^{1,a)}, Akinobu Irie

^{1}and Gin-ichiro Oya

^{1}

### Abstract

Shapiro steps in (BSCCO) high-superconductor intrinsic Josephson junctions (IJJs) embedded in the external magnetic field applied parallel to the junction surface have been numerically studied by using the unified theory including both the electric and magnetic field couplings between neighboring Josephson junctions. It has been found that (I) the effect of the electric field coupling between neighboring Josephson junctions is not so crucial in the Shapiro steps of BSCCO IJJs, (II) the overall profiles of the characteristics including Shapiro steps calculated for the same external magnetic induction are fairly similar with each other even if the junction lengths differ from each other, (III) the height of the Shapiro step decreases with increasing the external magnetic field but its magnetic field dependence does not so strongly depend on the number of stacked junctions, (IV) a Shapiro step device, which is *hard* for the external magnetic disturbance, can be made when the value of the product is selected to be small for BSCCO IJJs, and (V) the Shapiro step and the flux flow could be treated separately under the presence of the external magnetic field.

I. INTRODUCTION

II. THEORETICAL

III. RESULTS AND DISCUSSION

A. Effect of charge coupling

B. Effect of

C. Effect of

D. Effect of

E. Effect of product

F. Effect of

IV. SUMMARY

### Key Topics

- Josephson junctions
- 127.0
- Magnetic fields
- 30.0
- Magnetic induction
- 18.0
- Critical currents
- 10.0
- Tunneling
- 10.0

## Figures

The dc characteristics of BSCCO IJJs calculated for the case that the number of stacked junctions, , the critical current density , the plasma frequency , the external ac modulation frequency , the junction length along the axis, the junction cross section , and the shunt resistance per a junction, , are 5, , 122 GHz, 200 GHz, , , and , i.e., , respectively, and the normalized amplitude of the external alternating modulation current is set to 1, tentatively. In the present paper, we write the external magnetic induction parallel to the axis as , where is calculated as using a flux quantum , barrier thickness , and junction length . The is 0.345 T for and . The characteristics calculated for no external magnetic field, i.e., , are shown in (a)–(c) and those for , i.e., , are in (d)–(f), respectively. (a) and (d) are the case of the Debye screening length , i.e., no charge coupling case, (b) and (e) are that of , and (c) and (f) are for , where the typical value of is for BSCCO IJJs (Ref. 29). Note that the characteristics calculated for the 2@5 and 3@5 junctions have been shifted by 1 and 2 along the vertical axis, respectively, and the reduced voltage defined by Eq. (15) has been shifted by 3 in order to see the calculated results much easier. The open symbol shows the result calculated with the increasing order of the normalized external dc and the solid one is that due to the decreasing one. The horizontal axis indicates the dc voltage normalized to which is for , and the vertical one shows the external dc normalized to the critical curent at zero magnetic field, .

The dc characteristics of BSCCO IJJs calculated for the case that the number of stacked junctions, , the critical current density , the plasma frequency , the external ac modulation frequency , the junction length along the axis, the junction cross section , and the shunt resistance per a junction, , are 5, , 122 GHz, 200 GHz, , , and , i.e., , respectively, and the normalized amplitude of the external alternating modulation current is set to 1, tentatively. In the present paper, we write the external magnetic induction parallel to the axis as , where is calculated as using a flux quantum , barrier thickness , and junction length . The is 0.345 T for and . The characteristics calculated for no external magnetic field, i.e., , are shown in (a)–(c) and those for , i.e., , are in (d)–(f), respectively. (a) and (d) are the case of the Debye screening length , i.e., no charge coupling case, (b) and (e) are that of , and (c) and (f) are for , where the typical value of is for BSCCO IJJs (Ref. 29). Note that the characteristics calculated for the 2@5 and 3@5 junctions have been shifted by 1 and 2 along the vertical axis, respectively, and the reduced voltage defined by Eq. (15) has been shifted by 3 in order to see the calculated results much easier. The open symbol shows the result calculated with the increasing order of the normalized external dc and the solid one is that due to the decreasing one. The horizontal axis indicates the dc voltage normalized to which is for , and the vertical one shows the external dc normalized to the critical curent at zero magnetic field, .

The dc characteristics of BSCCO IJJs calculated at the same condition as in Figs. 1(a) and 1(d), that is, the effect of charge coupling is not taken into account. In this figure, three values, 2.5, 5, and , are selected so that the corresponding are 0.689, 0.345, and 0.172 T, respectively. The and values denoted as of (a), (b), (c), (d), (e), (f), (g), (h), and (i) are (2.5,0.5), (2.5,1), (2.5,2), (5.05), (5,1), (5,2), (10,0.5), (10,1), and (10,2), respectively, so that for the external magnetic induction in units of T, the relation “” is found.

The dc characteristics of BSCCO IJJs calculated at the same condition as in Figs. 1(a) and 1(d), that is, the effect of charge coupling is not taken into account. In this figure, three values, 2.5, 5, and , are selected so that the corresponding are 0.689, 0.345, and 0.172 T, respectively. The and values denoted as of (a), (b), (c), (d), (e), (f), (g), (h), and (i) are (2.5,0.5), (2.5,1), (2.5,2), (5.05), (5,1), (5,2), (10,0.5), (10,1), and (10,2), respectively, so that for the external magnetic induction in units of T, the relation “” is found.

The dc characteristics with no external magnetic field, i.e., , calculated for , , , and . The effect of charge coupling is not taken into account. (i), (ii), and (iii) are those for , 1, and 2 and are denoted as , (1,0), and (2,0), respectively. Note that characteristics (ii) and (iii) have been shifted by 1 and 2 along the vertical axis, and that (ii) is equal to characteristics (b) in Fig. 10 in our previous paper (Ref. 18). It is also noted that no external magnetic field is applied in the present case, so that all the SIS junctions show exactly the same characteristics because the no charge coupling case is considered. Namely, the relation “” is satisfied for the dc characteristics of the respective junctions. The horizontal axis indicates the dc voltage normalized to , and the vertical one shows the normalized dc .

The dc characteristics with no external magnetic field, i.e., , calculated for , , , and . The effect of charge coupling is not taken into account. (i), (ii), and (iii) are those for , 1, and 2 and are denoted as , (1,0), and (2,0), respectively. Note that characteristics (ii) and (iii) have been shifted by 1 and 2 along the vertical axis, and that (ii) is equal to characteristics (b) in Fig. 10 in our previous paper (Ref. 18). It is also noted that no external magnetic field is applied in the present case, so that all the SIS junctions show exactly the same characteristics because the no charge coupling case is considered. Namely, the relation “” is satisfied for the dc characteristics of the respective junctions. The horizontal axis indicates the dc voltage normalized to , and the vertical one shows the normalized dc .

The same as in Figs. 2(d)–2(f) except for . The and values denoted as of (a), (b), (c), (d), (e), and (f) are (0,0.5), (0,1), (0,2), (2,0.5), (2,1), and (2,2), respectively. (a), (b), and (c) should be compared with Figs. 2(d)–2(f), respectively.

The same as in Figs. 2(d)–2(f) except for . The and values denoted as of (a), (b), (c), (d), (e), and (f) are (0,0.5), (0,1), (0,2), (2,0.5), (2,1), and (2,2), respectively. (a), (b), and (c) should be compared with Figs. 2(d)–2(f), respectively.

The same as in Fig. 2(e) except for . The dc characteristics for the 1@9, 2@9, and 3@9 junctions are shown in (a), those for the 4@9 and 5@9 ones are in (b), those for the 1@10, 2@10, and 3@10 ones are in (c), and those for the 4@10 and 5@10 ones are in (d). The reduced voltages defined by Eq. (15) are also shown in (b) and (d) for and 10.

The same as in Fig. 2(e) except for . The dc characteristics for the 1@9, 2@9, and 3@9 junctions are shown in (a), those for the 4@9 and 5@9 ones are in (b), those for the 1@10, 2@10, and 3@10 ones are in (c), and those for the 4@10 and 5@10 ones are in (d). The reduced voltages defined by Eq. (15) are also shown in (b) and (d) for and 10.

The dc characteristics of surface junctions with , which are shown in (a) and (b), and the reduced voltages defined by Eq. (15) shown in (c) and (d), where the calculation condition is that , , , , , , and , and that the effect of charge coupling is not taken into account. The results for the odd number of junctions are shown in (a) and (c) and those for the even ones are in (b) and (d).

The dc characteristics of surface junctions with , which are shown in (a) and (b), and the reduced voltages defined by Eq. (15) shown in (c) and (d), where the calculation condition is that , , , , , , and , and that the effect of charge coupling is not taken into account. The results for the odd number of junctions are shown in (a) and (c) and those for the even ones are in (b) and (d).

The same as in Fig. 2(e) but for .

The same as in Fig. 2(e) but for .

The dc characteristics of a single junction calculated for five cases such that has been set to (a) 1, (b) 2, (c) 3, (d) 4, and (e) . Except for the values of , the calculation condition is the same as that in Fig. 2(e), i.e., , , , , , and [thus ], and the effect of charge coupling is not taken into account. In the respective figure, the values of , which define the external magnetic induction as , were set to 0, 0.2, 0.5, 1, and 2.

The dc characteristics of a single junction calculated for five cases such that has been set to (a) 1, (b) 2, (c) 3, (d) 4, and (e) . Except for the values of , the calculation condition is the same as that in Fig. 2(e), i.e., , , , , , and [thus ], and the effect of charge coupling is not taken into account. In the respective figure, the values of , which define the external magnetic induction as , were set to 0, 0.2, 0.5, 1, and 2.

The dc characteristics calculated for , , , , , , , and . (i), (ii), and (iii) are those for , 5, and , respectively. The effect of charge coupling is not taken into account.

The dc characteristics calculated for , , , , , , , and . (i), (ii), and (iii) are those for , 5, and , respectively. The effect of charge coupling is not taken into account.

The dc characteristics calculated for , , , , , , , and , i.e., . As the values of the shunt resistance per a junction, 1, 2, 3, 4, and have been selected. (a), (b), (c), (d), and (e) are for , (5,2), (5,3), (5,4), and (5,5), respectively. The effect of charge coupling is not taken into account.

The dc characteristics calculated for , , , , , , , and , i.e., . As the values of the shunt resistance per a junction, 1, 2, 3, 4, and have been selected. (a), (b), (c), (d), and (e) are for , (5,2), (5,3), (5,4), and (5,5), respectively. The effect of charge coupling is not taken into account.

The same as in Fig. 10 but for . Namely, (a), (b), (c), (d), and (e) are for , (6,2), (6,3), (6,4), and (6,5), respectively.

The same as in Fig. 10 but for . Namely, (a), (b), (c), (d), and (e) are for , (6,2), (6,3), (6,4), and (6,5), respectively.

The dc characteristics calculated for , , , , , , , and , i.e., . (a) is for the 1@1, 1@2, 1@3, 1@4, and 1@5 surface junctions, and (b) is for the 2@3 central, 2@4 intermediate, 2@5 intermediate, and 3@5 central junctions. The effect of charge coupling is not taken into account.

The dc characteristics calculated for , , , , , , , and , i.e., . (a) is for the 1@1, 1@2, 1@3, 1@4, and 1@5 surface junctions, and (b) is for the 2@3 central, 2@4 intermediate, 2@5 intermediate, and 3@5 central junctions. The effect of charge coupling is not taken into account.

Normalized height of the first-order Shapiro step and the normalized critical current as a function of and . In this figure, the values of have been selected from 0 to 2 with 0.1 step, and two values, 1 and 5, have been adopted as the number of junctions. The calculation condition is that , , , , , and . As already stated, the external magnetic induction is defined by and the is 0.345 T because of . The and are indicated by open and solid circles, respectively, and the and are drawn by open and solid squares, respectively. In the present paper, the effect of thermal noise has been always taken into account in all the calculations, so it must be stated that both the values of and are somewhat smaller than 1. The effect of charge coupling is not taken into account.

Normalized height of the first-order Shapiro step and the normalized critical current as a function of and . In this figure, the values of have been selected from 0 to 2 with 0.1 step, and two values, 1 and 5, have been adopted as the number of junctions. The calculation condition is that , , , , , and . As already stated, the external magnetic induction is defined by and the is 0.345 T because of . The and are indicated by open and solid circles, respectively, and the and are drawn by open and solid squares, respectively. In the present paper, the effect of thermal noise has been always taken into account in all the calculations, so it must be stated that both the values of and are somewhat smaller than 1. The effect of charge coupling is not taken into account.

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