Linewidth of Bloch oscillations in small Josephson junctions

Physica B 203 (1994) 376-380

ELSEVIER

Linewidth of Bloch oscillations in small Josephson junctions Leonid Kuzmin a'b'¢'*, Yuri Pashkin a'd, Alexander Zorin b'c, Tord Claeson a aDepartment of Physics, Chalmers University of Technology and GOteborg University, S-41296 GSteborg, Sweden bphysikaliseh-Technische Bundesanstalt, D-38116 Braunschweig, Germany CLaboratory of Cryoelectronics, Moscow State University, Moscow 119899, Russian Federation dLebedev Physical Institute, Russian Academy of Sciences, Moscow I 17924, Russian Federation

Abstract The line width of Bloch oscillations has been studied in small capacitance AI-(A1)PbAu Josephson tunnel junctions which were isolated from the low-impedance electromagnetic environment by miniature high-ohmic metallic resistors placed very close to the junctions. Under irradiation by microwaves of frequency f i n the range of 300-4000 MHz the I - V curve showed appearance of steps at I = ___2ef in correspondence with the Bloch relation. The line width of the oscillations was assumed to be the same as the width of peaks in differential resistance d V / d I under irradiation and was analyzed using a weak-signal response technique. The line width as a function of temperature flattened out at low temperatures and its level depended on frequency and hence on the biasing current. An analysis shows that the line width is basically determined by thermal noise in the resistors and the observed low-temperature plateau can be explained in terms of a hot-electron effect in the resistors.

1. Introduction At low temperature, the dynamics of superconducting tunnel junctions with small capacitance, C ~. e2/kBT, and finite Josephson coupling energy Ej ~ (h/2e) Ic, is governed by Coulomb interaction. It can result in correlated tunneling of individual Cooper pairs (see Refs. [ I, 21 for reviews). The necessary condition for achieving such a regime in a single junction is its biasing through small-size high-ohmic resistors to protect the junction from shunting by low environmental impedance. Then the coherent tunneling of Cooper pairs in such a current-biased Josephson junction, the so-called "Bloch oscillations", occur with a frequency proportional to the bias current, fBIooh = I/2e, * Corresponding author.

(1)

in contrast to the ordinary Josephson oscillations the frequency of which is proportional to the DC voltage across the junction, fj = 2eV/h. The Bioch oscillations (although strongly depressed by noise) have been observed in single Josephson junctions attached to smallsize resistive leads and subjected to microwave radiation of arbitrary power [3, 4]. The main difficulty in such experiments was to fabricate these thin-film strips with a sheet resistance of up to 1-2 kf~/square and to place them near the junction. It is desirable to make their resistance Rs ~> Ro =- h/4e 2 ~ 6.47 ktq and keep their length as short as 10-301xm. Noise and parasitic capacitance then can be low enough not to deteriorate BIoch oscillations. Recently, remarkable technological progress has given the possibility to create resistors with almost aribitrary parameters and it has given hope to improve the observability of the effect (in particular, to squeeze the line width of Bioch oscillations). However, the latest experiments have shown the line width to be

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L. Kuzmin et al./Physica B 203 (1994) 376-380

broader than expected. Thus the analysis of possible reason and ways of their elimination in further experiments are focused upon in the present paper.

377

In the experiment, it is more convenient to measure the derivative d V / d I which is contributed by the response term, d(A V)/d(Al) oc (12 - AI2)/(AI 2 + / 2 ) 2 ,

(4)

2. Method The method we used to detect Bloch-type oscillations is based on the property of an oscillator to react selectively to an external harmonic signal. For example, in the case of a small signal of amplitude I~ (of frequency ~o = 2xJ) the quadratic video response (change of DC voltage AV) of an overdamped Josephson junction is expressed by the formula (see, e.g., Ref. 1-5, ch. 10]) A V -- kvI~, 2

kv oc AI(AI 2 + lr2),

(2)

where AI is the DC current deviation from the value I at which the junction generates frequency fj = f The most remarkable feature is that the half-width of the resonance lr is independent on the amplitude of the signal Io and is proportional to the line width of the oscillations. In other terms, the response (2) describes an undeveloped (i.e., a strongly depressed by noise) Shapiro step in the I - V characteristic caused by external irradiation. Thus the step flattens out at large signal levels, its size becomes proportional to the amplitude I,~ and the relation (2) breaks down. In contrast to a Josephson oscillator, the dynamics of a Bloch junction is more complex even if its quasiparticle conductance is negligibly small compared to the conductance of the leads, R~-1, i.e., when tunneling of single electrons can be neglected [6, 7, 2]. In particular, besides the quasicharge q variable (conjugate to the Josephson phase ~b)the Bloch state should be supplemented with the band number. However, the system remains in the lowest band if the Josephson coupling energy Ej is not much less than the Coulomb energy Ec - e2/2C (this prevents Zener tunneling from the lowest to the upper bands at decent value of the biasing current) and if temperature is low enough not to cause unwanted thermal excitation of upper bands. In that case, it can be described by the Langevin-type equation for quasicharge q, which is similar to that for the Josephson phase within the frame of the resistively shunted junction model (see Ref. [5, ch. 4]). The only difference is that the 2rt-periodic Josephson sin ~b-term is replaced by a 2e-periodic term (in q-domain), the Bloch term, whose waveform gradually changes from sinus (at E: >> Ec) to saw shape (at Ej '~ Ec). A simple analysis shows that the response of the junction to a small signal is expressed by Eq. (2) with Ir = 2e ( r / 2 n ) ,

(3)

where F is the half-line width of the autonomous Bloch oscillations of frequency given by Eq. (1).

which is positively peaked around the resonance fmoch = f Mathematically, lr is expressed via the width Alo.s of the resulting peak in Eq. (4) taken at the midlevel of the peak-to-peak span as 2Ir m 1.84Alo.s and therefore it can be measured in the experiment. On the other hand, for the case under consideration (Rs ~> R o and for I well above the "Bloch nose" value of Vb/Rs, but still below the region of Zener tunneling) the theory [6] yields the current-independent value F = (x/e) 2 kB T/Rs,

(5)

which is valid in the thermal limit, hF ~ kB T.

3. Experiments and results The single Josephson junction and the thin film resistors were made in the same vacuum cycle using a three layer shadow evaporation technique described elsewhere [3, 4]. We used the configuration shown in Fig. 1 to diminish a stray capacitance between the electrodes and hence, improve the junction properties at high frequencies. The junctions were made of A1-A1Ox-AI (100.A)PbAu with an area of 0.01 pm 2. This resulted in a junction capacitance C ~ 0.5 fF as estimated from the geometrical sizes. The use of lead alloy instead of pure aluminium in the top electrode enabled us to increase slightly its superconducting gap and hence the coupling energy. The Cr resistors in the leads (Fig. 1) had length L of 10 ~tm (sample N1) and 30 ~tm (sample N2), width 0.1 Ixm and thickness 60 ?~. That resulted in the resistance Rs = 130 kf2 (sample N1) and 450 kf~ (sample N2). An I - V curve of the junction N1 with normal resistance R = 7 kf~ and gap voltage V2~ = 450 ~tV is shown in

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L. Kuzmin et al./Physica B 203 (1994) 376-380

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Fig. 2. (a) An l - V curve of the circuit with A1/AIOx/A1 (100 #,)PbAu junction N1 showing the Coulomb blockade of Cooper pair tunneling within + Vb = 40 laV, T = 60 mK. (b) A blow-up of the low-bias region of the I V curve with pronounced "Bloch nose". Fig. 2. Due to a technical reason, I - V curves were measured in a two-point configuration and the voltage across the resistors was substracted later using computer processing. The I - V curves show a well-pronounced C o u l o m b blockade of C o o p e r pair tunneling Vb = 40 ~tV and q u a s i - J o s e p h s o n current at the level of 1.5 nA. The Josephson coupling energy estimated from R and V2~ e q u a l s E j = 100 txeV, the charging energy estimated from the geometrical capacitance Ec = 160 IxeV. The C o u l o m b blockade voltage Vb = 40 txV (see Fig. 2(b)) is lower than that estimated from theory Vb'h ~ 70 ~tV for given Ej/Ec ~ 0.6 at T = 0. This disagreement is typical for our experiment and may be explained by rounding of Bloch nose by thermal fluctuations and by additional influence of the shunt resistance, The Bloch oscillations were analyzed using a "standard" method of irradiating the junctions with an R F signal. The R F signal was applied through one of the contact pads. Typical I - V curves and differential

Fig. 3. (a) A set of the I - V curves of the junction N1 for several values of the RF power (increasing value from left to right: - oo, - 4, - 2, 0, 2, 4, 6, 7, 8, 9 and 10dB) at f = 4 GHz. (b) Differential resistance dV/dl (horizontal axis) as a function of current I (vertical axis) for the same values of the applied microwave power. The arrows show the expected peak positions in accordance with the Bloch relation I = + 2ef. Both I - V and dV/dl curves are shifted in horizontal axis. T = 60 mK. resistances, d V / d I , for different amplitudes of external irradiation of frequency 4 G H z are shown in Fig. 3. One can see the appearance of steps in the I V curves and maxima in the differential resistance at I = + 2 e f w h e r e the frequencies of the Bloch oscillations and the external signal are equal. The D C positions of the peaks are independent of R F amplitude as shown in Fig. 3(a) and (b). The temperature dependence of d V / d l on I for the amplitude of irradiation 8 dB is shown in Fig. 4. In Fig. 5, we have plotted the full Bloch line width (2 Ir), determined as described above at half peak values and expressed in current units.

4. D i s c u s s i o n

The I - V and d V / d I curves show characteristic steps and peaks which are positioned at I = + 2efand therefore

L. Kuzmin et al./Physica B 203 (1994) 376-380

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are definitely explained by interaction of internal Bloch oscillations with an external RF signal. The peaks' width increases with increasing temperature and there is a qualitative agreement with the theoretical dependence given by Eq. (5) for high temperatures. At lower T the linearity breaks down and the line width flattens out. This behavior is typical at both high and low frequencies and the range and the height of the low-temperature plateaus increase with frequency and hence with DC current. The frequency-dependent position of a plateau makes it unlikely to be caused by an external electromagnetic noise. However, in order to exclude this reason we have measured the temperature dependence of the differential resistance of the junction at zero bias of one of the samples. No saturation at low temperature was observed (see Fig. 5(b)). This indicates that the line width plateaus are not due to impinging noise. Quantum (or zero-point) fluctuations could be a reason of noise giving a larger line width than the equilibrium thermal fluctuations at low T. This seems to be most likely for the case of higher frequencies, say 4 GHz, when formally hf/2kB is about 100 mK. However, only noise components in the range [0, F] are essential in influencing F itself (see Refs. [5, 6]). Therefore even in the worst case of the largest F the value ofhF/2kB is less then 60 inK, that is still well below the transition temperature (100-250 mK). On the other hand, the low-frequency component of the noise power could be somewhat enhanced due to the down-conversion of quantum noise from the frequency of oscillation and its higher harmonics [8]. In our case for the biasing current I well above Vb/R, the conversion coefficients are small [6] and the quantum contribution can be omitted. Thus, we conclude that the most probable reason of the observed dependence of F versus T is overheating of the electron system in the thin-film resistors at low tern-

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Fig. 5. A comparison of the experimental temperature dependences of Bloch line width in current units 21 contact pads. Typical I-V curves and differential with those due to external shunt resistance including the hot electron effect in resistors (Eq (6)): (a) sample NI with Ej = 100tteV and Ec = 160ttV; (b) sample N2 with Ej = 20 ~eV and Ec = 120 laV. Zero-bias resistance R• was measured to exclude the external noise as one of the reasons of line width flattening at low temperatures. perature. It results in an enhanced power of fluctuations which can be characterized by an effective electron temperature T~ [9]. Because of a weak electron-phonon coupling in metals at low T, the electron gas is overheated by the electric field and its temperature can essentially exceed the phonon temperature Tp (which in case of small Kapitza boundary resistance is equal to the temperature of the substrate T) [10]. It is expressed quantitatively as [12]

Te = ( P / Z V + T~) x/',

(6)

where P = 12Rs is the dissipated power, V the volume of the resistor and X a material constant. According to Eq. (6)

380

L. Kuzmin et al./Physica B 203 (1994) 376-380

the minimum electron temperature of the microstrip resistor is determined by the dissipated power and the material properties as T ~i, = ( P / S V) 1Is. The parameter • depends on the electronic heat capacity and on the electron-phonon scattering rate which are both determined in a complex way by the structure of the Fermi surface. Unfortunately, the accumulated experimental data on r are limited to a few metals and alloys. In particular for pure copper S = S c u 2 x 1 0 9 W m - a K - s [11]. Due to a 1/5 power law the dependence on material is quite weak. Therefore we have taken this value to estimate the effect in chromium resistors. The volume of a resistor was well defined and in combination with the chosen values of,~ it gave the value of T c~" in the range 150-300mK for various samples and currents in the range 0.1-1.3 nA. This temperature is considerably higher than the lowest operating temperature. We substitute the values of Te into Eq. (5) choosing a proper value of Rs for our circuits and compare the theoretically estimated temperature dependence of the line width (Fig. 5) with the experimental data for both samples. Theoretical curves describe qualitatively the experimental dependencies for different frequencies and in different regions of currents for samples 1 and 2. A more accurate comparison demands an analysis taking into account the arbitrary environmental impedance Z (to) [7] instead of the Ohmic case [6]. We believe, however, that using this method we determine the main reason for broadening of the Bloch line width due to the hot electron effect in the resistors.

5. Conclusion We have carried out a set of measurements of the line width of Bloch oscillations in small Josephson junctions biased through small-size high-ohmic resistors. We have shown that the line width, which we associate with the width of the DC response of the junction to small-signal RF radiation, does not decrease with decreasing temperature below 100-200inK. This behavior can be explained by a hot electron effect in the resistors. At first glance it seems hard to cool electrons in such a small volume, because a larger geometrical size of resistors

inevitably increases the stray capacitance which shunts the junction itself. Thus we see the solution of this problem in two ways: (i) a series of thick "cooling fins" connected to the resistor with space period, say 0.5 pro, or (ii) small resistors connected to very massive pads (say 100 x 100 x 0.3 pm3). Our estimations show (see also Ref. [11]) that such resistors will provide a lowering of Te down to the 50 mK level and do not substantially increase the stray capacitance. We believe that this should reduce fluctuations and make the effect of Bloch oscillations much stronger.

Acknowledgements The work was supported by the Nobel Foundation, by the Swedish NFR and TFR, and by the Swedish Russian Academy of Sciences cooperation program. The samples were fabricated at the Swedish Nanometer Laboratory.

References [1] H. Grabert and M.H. Devoret (eds.), Single Charge Tunneling, NATO ASI Series B, Vol. 294 (Plenum, New York, 1991). [2] D.V. Averin and K.K. Likharev, in: Mesoscopic Phenomena in Solids, eds. B.L. Altshuler, P.A. Lee and R.A. Webb (North-Holland, Amsterdam, 1991) p. 167. I-3] L.S. Kuzmin and D.B. Haviland, Phys. Rev. Lett. 67 (1991) 2890. [4] L.S. Kuzmin, IEEE Trans. Appl. Superconductivity 3 (1993) 1983. 1-5] K.K Likharev, Dynamics ofJosephson Junctions and Circuits (Gordon and Breach, New York, 1986). 1-6] K.K Likharev and A.B. Zorin, J. Low Temp. Phys. 59 (1985) 347. I-7] D.S. Golubev and A.D. Zaikin, Phys. Rev. B 56 (1992) 10903. [8] R.H. Koch, D.J. Van Harlingen and J. Clarke, Phys. Rev. Lett. 45 (1980) 2132. 1-9] M.R. Arai, Appl. Phys. Lett. 42 (1983) 906. [10] M.L. Roukes, M.R. Freeman, R.S. Germain, R.C. Richardson and M.B. Ketchen, Phys. Rev. Lett. 55 (1985) 422. 1-1l] F.C. Wellstood, C. Urbina and J. Clarke, Phys. Rev. B 49 0994) 5942.