (the expected traversal time through the barrier of a tunneling electron (3» with global rule. In our much larger devices, if this timescale were really relevant local rules would apply. In Fig. 2 the current versus gate voltage (1-Vg) is shown for several arrays. At fixed Vg the slope iNg/dl can be used to calculate the charge noise l1n on the central island from the observed current noise in as qn=inCgdVg/dI. Preliminary measurements for a double junction (curve a) yielded 1.5·IO-4e between 2 and 200 Hz, which is one of the lowest values reported so far, but still an order of magnitude higher than expected (1). Curve b shows an example of telegraph noise that we observed in several devices. Effectively two curves were measured, with jumps between the two. Obviously at least one of the two curves corresponds to offset charging of the junctions. Generally offset charging of junctions (polarization of the island between junctions with a. non-integer charge) results from trapped charges, e.g. near the junctions. For curve b apparently a charge could jump between two positions. For more than 2 junctions (c-g), we observe fine structure similar to results for arrays of superconducting junctions frustrated by a magnetic field (12). If we define a frustration CgVgte, then the number of current dips in one period is equal to the number of islands between the junctions, the equivalent of the width in unit cells of an array of superconducting junctions. Curves d end e show beating, probably due to differences in electrode self capacitances. The difference of the curves for the various 5 junction arrays is another indication for offset charging of the junctions. Also the observation that the threshold voltage for more than 2 junctions is systematically less than the expected (n-l)e/2C can thus be understood (inset of Fig. I).
/.mz
4. CONCLUSIONS We determined a lower limit of order 10- 13 S for the tunneling time in small planar tunnel junctions. The possibility of offset charges in these systems should be seriously considered. ACKNOWLEDGEMENTS We thank M. Devoret, D. Esteve, U. Geigenmiiller, K Likharev, M. Peters and G. Schon for valuable
-2
-1
2
FIGURE 2 Current versus gate voltage Vg for several arrays with junction resistance between 50 and 100 ill. For each curve the bias voltage is fixed. (a) and (b) 2 junctions. (c) 3 junctions. (d)-(g) 5 junctions. (d) and (e) are I-Vg curves of the same device, at Vg differing by 10 periods. discussions. This work was supported by the Dutch Foundation for Fundamental research on Matter (FOM) and L. Kouwenhoven. REFERENCES (1) See the review by D.V. Averin and KK Likharev, Single electronics, in: Quantum effects in small disordered systems, eds B.L. Al'tshuler, P.A. Lee, and R.A. Webb (Elsevier, Amsterdam) to be published. (2) T.A. Fulton and G.J. Dolan, Phys. Rev. Lett. 59, (1987) 109 have reported experiments on a double planar junction. (3) M. Biittiker, and R. Landauer, ffiM J. Res. Dev. 30 (1986) 451. (4) J.E. Mooij et al., Charge-anticharge unbinding in arrays of small tunnel junction!Y, preprint submitted to Phys. Rev. Lett.. (5) V.F. Anderegg et al., Frequency-determined current in a turnstile device for single electrons, this volume. (6) R. Wilkins et al., Phys. Rev. Lett. 63 (1989) 801. (7) U. Geigenmiiller and G. SchOn, Europhys. Lett. 10 (1989) 765. (8) L.J. Geerligs et ai., Europhys, Lett. 10 (1989) 79. P. Delsing et ai., Phys. Rev. Lett. 63 (1989) ll80. (9) (10) D. Esteve, private communication. (ll) Yu. V. Nazarov, JETP Lett. 49 (1989) 126. (12) D.J. van Harlingen et ai., Physica B152 (1988) 134.