Patent Application: US-201514595559-A

Abstract:
a superconducting quantum interference devices comprises a superconducting inductive loop with at least two josephson junction , whereby a magnetic flux coupled into the inductive loop produces a modulated response up through radio frequencies . series and parallel arrays of squids can increase the dynamic range , output , and linearity , while maintaining bandwidth . several approaches to achieving a linear triangle - wave transfer function are presented , including harmonic superposition of squid cells , differential serial arrays with magnetic frustration , and a novel bi - squid cell comprised of a nonlinear josephson inductance shunting the linear coupling inductance . total harmonic distortion of less than − 120 db can be achieved in optimum cases .

Description:
as described above , a major part of the invention comprises a new squid cell , the bi - squid . the dc squid , modified by adding a josephson junction shunting the loop inductance , provides extremely high linearity with the proper selection of parameters . this is somewhat surprising , since a josephson junction presents a nonlinear inductance . however , the junction nonlinearity is able to compensate the nonlinearity of the device in order to achieve an improved linearity close to 120 db for significant loop inductances ( which are necessary to achieve large coupling to external signals ). it is to be understood by those skilled in the art that any other nonlinear reactance that functions in a similar way would have a similar effect on reducing the nonlinearity of the system transfer function . the linearity dependence of the shunt junction i c3 on critical current at different inductances of the squid loop is shown in fig5 . the linearity is calculated using a single - tone sinusoidal flux input ( of amplitude a / a max = 0 . 2 , where a max corresponds to the flux amplitude φ 0 / 4 ), and measuring the total harmonic distortion in db . this result shows that the linearity is sharply peaked for each value of l = li c / φ 0 , but with different optimized values of i c3 . very large values of linearity as high as ˜ 120 db are achievable . fig6 shows how the linearity parameter varies as a function of the signal amplitude for other parameters fixed . the linearity decreases as the signal approaches the maximum value . a serial array of bi - squids can be implemented to increase the dynamic range up to a value comparable with the response linearity . moreover , a serial sqif providing a single ( non - periodic ) voltage response with a single triangular dip at zero magnetic flux can be implemented . single bi - squids , serial arrays of bi - squids , and a prototype of an active electrically small antenna based on a bi - squid - array were designed , fabricated and tested , using a 4 . 5 ka / cm 2 nb hypres process ( hypres inc ., elmsford n . y .). the layout design of the chips with these elements was made before the completion of the numerical simulations aimed at the optimization of the circuit parameters , in particular before obtaining the results presented in fig5 were obtained . therefore the critical currents of all josephson junctions in bi - squids were chosen equal ( i c1 = i c2 = i c3 = i c ) while the optimal shunt shunting junction critical current should be somewhat less for the implemented inductance parameter l = 1 . 4 . fig7 a shows the schematic equivalent circuit of the bi - squid for both the fabricated single bi - squid and the serial array of bi - squids , for amplifier applications . to apply magnetic flux , a control strip line coupled magnetically with an additional transformer loop was used . the coupling loop with high inductance l ex is connected in parallel to inductance l in and therefore practically does not change the interferometer inductance . fig7 b shows the corresponding equivalent circuit of the bi - squid for an electrically small antenna . the voltage response of the bi - squid to applied flux ( as measured in current units ) is shown in fig8 . the applied bias current was slightly more than 2i c for the bi - squid . the shunt junction critical current is not optimal at the implemented inductance parameter l = 1 . 4 . as a result , the observed voltage response is not perfectly linear , although it shows a clear triangular shape . the measured transfer function closely coincides with simulations , however . as for the small hysteresis at the flux value close to ± φ 0 / 2 , this indicates that effective inductance parameter of a single - junction squid l *≡ l · i c3 ≡ 2πli c3 / φ 0 is more than 1 and hence the static phase diagram becomes hysteretic . the voltage response of the 12 - element bi - squid array is presented in fig9 a , and looks virtually identical to that for a single bi - squid . the applied bias current was slightly more than the critical current of the array . the voltage response linearity of both bi - squid and bi - squid serial array can be further improved by means of differential connection of two identical bi - squids or serial arrays oppositely frustrated by half a flux quantum . this improvement results from cancellation of all even harmonics of the individual responses . fig9 b shows the voltage response of the differential scheme of two serial arrays of 12 bi - squids frustrated by half a flux quantum as well as the source responses of the arrays . the arrays are biased about 10 % above their critical currents . the differential scheme of two parallel sqifs oppositely frustrated by an applied magnetic field δb ( see fig3 ) is able to provide extremely linear voltage response in case of a proper choice of the sqif structure . in the limit of vanishing inductances l of the interferometer cells , one can use an analytical relation for the parallel sqif response [ 3 ]-[ 5 ]: where i b is the bias current , i c is the total critical current , k is the number of josephson junctions , and a m is the area of the m - th interferometer cell . for sufficiently large k , one can use integration instead of summation , and relation ( 5 ) can be transformed as follows : a solution for the specific distribution of the interferometer cell areas a ( x ) along the sqif - structure ( 0 & lt ; x & lt ; l ) to make the differential circuit voltage response relations ( 5 )-( 9 ) allow derivation of master equations and minimizing the resulting functional to obtain an optimal distribution a ( x ). one can use an iterative algorithm to find the problem solution , starting from some initial approximation ( see fig2 ). in the case of finite inductances l of the interferometer cells , the sqif response v ( b ) has to be calculated by means of numerical simulation , using in particular the well known software pscan [ 18 ]. the problem can have more than one solution . various analytical approximations for the problem solution at l = 0 are found ; the best one is as follows : a ( x )/ a σ = 1 . 2 − 0 . 48 sin 3 ( π x ), ( 10 ) fig1 a and 10b show both the cell area distribution ( fig1 a ) ( 10 ) and the differential circuit voltage response ( fig1 b ). linearity of the voltage response within the shaded central area is as high as 101 db . to estimate the linearity a sin - like input signal was applied , and the spectrum of the output signal then studied . a ratio of the basic harmonic to the maximal higher one was used to characterize the response linearity . it was found that a very high linearity can be obtained using a relatively small number n of sqif cells with areas fitted to ( 10 ). fig1 a shows that the linearity increases rapidly with the number n and at n & gt ; 35 reaches a plateau where the linearity is as high as 101 db . as for the impact of technological spread in the cell areas , fig1 b shows that the tolerable spread is about 4 % at n = 36 ; and then the linearity decreases with the spread value . approximately the same result was obtained for the spread in critical currents of josephson junctions . a further increase of n can be used to decrease the impact of the technological spread in the sqif circuit parameters as well as to increase the dynamic range proportional to √{ square root over ( n )} up to the linearity level obtained . both the dynamic range and the output signal amplitude can be additionally increased by connection of the differential sqif structures in series , i . e ., by providing a two - dimensional differential serial - parallel sqif structure ( see fig1 ). the number k of the elements connected in series is responsible for the output signal amplitude , while the total number of josephson junctions n *= n · k is responsible for the dynamic range of the structure . by varying the number of elements connected in parallel ( n ) and in series ( k ), one can change the impedance of the structure over a wide range . at the same time , there are several problems which should be solved to realize the potentially high performance of the amplifier or antenna . first of all , one should note that the optimal specific structure of the parallel sqif reported in [ 5 ] was determined based on the ideal rsj model of josephson junctions and for the case of vanishing coupling inductances ( l = 0 ). deviations of junctions and inductors from ideal theoretical behavior will hinder the linearity of the real structure fabricated . there are two general approaches to the problem solution : ( i ) to provide the closest approach of the experimental josephson - junction characteristics to the ones given by the rsj model and ( ii ) to synthesize an optimal sqif structure founded in experimental josephson - junction characteristics by means of numerical simulation technique ( for example by software pscan [ 18 ]) and an iterative algorithm ( fig2 ). indeed , an optimal strategy may be based on a combination of schemes . in particular , as for the coupling inductance l , the negative influence of the finite value of l on the voltage response linearity can be reduced by shunting resistors r sh connected in parallel to the inductances . due to the fact that the impedance of the rl circuit becomes low enough at the josephson oscillation frequency , the parallel array voltage response approaches that for smaller and smaller inductance with the decrease of r sh down to some optimal resistance value depending on the normalized inductance l ; further increase in r sh leads to some other linearity distortions . therefore , the most effective method is synthesis of an optimal sqif structure with the cell area distribution a ( x ) optimized for the finite value of l . in this case one should use a high performance numerical simulation technique ( e . g ., pscan software [ 18 ]) for calculation of the sqif voltage response v ( φ ) in every cycle of the iterative algorithm ( fig2 ), which has to be used to solve the master equation . the shunting technique efficiency is confirmed by results of numerical simulations presented in fig1 a and 13b . one can see that at r sh ≈ 0 . 1r n ( where r n is josephson junction normal resistance ), the voltage response of the parallel array of 6 junctions with l = 2πi c l / φ 0 = 1 approaches that for vanishing coupling inductances . as a consequence , the required linear voltage response of the differential scheme of two parallel sqifs with n = 20 and coupling inductances l = 1 each shunted by resistor r sh = 0 . 1r n are observed . in the case of a serial sqif including n dc squids , the thermal noise voltage v f across the serial structure is proportional to square root of n , while the voltage response amplitude v max ( φ ) and the transfer factor b =∂ v /∂ φ both are about proportional to n . this means that the dynamic range d = v max ( φ )/ v f increases as n 1 / 2 . as for the parallel sqif , in the case of vanishing coupling inductances ( l = 0 ), the dynamic range is also proportional to square root of number of junctions n . in fact , the thermal noise voltage v f across the parallel structure decreases with the square root of n , while the voltage response amplitude v max ( φ ) remains constant and the transfer factor b =∂ v /∂ φ increases as about n . a sqif - like structure is characterized by a superior broadband frequency response from dc up to approximately 0 . 1 · ω c , where ω c is characteristic josephson frequency [ 13 ]. therefore , a further increase in characteristic voltage v c of josephson junctions by implementation in niobium technology with higher critical current density , or by use of high - t c superconductors , promises an extension of the frequency band up to several tens of gigahertz . moreover , the sqif eliminates high interference , and it sufficiently decreases the well known saturation problem of squid - based systems . therefore , sqif - based systems can easily operate in a normal lab environment . an approach to synthesis of multi - squid serial structures has been reported , capable of providing periodic high linearity voltage response [ 11 , 12 ]. the approach is based on the formation of serial structures which are capable of providing periodic triangular voltage response to a magnetic field b . using interferometer cells with a harmonic voltage response , one can synthesize a serial array consisting of many groups of identical interferometers , each group providing a specific spectral component of the resulting voltage response of the array . according to estimations , the response linearity reaches 120 db , if the number of the groups is as high as about 165 . the second way to synthesize a highly linearity array structure is through implementation of a differential scheme of two serial arrays of dc interferometers biased by current i b = i c ( critical current biasing ), where i c is the interferometer critical current . according to an embodiment , a more advanced system is provided comprising one - and two - dimensional multi - element structures characterized by sqif - like high linearity voltage response . the structures are based on use of a differential scheme of two magnetically frustrated parallel sqifs , with both a specific cell area distribution a ( x ) along array and a critical current biasing ( see fig3 ). optimization of the cell area distribution allows an increase of the voltage response linearity up to the levels required . this optimization can be performed numerically by solution of a master equation with the aid of an iterative algorithm . a multi - element structure synthesized according to the present embodiments can be used , for example , to provide high performance amplifiers . the proposed two - dimensional structure can also used as an active antenna device . the efficiency of the antenna can be significantly increased by combining it with a reflecting parabolic antenna . by varying the number of elements connected in parallel ( n ) and in series ( k ), one can set the impedance to a value needed to optimally match the antenna load used . the high expectation for the multi - element sqif - like structures is based on estimations based on idealized structures , as well as on the voltage response characteristics calculated with use of rsj model . however , the true characteristics of the actually realized array structures may be different . limitations imposed by finite coupling inductances and stray capacitances are discussed below . the finite value of coupling inductances 1 between josephson junctions in a parallel array is of importance for all principal characteristics of the array , because of limitations on the coupling radius . the finite coupling radius limits an increase of both the dynamic range and the transfer factor dv / dφ with increase of number of junctions n . to study the noise characteristics in a clearer and more powerful manner , one can perform numerical simulation of a parallel array of the inductively coupled resistors r n , each connected to an individual source of white - noise current . fig1 shows an active electrically small antenna based on two - dimensional differential serial - parallel sqif - structure ( the filled structure in central part of chip ). sqif sections are connected by strips of normal metal . the chip contains a regular matrix of identical blocks of parallel sqifs , to provide a homogeneous magnetic field distribution in the center part of the chip . the inset shows such a block with a parallel sqif . the shown sqif is topology - oriented for a high - tc superconductor technology . fig1 shows the dependence of the low - frequency spectral density s v ( 0 ) of the resistor voltage noise on the number of resistors n at different values of normalized coupling inductance l . the data are presented for normalized frequency ω / ω c = 10 − 3 corresponding closely to the signal frequency range in a squid / sqif amplifier ( here ω c is characteristic josephson frequency ). within the coupling radius , the spectral density s v ( 0 ) decreases as 1 / n and then it comes to a constant value when number n becomes more than coupling radius depending on l . fig1 shows the spectral density s v ( ω ) versus normalized frequency ranged from 0 . 01 to 1 for parallel array of 30 resistors r n . at both coupling inductances l = 3 and l = 1 , the spectral density s v ( ω ) monotonically increases with frequency and remains constant at l = 0 . 001 . it reflects a decrease in coupling radius with frequency for both inductances l = 3 and l = 1 , as well as the fact that the coupling radius at l = 0 . 01 exceeds the size of the array of 30 elements in the entire frequency range . one can see that implementation of noiseless resistors r sh = 0 . 1r n shunting the inductances l = 1 stops both the coupling radius decrease and the noise spectral density increase at ω / ω c ≧ 0 . 1 ( see dashed line in fig1 ). a proper account of the respective noises of the shunting resistors will lift the curve about two times as high , as if the spectral density for the noise current sources connected to basic resistors r n becomes more by factor k ≈ 4 - 5 . fig1 shows the dependence of the normalized transfer factor b = dv / dφ for a parallel array of josephson junctions versus number of junctions n at different normalized values l of coupling inductances . the dashed curve shows the transfer factor dependence for l = 1 when all the coupling inductances are shunted by resistors r sh = r n . the observed saturation in the transfer factor , depending on n , is reached when number of junctions exceeds the coupling radius at frequency ω / ω c ˜ 1 . in such a way , increases in dynamic range d = v max ( φ )/ v f with the number n of josephson junctions in a parallel array are limited by the coupling radius at finite coupling inductances . shunting of the inductances for improving linearity of the differential sqif voltage response does not really change the dynamic range . in fact , the observed increase in the voltage response amplitude v max ( φ ) ( see fig1 ) is compensated by an increase in v f owing to the noise of the shunts . in the case of an unloaded serial array of dc squids , the dynamic range does actually increase with the number n of interferometer cells . nevertheless , in reality , stray capacitances and load impedance are both able to substantially change the i - v curve of the array , and hence the amplitude v max and form of the array voltage response . the decrease in v max leads to a proportional decrease in dynamic range . the change in the voltage response curve reduces linearity of the whole array structure . fig1 a and 19b shows the typical impact of the stray capacitances on i - v curve of the serial array of dc squids . the contribution of the stray capacitance of each squid increases with the squid position from ground to signal terminal . stray capacitances cause the i - v curve to appear similar to a hysteresis curve , as well as to form one or even more undesired features on the i - v curve . the features shown result from a phase - locking phenomenon . in the solid line labeled a of fig1 b , the features of the i - v curves of the array cells do not coincide because of different “ capacitive loads .” in the dashed line labeled b , the features of all the individual i - v curves coincide because of mutual phase - locking of the josephson - junction oscillations . the fabrication of serial arrays based on standard niobium technology using two superconducting screens is accompanied by undesirably high stray capacitances ( see fig2 a ). to essentially decrease the impact of the capacitance , individual double screening may be used for each squid as shown in fig2 b and fig2 c . both schemes are characterized by the i - v curve b in fig1 b , but the latter one provides lower inductances of the strips which connect the squid cells . advantages of one - and two - dimensional sqif - like structures for microwave applications as high - performance amplifying devices are readily apparent from their ability to provide an increase in dynamic range with a number of elements as well as high linearity when employing a properly specified array structure . linearity can be especially enhanced using cells comprising the bi - squid structure . at the same time , there are some fundamental limitations imposed by finite coupling inductances , stray capacitances and parasitic couplings . therefore , implementation of high - performance devices preferably employs careful and detailed analysis of the multi - element array structure , taking into consideration all the true parameters including all parasitic parameters and couplings . a differential scheme comprising two magnetically frustrated parallel sqifs is developed to obtain a highly linear single - peak voltage response . the response linearity can be increased up to 120 db by means of a set of properly specified cell area distribution of the sqifs . the high linearity is attainable with a relatively small number of junctions . such a circuit provides a high - performance two - dimensional serial - parallel sqif - like array . varying the number of elements connected in parallel , and in series , permits setting the impedance value needed to solve the problem related to negative impact of the load used . the synthesized structures can be used to design high - efficiency amplifiers and electrically small active antennae for use in the gigahertz frequency range . the efficiency of the antenna can be significantly increased by combination with a reflecting parabolic antenna . it should be appreciated that changes could be made to the embodiments described above without departing from the inventive concepts thereof . it should be understood , therefore , that this invention is not limited to the particular embodiments disclosed , but it is intended to cover modifications within the spirit and scope of the present invention as defined by the appended claims . v . k . kornev , i . i . soloviev , n . v . klenov , and o . a . mukhanov , “ synthesis of high linearity array structures ,” superconducting science and technology ( sust ), vol . 20 , 2007 , p . s362 - 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