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Electronic noise of superconducting tunnel junction detectors
Nuclear Instruments and Methods in Physics Research A 338 (1994) 458-466 North-Holland
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
Electronic noise of superconducting tunnel junction detectors J. Jochum, H. Kraus, M . Gutsche, B. Kemmather, F. v. Feilitzsch and R.L. Mössbauer Physik Department E15, Technische Universität München, James-Franck-Str., 85748 Garching bei München, Germany
(Received 6 August 1993) The optimal signal to noise ratio for detectors based on superconducting tunnel junctions is calculated and compared for the cases of a detector consisting of one single tunnel junction, as well as of series and of parallel connections of such tunnel junctions. The influence of I/ f noise and its dependence on the dynamical resistance of tunnel junctions is discussed quantitatively . A single tunnel junction yields the minimum equivalent noise charge . Such a tunnel junction exhibits the best signal to noise ratio if the signal charge is independent of detector size . In case, signal charge increases with detector size, a parallel or a series connection of tunnel junctions would provide the optimum signal to noise ratio . The equivalent noise charge and the respective signal to noise ratio are deduced as functions of tunnel junction parameters such as tunneling time, quasiparticle lifetime, etc. 1. Introduction In many fields of physics, for example in neutrino physics, in X-ray astronomy, or in the search for dark matter candidates, progress depends on improved detection techniques, in particular lower energy threshold and higher energy resolution . Substantial improvement is anticipated by utilizing superconductors exhibiting much smaller energy gaps than semiconductors . The development of such new detectors based on superconducting tunnel junctions is indeed pursued by many groups [1-4]. At present the best tunnel junction detectors are already superior in energy resolution compared to state of the art semiconductor detectors, but are still far from their statistical limit [4]. The energy resolution is at present still dominated by the ratio between signal height and electronic noise. Processes affecting the signal height are discussed in a separate paper [5]. In this paper we shall investigate the dependence of electronic noise on detector parameters such as detector capacitance, detector resistance, lifetime of excess quasiparticles, and quasiparticle tunneling time . Most applications require larger sensitive detector areas than those attained with prototype tunnel junctions during earlier experiments . In efforts to combine larger signal to noise ratios with large detector areas, we checked various possibilities : series and parallel arrays of tunnel junctions and single tunnel junctions involving separate superconducting absorbers engaging quasiparticle trapping [4,6-8]. We call these detectors : serial or parallel junction detectors and single junction detectors. In this context we shall treat in section 2
signal to noise ratio for a superconducting tunnel junction detector combined with its preamplifier . In section 3 we compare the signal to noise ratio for single junction detectors, and for serial or parallel junction detectors. Section 4 deals with the influence of tunnel junction parameters on the signal to noise ratio.
2. Signal to noise ratio .BPS Absorption of energy yields free charge carriers in a detector based on superconductors as well as in a detector based on semiconductors . Depending on the physics of the detector, a total charge Qt , which is a fraction of the initial charge Qo causes the detector signal. In superconducting tunnel junctions the signal is generated by tunneling of the free charge carriers, the so called "quasiparticles" . A voltage signal proportional to the tunneled charge Qt , and thus to the incident energy, results at the output of a preamplifier . The Fourier transform of this signal is described by Q,S(w). In addition there is noise at the preamplifier output described by the spectral density J'(&)). The accuracy of energy measurement is determined by the ratio between signal height and noise. To optimize the signal to noise ratio the preamplifier output signal is filtered . Depending on the filter, a certain signal shape results, whence filtering is commonly referred to as "shaping". There is always some noise passing the filter and resulting in an uncertainty in the determination of the signal height and thus in the determination of the absorbed energy . The accuracy of the energy
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J. Jochum et al. I Nucl. Instr. and Meth . in Phys . Res. 338 (1994) 458-466
determination is limited by the signal to noise ratio `'fsn : signal height `Rsn = root mean square of noise The amount of charge leading to a signal to noise ratio of Msn = 1 is a measure for noise and is called the "equivalent noise charge" Qn. The highest signal to noise ratio synonymously with the smallest equivalent noise charge can be achieved by application of the "optimum shaper" and is given by [9]: S2 (ù» z - Q1 dù), sn -
h
2Tr _~ .~(co)
Qn -
Qt Msn
(lb)
'
In the superconducting films of the detectors excess quasiparticles are created by Cooper pair breaking after energy absorption . In a tunnel junction these excess quasiparticles produce a tunnel current It(t) = Qo e_t / TP . It(t) Tt Qo is the charge, due to the initial number of excess quasiparticles, T t is the tunneling time, and TP is the pulse length given by the effective decay time constant of quasiparticles in the films of a tunnel junction [5]. The Fourier transform of the tunnel current signal It(t) is given by Qt It(w) 1 + i(,)TP'
Additional parasitic capacitances may have to be included in Cg. In a charge sensitive preamplifier, Zf(w) consists of a capacitance Cf in parallel with a resistance R f , yielding the preamplifier integration time Tf = Cf R f. In case of a single junction detector, its impedance Zd(w) consists of the tunnel junction capacitance Cj in parallel with the dynamical resistance Rj at the bias point. The signal current It arises from the tunnel junction, in which energy is absorbed. Consequently, for a single junction detector, or for a parallel junction detector (Fig. la), the signal current source is located across the entire detector impedance, whereas in the case of a serial junction detector (Fig. lb) it appears across the impedance of the energy absorbing tunnel junction only. The signal Qt S(a)) in these cases is Q,S(w)
P(w)It(-), single or parallel junction detector, 1 MP(w)It(w)~ serial junction detector with M tunnel junctions.
As an example we give the transfer function of a pure capacitive detector and preamplifier: P(w) = Go/[iw(Cd + GoCf + Cg)]. The capacitance Cd of a tunnel junction detector and its output signal voltage Vmax divided by the open loop gain is then given by Cj MCj,
with the signal charge Qt
TV
_
Tt
Ci/M,
Q0+
where the factor Tp /Tt arises from the competition between quasiparticle decay and quasiparticle tunneling. To obtain the signal S(a)) at the preamplifier output, we need the product I,(w)-P(w), where the transfer function P(w) of the preamplifier loaded with the impedance Zi(w) is given by _ GOZi(w)Zf(w) . (Sa) GoZi(w) + Zf(w) Z,(w) and G o are the feedback impedance of the preamplifier and its open loop gain, respectively. The impedance Zi(w) is the parallel combination of the detector impedance Zd(w) and the impedance of the gate capacitance Cg of the input FET P(w)
Zi(w) =
1Zd(w) +
459
(5b)
Vmax
Go
Qt _ Qt Ci +C' Cd +C'' single junction detector, Qt __ Qt MC j +C' Cd +C'' parallel junction detector, _ _ 1 Qt Qt Cj + MC' M . C d + C' ' serial junction detector,
where we have used the abbreviation C'= GoCf + Cg. The result for the serial junction detector has also been given in ref. [7]. Assuming a situation where Ci > MC' holds (voltage sensitive measurement), one obtains the same voltage for a single junction detector and for a serial junction detector, provided the signal charges Qt are equal . However, a factor 1/M appears in the signal height of a serial junction detector, because the total detector capacitance Cd rather than the
J. Joehum et al. I Nucl. Instr. and Meth. in Phys. Res. 338 (1994) 458-466
46 0
If = (Vo - Vt)/Zf`VO/Z,
It -Id+h o -Id = -Vi/Zd=VO/(GOZd)
Vo-QtS(W)= GoZd ZZf It
for a single junction detector, parallel junction detector, and serial junction detector, respectively . The various noise sources are shown in Fig . 2. The white noise a is caused by the thermal noise in the channel of the input FET. It appears as a voltage noise in series to the detector and a is therefore called "series noise" . The current noise b, independent of frequency, is caused by the thermal noise of the resistors in the circuit, the shot noise of the detector bias current and the gate leakage current of the input FET. As these noise currents flow in parallel to the detector current, b is called "parallel noise" . An additional contribution to the series noise a is 1/f noise described by c/ lto l, which has several sources. The total noise power spectrum present at the preamplifier output is then given by
r
X(to) = (b + (a + I
c i = Cd
Zi
Fig. 1. (a) In a single junction detector or a parallel junctions detector, the entire detector with the impedance Zd (w) acts as source of a signal current It . The signal current is split into Id flowing through the detector and If flowing through the preamplifier . The preamplifier transfer function corresponds to the impedance P(w) which, when multiplied by the signal current, yields the signal voltage at the preamplifier output . (b) In a serial junction detector only the tunnel junction where absorption occurs acts as source of the signal current, which now is split into If flowing only through this particular tunnel junction and If flowing through the preamplifier and the other tunnel junctions of the detector. As the total detector impedance Zd(w)= MZj (w) enters into the transfer function, and not only the impedance of one tunnel junction Zj (w), a factor 1/M, compared to the case in Fig. la, arises in the calculation of output signal . capacitance Ci of a single tunnel junction enters into the transfer function . For the serial junction detector the same factor 1/M appears also in Eq . (6), because the total detector impedance Zd (w) rather than the single junction impedance Zj (o)) enters into the transfer function . In the general case, the detector impedances Zd((o) are Zj(w), Zj(w)/M and MZ,(w)
+Cg +Cf ;
c
1
+ WZ Ci
)1 Rz 1 1 1 -=-+Ri
Rd
l jp 2 (6)),
(7)
Rf
The feedback components (R f , Cf ) are negligible against the parallel electrical detector parameters (R d , Cd ) because the open loop gain Go is typically 5000 or higher, i.e . C i = Cd + Cg and R i = R d . A typical noise spectrum of a charge sensitive preamplifier is shown in Fig. 3. Both SZ(w) and .1V(to) are proportional to PZ(to), whence the preamplifier transfer function P(w) drops out of the optimal signal to noise ratio (cf. Eq . (1)). By consequence, the optimal signal to noise ratio is independent of the preamplifier (charge, current, or volt-
V n-Vni -V_ z Vn-V na :
Fig. 2. The noise current in parallel to the signal current resulting from thermal noise of bias and feedback resistors and from shot noise of bias and FET gate leakage currents is described by the spectral power density b. The thermal noise in the FET channel is equivalent to a noise voltage at the FET gate in series with the detector . This white series noise added to the 1 /f noise of the input FET is denoted by the spectral power density a + c/to .
J. Jochum et al. /Nucl. Instr. and Meth . in Phys. Res. 338 (1994) 458-466
6 10 101 105 10 3
frequency tv [Hzl 1%T s Fig. 3. Typical noise spectrum of a charge sensitive preamplifier. The dotted lines show the noise spectrum according to Eq . (7). The high frequency part of the spectrum W > TS t is given by the series noise a of the input FET. The low frequency part is determined by the parallel noise constant b. age sensitive) type . The constants a, b, and c, however, will be affected by the actual circuitry and components of the preamplifier . The optimal signal to noise ratio resulting from signal (6) and noise (7) is
1
X(RZ
d
1
-Qt
+
J///
I
TrK 2 bQ 0 [dw] L(
1+W2Ts
account. A low value of R d increases the contribution of the effective parallel noise ba . The parameter p(0 < p < 1) describes the influence of the detector resistance R d on effective parallel noise and on the contribution of 1/f noise. In noise formulae deduced for semiconductor detectors this parameter is often neglected due to the high values of R d . For detectors using superconducting tunnel junctions p :A 0 whence the influence of R d cannot be neglected . To estimate the influence of 1/f noise, we calculate the signal to noise ratio in the presence of 1/f noise (c :;,- 0) and compare with the case c = 0. The limits for c, above which the existence of 1/f noise would reduce the signal to noise ratio by more than 10%, are shown in Fig. 4, depending on pulse length T P and detector resistance parameter p. The limits for c become more and more stringent for decreasing R d or increasing p. The detector resistance R d determines in a significant way the influence of 1/f noise on the signal to noise ratio. For an example we -z0 take typical values of our experiments, a = 2.6 X 10 -28 AZ S,C V2 s,b=2X10 ;=1 .6nF,c=5X10-tb V2, and TP = 50 Ws, yielding ry < 19 Ws, TP /TS > 2.6, and cTs /a < 0.34 . 1/f noise can be neglected on a 10% level for p < 0.22, equivalent to R d > 50 kit ; compare Fig. 4. A dynamical resistance of R d > 50 kft can quite easily be achieved with our aluminum tunnel junctions. Hence, 1/f noise may be disregarded in our experiments. The ratio & = Ri/Rnn between dynamical resistance R i and the normalconducting resistance Rnn (at voltages Vb >> 2d/e) is an indicator of tunnel junction quality whence we call er the "quality factor". The normalconducting resistance R nn is proportional to the
1I t t/z \1 + +WZTP) a p (1 W Z Tdet) I(1 I
c
with ba = b +
ll
+W 2(Cd +Cg ) 2 l }(1 +WZT P )
46 1
a Rzd
2
,
effective parallel noise,
Ts = ~( alba) (Cd + Cg), ?det = Rd(Cd+Cg ), T5
p=-=1/ Tdet
V
b
1 + -R zd .
a
The number of the elements connected in series is K= 1 for a single or a parallel junction detector and K=M for a serial junction detector of M elements . For a CR-RC shaper with equal time constants for integration and differentiation Ts represents the optimal shaping time in the case of a charge sensitive preamplifier and for Ts >> T P [10] . For our tunnel junction detectors T s :5 TP applies, and thus the influence of T P on the signal to noise ratio must be taken into
Fig. 4. The influence of 1/f noise on the total noise depends on the signal pulse length TP and on the dynamical resistance of the tunnel junction at the bias point, represented by the parameter p, according to Eq. (8). For values of c TS /a below a curve given by the respective parameter p, the presence of 1/f noise reduces the signal to noise ratio by less than 10%. For lower values of the dynamical resistances R d (higher p) the tolerable values of c become smaller. The marked point is discussed in the text.
462
J. Jochum et at I Nucl. Instr. and Meth . in Phys. Res. 338 (1994) 458-466
tunneling time Tl of quasiparticles and is inversely proportional to the tunnel area A, Hence, the dynamical resistance of a tunnel junction is R j a d'7t /A t . To increase the signal charge Q, the tunneling time T, has to be minimized (cf. Eq. (4)). To optimize the signal to noise ratio, a high quality factor a is essential .
3 . Series and parallel connected tunnel junctions We shall now compare the various possibilities to enlarge the sensitive area of detectors: trapping detectors, parallel junction detectors, and serial junction detectors. An alternative would be to use tunnel junctions of large area with consequently large detector capacitances. Being equivalent to many tunnel junctions of small area ("normal sized" junctions) connected in parallel, we shall refer to large tunnel junctions as a parallel junction detector . To scrutinize the performance of the various detector types we shall now compare their equivalent noise charges . As a reference we quote the equivalent noise charge produced by a single "normal sized" tunnel junction (index j) . Exploiting quasiparticle trapping [4], the sensitive detector area can be enhanced significantly employing only a small number of normal sized tunnel junctions . The equivalent noise charge of a single junction detector is thus a measure for the equivalent noise charge of a trapping detector. According to Eq . (8) the optimal signal to noise ratio Msn and the equivalent noise charge Qn , neglecting 1/f noise, are given by 2 Qc
,rrK 2 b a f~ (1 + C ) 2TS)(1 + 1 Qt K Qn
1/2
d
c) 2TP
1/2
2ba (TS +'r,) }
'
(l0a)
= K~26a (TS + rp) =K
4ab a Ci + 2b aT p .
(10b)
The first term of Qn under the square root (2b aT 5 ) can be interpreted as the noise charge collected during the optimal shaping time r, . The second term (26 aT p ) is associated with the additional noise charge generated by the effective current noise during the time Tp required for the collection of the signal charge . Both contributions depend on the effective parallel noise ba . The first term arises due to the FET series noise a and the second term due to the nonvanishing pulse length
,r p . Therefore, we shall call the first term the FET noise term and the second one the pulse noise term . To assess the noise charges Q n for different detector types, the dependence of the values a, ba , c, Ci and TP on the detector type and on its parameters has to be determined : Pulse length T P : TP depends on quasiparticle lifetime and tunneling time and thus on the purity of the films of the tunnel junctions and on the quality of the tunnel barrier . Both are independent of detector type, implying equal pulse lengths for single, serial, and parallel junction detectors . White series noise constant a : For a JFET a is given by a=
2kT gm
0 .65 .
(11)
T, k, and g m are the temperature of the FET, the Boltzmann constant, and the FET forward transconductance, respectively [ll] . gn, is proportional to the gate capacitance C g , whence the FET noise term to the equivalent noise charge Qn (Eq . (10), with b a = b) is 1 4abC ;a f(Cd +C g ) .
(12)
In order to minimize this noise term, the gate capacitance of an FET has to be equal to the detector capacitance . For a properly matched FET the series noise a then becomes a a 1/Cd .
(14)
Minimizing the FET noise term in the equivalent noise charge, Eq . (12), the contribution of a/R2 to the effective parallel noise was neglected . If this is taken into account, the FET gate capacitance C g has to be chosen somewhat larger than the detector capacitance Cd . The changes, however, which arise from this modified FET matching, are of less importance, especially for the values of interest here . Input capacitance Ci : Ci = Cd + Cg . With an FET matched to the detector one obtains Ci = 2Cd . 1 /f noise constant c : The constant c depends on the gate size and thus on the gate capacitance in the same way as constant a [12] . c a 1/Cg .
(15a)
For the case of CR-RC shapers having equal time constants r, for differentiation and integration, 1/f noise contributes to Q n with a term proportional to cCi [13] . This contribution is likewise minimized by the condition C g = Cd , yielding c a 1/Cd .
(15b)
46 3
J. Jochum et al. I Nucl. Instr. and Meth. in Phys. Res. 338 (1994) 458-466
The parallel noise constant b : The parallel noise consists of the thermal noise of the feedback and bias resistors Rf and R b as well as on the shot noise of the detector bias current Ib and the FET gate leakage current Ig. b = 2kT(Rf'+Rb')+e(Ib +Ig ) .
(16)
By lowering the operating temperature T of the feedback resistor, the bias resistor, and the input FET, these contributions to parallel noise can be reduced substantially. The parallel noise is then dominated by the bias current contribution . Provided tunnel junctions are equal, each of them requires the same bias current Ib. Hence, a detector consisting of M tunnel junctions connected in parallel requires a bias current MIb. Consequently, the parallel noise constant of a parallel junction detector is M times larger than for a single junction detector . M series connected tunnel junctions have to be biased with the current Ib of a single tunnel junction . Thus, each tunnel junction exhibits a shot noise b = eIb. The noise current source is parallel to the respective tunnel junction in the same way as the signal current is in Fig. lb. The corresponding noise power density at the preamplifier output is proportional to bIMZ (cf. Eq . (6)). As there are a number of M such noise current sources, this leads to a noise power density proportional to b/M. Compared to a single tunnel junction, the parallel noise constant of M series connected tunnel junctions is therefore reduced by a factor M. To give an example for typical noise contributions we refer to an aluminum tunnel junction as used in ref. 4, with a tunnel area of At = 1.6 X 10 -4 cm', exhibiting a capacitance of Ci = 0.8 nF (by parallel plate model) and a pulse length of rp = 50 ws . A properly matched FET for this tunnel junction is the NJ 3600 exhibiting a forward transconductance of gn, = 200 mfZ- ' (at a drain current of Ids = 20 mA). The result-20 V2 s. ing series noise (cf. Eq . (11)) is a = 2.6 X 10 noise constant of this FET is approximately The 1/f c = 5 X 10 -16 V2. For tunnel junctions of this area A t a dynamical resistance in the range of Ri >_ 50 kfl can easily be achieved . If such a tunnel junction is biased with a typical voltage of Vb = A/4 e, the bias current is about Ib = 1 nA (Fig . 5), causing a shot noise of b = 2 X 10 -28 AZ s, Eq . (17) . Thus, -r s = 19 ws and the two noise contributions to the equivalent noise charge Qn are: FET noise term 2bjs = 543 electrons (rms) and pulse noise term 2b a,p = 906 electrons (rms). We shall now evaluate the equivalent noise charges Qn for the different detector types. The quantities ail bj, bail cj, Tsi, and pj, specified by the index j, denote the values of a single "normal sized" tunnel junction
Fig. 5. Current-voltage characteristics of a tunnel junction with a dynamical resistance of Rj = 50 kf and a tunnel area of At = 1 .6 x 10 -4 cm2. The bias point has to be chosen at the largest possible dynamical resistance. A bias current in the range of Ib = 1 nA is typical for our aluminum tunnel junctions of this size . detector as defined in Eq . (8). Cj , Rj are the capacitance and the dynamical resistance of a normal sized tunnel junction, respectively . The corresponding values for the parallel connection or the series connection of M such tunnel junctions are compiled in Table 1. With the help of Eq . (10) and the values of Table 1, the minimal equivalent noise charges for the different detector types are compared : 4a jbaj 2Cj + 2b ai Tp = Qnj, single junction, Qn =
4a~b at 2MC, + 2Mbaj T p
= V1~~
Qnj,
= Y1~~
Qnj ,
parallel connection, M 4a ibat 2 V M + 2 M 7-1, series connection .
(18)
Since the equivalent noise charges Qt, are calculated under optimal conditions, they represent the lowest possible limits . The lowest equivalent noise charge Qn is obtained for a single junction detector . The parallel and series connections of M tunnel junctions exhibit noise charges which are in both cases larger by a factor of ~-M _. The best signal to noise ratio is obtained in the case of a single junction detector, in a situation where the amount of energy deposited is independent of detector size, e.g . in the case of X-ray detection. By contrast a different situation arises, if the amount of energy detected increases in proportion to the sensitive area of a tunnel junction detector, e.g. in the case of phonon detection [14,15]. In this situation the signal charge Qt
independent 1/f Dependence parallel adetector advantage independent energy noise detector quantities single the term can be noise junction 1noise Mi to connection to tunnel of to therefore estimate ratio junction connection however, charge conditions, parameters M ratio detector detector), than noise area noise matched easily with area film equivalent Tp, atunnel noise especially charge the affected tunnel or junction times charges nonvanishing absorbed type isAn p,junction series exhibits determining materials of for abut required taken ratios ratio, of ofsingle visualize of injunctions r,ratio the in the detector conclude of smaller (serial of the may junction aand increase employing the signal larger toas such by single aconnections detector this ifnoise of energy the the detector, detector into are the sensitive tunnel given well low ajunction detector increases the the be the parallel higher gap that junction, ato factor enhanced best the for comparison by detector account the turns junction equivalent of charges FET 1/f by detector ratio detector that, as element gaps noise aratio by junction equivalent Jochum quasiparticle normal parallel the A best afactor on asignal type detector detector signal [4] VMi/M noise, matching Eq out factor and detector ifof parallel of of in relations the +ratio the parallel reduces capacitance, do detector /M detector noise only A M tunnel type to Cg et(8) (cf proportion size serial Thus, parameters The or detector noise to trapping Mi/M not to albe tunnel qualitatively we energy noise VM, area has -'~25n by Table Inoise series With By or with has constants is for difficult Nucl change, trapping, junction consider influence between junction, junction vFM charge the in then serial for whence to charge, contrast, The and aan on compared junctions absorbed the terms depends lM compete serial junction detector an ratio Instr phonon 1)the ratio initially relative equivatounder tunnel equivK2ba of juncsame even FET compared of The and c/a the the pathe dethe has and an or or of A of isa /Mfor if i affects Phys the noise On determined = reducing FET =equivalent tunneling limited have tunnel to bias determined between (20-80) our length three term -QO/1/2ba(7s tothe be area noise reduce Q Res pulse /M signal etc noise of detector parallel ratio to +the bias noise both to noise aor resistors mainly ofTtThe aluminum different 338 initial contrary by If as junction baTp chance immediately will the optimal by to tunneling the length signal 7p ws, current time can an to the noise the effective ba term signal "back (1994) by influence ratio quasiparticle isthe be noise noise determined excess tunnel by At operating r,isexcess and signal the electronic be /M 7tshorter, detector and to =given charge elucidated of 7p charge the if signal tunnel 458-466 +shaping, parameters, present back is charge 19 illustrated tunneling" quasiparticle (cf tunnel ba, 7p) time thus quasiparticles ratio isQn parallel then density back the lifetime once charge quasiparticles, ws area aby of r,Eq tunneling might [16] types critical to junctions, (eqs temperatures involving pulse rt, see the increases noise, input by Qt our the tunneling, dominated trapping repeatedly they by in(16)) noise isThis given noise of sections the by signal and (10), of this eventually ismeans large The contribution energy like length parameter, quasiparticles whence have lifetime FET determined considering the shot situation ratio By would section equivalent both by ba (18)) the (cf pulse tunneling compared [41, Eq to quasiparticles tunneled, (cf 2the and by the reducing on resolution of noise pulse of Eq and itof noise quasipartilifetime 7, (4) Under dominate the leave the aEq tunneling length isthis feedback effective (8)) because isaand of 3) proper imporand of by length "back signal single often pulse (19)) noise stays time, ratio electo The way and the the the the the by 7, isT,is
464
J.
.
.
.
Meth. n
.
.
Table The Single Parallel Series
may detector alent tectors, signal to sensitive the single using lent to However, created signal with absorber To effect signal properly rameters are only One different if We is optimal obtained the sensitive tion trapping detection, parallel
K 1 1 M
Cd 2Ci 2MCi 2Ci
a ai ai Mai
.
. . . . . .
. .
.
.
.
. . .
4. junction The
on trapping individual
Rd Ri Ri MRi
.
b bi Mbi bi
ba bdi Mb,i bai
c ci ci/M Mci
the lifetime . The ratio fraction the condition tronic junction 7 sn Tp In 7p, quasiparticle (Tp signal noise still tant By and parallel would detector effective the The it charge . . to tunneling". confined geometry cles amplifies referred then T, . tunnel pulse and 7,7t 7, 7,,
P Pi Pi Pi
Tsi T,i 'r'i
.
.M,a .
.
.
.
(19)
.
; . .
.
.
.
.
.
.
.
.
.
.
: without with
.
(20)
J. Jochum et al. I Nucl. Instr. and Meth. in Phys. Res. 338 (1994) 458-466
Back tunneling was considered as an amplification process for tunnel junction detectors increasing the signal charge Qt by a gain factor gbt 71 gbt =1 + - . Tt
(21)
This gain 9bt, however, is due to a longer pulse length, Eq . (20), rather than a higher current, giving at the same time rise to a higher noise charge . The gain G bt associated with the signal to noise ratio is somewhat smaller : G bt
~
7",
(1 +
7'
SI T t ) (1 + Tt
gbt( 1 +
7t
(22a)
with Ts ,=TS T1/(Ts +T,) . We thus obtain gbt > Gbt >
gbt
(22b)
If the pulse noise term b aTP of the equivalent noise charge dominates, the noise charge is considerably increased by a longer pulse length . In this case, only the square root of the signal charge gain gt,t enters in the signal to noise ratio G bt = gbt . Back tunneling on the one hand maximizes the signal to noise ratio, while on the other hand it limits the maximum counting rate . We will nevertheless assume in the following the presence of back tunneling with TP =T1. At the usual low operating temperatures, the bias current of our tunnel junction detectors is caused by imperfections of the tunnel barrier and of the film quality . The bias current is then inversely proportional to the dynamical resistance R~ 1 of a tunnel junction (Fig . 5). Consequently we obtain for the effective parallel noise ba aA t /~f7t,
(23)
where Eq . (9) was employed. The series noise a for a properly matched FET is inversely proportional to the capacitance C j of the tunnel junction and hence to the tunnel area A t . a a 1/At .
(24)
Eventually, four parameters are decisive for the contribution of electronic noise to the signal to noise ratio of a tunnel junction detector : - the quasiparticle lifetime - the tunneling time Tt , - the quality factor of the tunnel junction &I = Rj/R , - the tunnel area A t . For the signal to noise ratio `Wsn (cf . Eq . (19)) we
465
obtain Qt
2bT P a if pulse noise term dominates, rP >> Ts Qt
,r2--b,.
a T,
V
A, 7, Tt
if FET noise term dominates, TP << 7s . (25) In our typical experimental situation, the pulse noise term dominates and T, enters into both, the noise and the signal charges (cf. Eq . (19) rP = r,), whence the signal to noise ratio becomes proportional to T, . If experimental progress permits a lower effective parallel noise ba or a shorter pulse length rP , the FET noise term might dominate. In this case -~Fsn is directly proportional to the quasiparticle lifetime r,, because T, enter only in the signal charge . The effective parallel noise ba enters differently into the FET noise term bars and the pulse noise term baTP , thus the quality factor & becomes more important in case the pulse noise term dominates . If the tunnel area A t is enlarged, the detector capacitance and consequently the FET noise term increases . Simultaneously the dynamical resistance reduces requiring a higher bias current, and thus leading to a higher effective parallel noise. In both cases the signal to noise ratio becomes therefore proportional to 1/ A t , necessitating a small tunnel area . On the contrary there exist lower limits for the tunnel junction size, due to the quasiparticle pair recombination, which would lead to a nonlinear energy response of the detector [5] .
5. Summary The signal to noise ratio for tunnel junction detectors was calculated depending on detector type (single, serial, or parallel junction detectors). The influence of 1/f noise was shown to depend upon pulse length and in particular on the dynamical resistance of the tunnel junctions at the bias point . The dynamical resistances required to suppress 1/f noise are technically feasible . The signal to noise ratio, which can be reached under optimal conditions such as bias and feedback resistors operated at low temperatures, an input FET matched to the detector, and optimal pulse shaping, was com-, pared for the cases of a single tunnel junction and a parallel or a series connection of M tunnel junctions . For a fixed signal charge, as prevailing in the case of X-ray detection, a single junction detector yields the
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J. Jochum et al. /Nucl. Instr. and Meth . in Phys. Res. 338 (1994) 458-466
best signal to noise ratio. The ratio becomes worse by a factor rM for a parallel or series connection of M tunnel junctions as compared to a single tunnel junction. If the signal charge increases with the sensitive area, as in the case of phonon detection, a parallel or serial junction detector or a trapping detector would be the better choices.
Back tunneling increases the signal charge but at the same time increases also the noise charge to some extent . Consequently, the gain in signal to noise ratio by implementation of back tunneling is smaller than anticipated by the gain in signal charge alone. To maximize the signal to noise ratio, a long quasiparticle lifetime rl, a short quasiparticle tunneling time
Tc , a high quality factor ci, and a small tunnel area A t
are essential. Under the conditions prevailing in our experiments, the signal to noise ratio depends on the square root of all these parameters . If the quality
factor is very large and the parallel noise is small due to a low bias current, the FET series noise matters and thus matching of the FET to the detector is crucial. Acknowledgement This work has been supported by the Bundesministerium für Forschung und Technologie. References [1] L. Brogiato, D.V. Camin and E. Fiorini (eds.), Proc. Low Temperature Detectors for Neutrinos and Dark Matter 111, Gran Sasso, L'Aquila, 1989 (Editions Frontières, Gif- sur-Yvette, 1990). [2] A. Barone, R. Cristiano and S. Pagano (eds .), Proc . X-ray Detection by Superconducting Tunnel Junctions, Naples, 1990 (World Scientific, Singapore, 1991).
[3] N.E . Booth and G.L . Salmon (eds.), Proc. Low Temperature Detectors for Neutrinos and Dark Matter IV, Oxford, 1991 (Editions Frontières, Gif- sur-Yvette, 1992). [4] H . Kraus, F. v. Feilitzsch, J. Jochum, R.L. M6ssbauer, Th . Peterreins and F. Prèbst, Phys . Lett . B 231 (1989) 195. [5] J. Jochum, H. Kraus, M. Gutsche, B. Kemmather, F. v. Feilitzsch and R.L. M6ssbauer, Ann. Phys . 7 (1993) 611. [6] D.J . Goldie, N.E. Booth and G.L . Salmon, Proc. Low Temperature Detectors for Neutrinos and Dark Matter IV, Oxford, 1991, eds. N.E . Booth and G.L . Salmon (Editions Frontières, Gif- sur-Yvette, 1992) p. 245. [7] M. Kurakado, A. Matsumura and T. Takahashi, Proc . X-ray Detection by Superconducting Tunnel Junctions, Naples, 1990, eds. A. Barone, R. Cristiano and S. Pagano (World Scientific, Singapore, 1991) p. 77. [8] J.P . Maneval, S. Labov, R.W . Bland, S.C . Dickson, K. Laws, R.T. Johnson, J. Lockhart, R. Martin, M.W . Simon, D.A . Stricker and R.M . Watson, Proc . Low Temperature Detectors for Neutrinos and Dark Matter IV, Oxford, 1991, eds. N.E . Booth and G.L . Salmon (Editions Frontières, Gif- sur-Yvette, 1992) p. 291 . [9] E. Gatti and P.F . Manfredi, La Rivista del Nuovo Cimento 9(3) (1986) . [10] M . Tsukuda, Nucl . Instr. and Meth . 14 (1961) 241. [11] R.S .C. Cobbold, Theory and applications of field effect transistors (Wiley/Interscience, New York, 1970). [12] P.O . Lauritzen, Solid-State Electrons 8 (1965) 41 . [13] E. Kowalski, Nuclear Electronics (Springer-Verlag, Berlin, Heidelberg, New York, 1970). [14] Th . Peterreins, J. Jochum, F. Prèbst, F. v. Feilitzsch, H. Kraus and R.L. M6ssbauer, J. Appl. Phys . 69(4) (1991) 1791 . [15] D.J . Goldie, Proc. X-ray Detection by Superconducting Tunnel Junctions, Naples, 1990, eds. A. Barone, R. Cristiano and S. Pagano (World Scientific, Singapore, 1991) P. 98 . [16] F. Prèbst, H. Kraus, Th . Peterreins and F. v. Feilitzsch, Nucl . Instr. and Meth . A 280 (1989) 251 .
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
Electronic noise of superconducting tunnel junction detectors J. Jochum, H. Kraus, M . Gutsche, B. Kemmather, F. v. Feilitzsch and R.L. Mössbauer Physik Department E15, Technische Universität München, James-Franck-Str., 85748 Garching bei München, Germany
(Received 6 August 1993) The optimal signal to noise ratio for detectors based on superconducting tunnel junctions is calculated and compared for the cases of a detector consisting of one single tunnel junction, as well as of series and of parallel connections of such tunnel junctions. The influence of I/ f noise and its dependence on the dynamical resistance of tunnel junctions is discussed quantitatively . A single tunnel junction yields the minimum equivalent noise charge . Such a tunnel junction exhibits the best signal to noise ratio if the signal charge is independent of detector size . In case, signal charge increases with detector size, a parallel or a series connection of tunnel junctions would provide the optimum signal to noise ratio . The equivalent noise charge and the respective signal to noise ratio are deduced as functions of tunnel junction parameters such as tunneling time, quasiparticle lifetime, etc. 1. Introduction In many fields of physics, for example in neutrino physics, in X-ray astronomy, or in the search for dark matter candidates, progress depends on improved detection techniques, in particular lower energy threshold and higher energy resolution . Substantial improvement is anticipated by utilizing superconductors exhibiting much smaller energy gaps than semiconductors . The development of such new detectors based on superconducting tunnel junctions is indeed pursued by many groups [1-4]. At present the best tunnel junction detectors are already superior in energy resolution compared to state of the art semiconductor detectors, but are still far from their statistical limit [4]. The energy resolution is at present still dominated by the ratio between signal height and electronic noise. Processes affecting the signal height are discussed in a separate paper [5]. In this paper we shall investigate the dependence of electronic noise on detector parameters such as detector capacitance, detector resistance, lifetime of excess quasiparticles, and quasiparticle tunneling time . Most applications require larger sensitive detector areas than those attained with prototype tunnel junctions during earlier experiments . In efforts to combine larger signal to noise ratios with large detector areas, we checked various possibilities : series and parallel arrays of tunnel junctions and single tunnel junctions involving separate superconducting absorbers engaging quasiparticle trapping [4,6-8]. We call these detectors : serial or parallel junction detectors and single junction detectors. In this context we shall treat in section 2
signal to noise ratio for a superconducting tunnel junction detector combined with its preamplifier . In section 3 we compare the signal to noise ratio for single junction detectors, and for serial or parallel junction detectors. Section 4 deals with the influence of tunnel junction parameters on the signal to noise ratio.
2. Signal to noise ratio .BPS Absorption of energy yields free charge carriers in a detector based on superconductors as well as in a detector based on semiconductors . Depending on the physics of the detector, a total charge Qt , which is a fraction of the initial charge Qo causes the detector signal. In superconducting tunnel junctions the signal is generated by tunneling of the free charge carriers, the so called "quasiparticles" . A voltage signal proportional to the tunneled charge Qt , and thus to the incident energy, results at the output of a preamplifier . The Fourier transform of this signal is described by Q,S(w). In addition there is noise at the preamplifier output described by the spectral density J'(&)). The accuracy of energy measurement is determined by the ratio between signal height and noise. To optimize the signal to noise ratio the preamplifier output signal is filtered . Depending on the filter, a certain signal shape results, whence filtering is commonly referred to as "shaping". There is always some noise passing the filter and resulting in an uncertainty in the determination of the signal height and thus in the determination of the absorbed energy . The accuracy of the energy
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J. Jochum et al. I Nucl. Instr. and Meth . in Phys . Res. 338 (1994) 458-466
determination is limited by the signal to noise ratio `'fsn : signal height `Rsn = root mean square of noise The amount of charge leading to a signal to noise ratio of Msn = 1 is a measure for noise and is called the "equivalent noise charge" Qn. The highest signal to noise ratio synonymously with the smallest equivalent noise charge can be achieved by application of the "optimum shaper" and is given by [9]: S2 (ù» z - Q1 dù), sn -
h
2Tr _~ .~(co)
Qn -
Qt Msn
(lb)
'
In the superconducting films of the detectors excess quasiparticles are created by Cooper pair breaking after energy absorption . In a tunnel junction these excess quasiparticles produce a tunnel current It(t) = Qo e_t / TP . It(t) Tt Qo is the charge, due to the initial number of excess quasiparticles, T t is the tunneling time, and TP is the pulse length given by the effective decay time constant of quasiparticles in the films of a tunnel junction [5]. The Fourier transform of the tunnel current signal It(t) is given by Qt It(w) 1 + i(,)TP'
Additional parasitic capacitances may have to be included in Cg. In a charge sensitive preamplifier, Zf(w) consists of a capacitance Cf in parallel with a resistance R f , yielding the preamplifier integration time Tf = Cf R f. In case of a single junction detector, its impedance Zd(w) consists of the tunnel junction capacitance Cj in parallel with the dynamical resistance Rj at the bias point. The signal current It arises from the tunnel junction, in which energy is absorbed. Consequently, for a single junction detector, or for a parallel junction detector (Fig. la), the signal current source is located across the entire detector impedance, whereas in the case of a serial junction detector (Fig. lb) it appears across the impedance of the energy absorbing tunnel junction only. The signal Qt S(a)) in these cases is Q,S(w)
P(w)It(-), single or parallel junction detector, 1 MP(w)It(w)~ serial junction detector with M tunnel junctions.
As an example we give the transfer function of a pure capacitive detector and preamplifier: P(w) = Go/[iw(Cd + GoCf + Cg)]. The capacitance Cd of a tunnel junction detector and its output signal voltage Vmax divided by the open loop gain is then given by Cj MCj,
with the signal charge Qt
TV
_
Tt
Ci/M,
Q0+
where the factor Tp /Tt arises from the competition between quasiparticle decay and quasiparticle tunneling. To obtain the signal S(a)) at the preamplifier output, we need the product I,(w)-P(w), where the transfer function P(w) of the preamplifier loaded with the impedance Zi(w) is given by _ GOZi(w)Zf(w) . (Sa) GoZi(w) + Zf(w) Z,(w) and G o are the feedback impedance of the preamplifier and its open loop gain, respectively. The impedance Zi(w) is the parallel combination of the detector impedance Zd(w) and the impedance of the gate capacitance Cg of the input FET P(w)
Zi(w) =
1Zd(w) +
459
(5b)
Vmax
Go
Qt _ Qt Ci +C' Cd +C'' single junction detector, Qt __ Qt MC j +C' Cd +C'' parallel junction detector, _ _ 1 Qt Qt Cj + MC' M . C d + C' ' serial junction detector,
where we have used the abbreviation C'= GoCf + Cg. The result for the serial junction detector has also been given in ref. [7]. Assuming a situation where Ci > MC' holds (voltage sensitive measurement), one obtains the same voltage for a single junction detector and for a serial junction detector, provided the signal charges Qt are equal . However, a factor 1/M appears in the signal height of a serial junction detector, because the total detector capacitance Cd rather than the
J. Joehum et al. I Nucl. Instr. and Meth. in Phys. Res. 338 (1994) 458-466
46 0
If = (Vo - Vt)/Zf`VO/Z,
It -Id+h o -Id = -Vi/Zd=VO/(GOZd)
Vo-QtS(W)= GoZd ZZf It
for a single junction detector, parallel junction detector, and serial junction detector, respectively . The various noise sources are shown in Fig . 2. The white noise a is caused by the thermal noise in the channel of the input FET. It appears as a voltage noise in series to the detector and a is therefore called "series noise" . The current noise b, independent of frequency, is caused by the thermal noise of the resistors in the circuit, the shot noise of the detector bias current and the gate leakage current of the input FET. As these noise currents flow in parallel to the detector current, b is called "parallel noise" . An additional contribution to the series noise a is 1/f noise described by c/ lto l, which has several sources. The total noise power spectrum present at the preamplifier output is then given by
r
X(to) = (b + (a + I
c i = Cd
Zi
Fig. 1. (a) In a single junction detector or a parallel junctions detector, the entire detector with the impedance Zd (w) acts as source of a signal current It . The signal current is split into Id flowing through the detector and If flowing through the preamplifier . The preamplifier transfer function corresponds to the impedance P(w) which, when multiplied by the signal current, yields the signal voltage at the preamplifier output . (b) In a serial junction detector only the tunnel junction where absorption occurs acts as source of the signal current, which now is split into If flowing only through this particular tunnel junction and If flowing through the preamplifier and the other tunnel junctions of the detector. As the total detector impedance Zd(w)= MZj (w) enters into the transfer function, and not only the impedance of one tunnel junction Zj (w), a factor 1/M, compared to the case in Fig. la, arises in the calculation of output signal . capacitance Ci of a single tunnel junction enters into the transfer function . For the serial junction detector the same factor 1/M appears also in Eq . (6), because the total detector impedance Zd (w) rather than the single junction impedance Zj (o)) enters into the transfer function . In the general case, the detector impedances Zd((o) are Zj(w), Zj(w)/M and MZ,(w)
+Cg +Cf ;
c
1
+ WZ Ci
)1 Rz 1 1 1 -=-+Ri
Rd
l jp 2 (6)),
(7)
Rf
The feedback components (R f , Cf ) are negligible against the parallel electrical detector parameters (R d , Cd ) because the open loop gain Go is typically 5000 or higher, i.e . C i = Cd + Cg and R i = R d . A typical noise spectrum of a charge sensitive preamplifier is shown in Fig. 3. Both SZ(w) and .1V(to) are proportional to PZ(to), whence the preamplifier transfer function P(w) drops out of the optimal signal to noise ratio (cf. Eq . (1)). By consequence, the optimal signal to noise ratio is independent of the preamplifier (charge, current, or volt-
V n-Vni -V_ z Vn-V na :
Fig. 2. The noise current in parallel to the signal current resulting from thermal noise of bias and feedback resistors and from shot noise of bias and FET gate leakage currents is described by the spectral power density b. The thermal noise in the FET channel is equivalent to a noise voltage at the FET gate in series with the detector . This white series noise added to the 1 /f noise of the input FET is denoted by the spectral power density a + c/to .
J. Jochum et al. /Nucl. Instr. and Meth . in Phys. Res. 338 (1994) 458-466
6 10 101 105 10 3
frequency tv [Hzl 1%T s Fig. 3. Typical noise spectrum of a charge sensitive preamplifier. The dotted lines show the noise spectrum according to Eq . (7). The high frequency part of the spectrum W > TS t is given by the series noise a of the input FET. The low frequency part is determined by the parallel noise constant b. age sensitive) type . The constants a, b, and c, however, will be affected by the actual circuitry and components of the preamplifier . The optimal signal to noise ratio resulting from signal (6) and noise (7) is
1
X(RZ
d
1
-Qt
+
J///
I
TrK 2 bQ 0 [dw] L(
1+W2Ts
account. A low value of R d increases the contribution of the effective parallel noise ba . The parameter p(0 < p < 1) describes the influence of the detector resistance R d on effective parallel noise and on the contribution of 1/f noise. In noise formulae deduced for semiconductor detectors this parameter is often neglected due to the high values of R d . For detectors using superconducting tunnel junctions p :A 0 whence the influence of R d cannot be neglected . To estimate the influence of 1/f noise, we calculate the signal to noise ratio in the presence of 1/f noise (c :;,- 0) and compare with the case c = 0. The limits for c, above which the existence of 1/f noise would reduce the signal to noise ratio by more than 10%, are shown in Fig. 4, depending on pulse length T P and detector resistance parameter p. The limits for c become more and more stringent for decreasing R d or increasing p. The detector resistance R d determines in a significant way the influence of 1/f noise on the signal to noise ratio. For an example we -z0 take typical values of our experiments, a = 2.6 X 10 -28 AZ S,C V2 s,b=2X10 ;=1 .6nF,c=5X10-tb V2, and TP = 50 Ws, yielding ry < 19 Ws, TP /TS > 2.6, and cTs /a < 0.34 . 1/f noise can be neglected on a 10% level for p < 0.22, equivalent to R d > 50 kit ; compare Fig. 4. A dynamical resistance of R d > 50 kft can quite easily be achieved with our aluminum tunnel junctions. Hence, 1/f noise may be disregarded in our experiments. The ratio & = Ri/Rnn between dynamical resistance R i and the normalconducting resistance Rnn (at voltages Vb >> 2d/e) is an indicator of tunnel junction quality whence we call er the "quality factor". The normalconducting resistance R nn is proportional to the
1I t t/z \1 + +WZTP) a p (1 W Z Tdet) I(1 I
c
with ba = b +
ll
+W 2(Cd +Cg ) 2 l }(1 +WZT P )
46 1
a Rzd
2
,
effective parallel noise,
Ts = ~( alba) (Cd + Cg), ?det = Rd(Cd+Cg ), T5
p=-=1/ Tdet
V
b
1 + -R zd .
a
The number of the elements connected in series is K= 1 for a single or a parallel junction detector and K=M for a serial junction detector of M elements . For a CR-RC shaper with equal time constants for integration and differentiation Ts represents the optimal shaping time in the case of a charge sensitive preamplifier and for Ts >> T P [10] . For our tunnel junction detectors T s :5 TP applies, and thus the influence of T P on the signal to noise ratio must be taken into
Fig. 4. The influence of 1/f noise on the total noise depends on the signal pulse length TP and on the dynamical resistance of the tunnel junction at the bias point, represented by the parameter p, according to Eq. (8). For values of c TS /a below a curve given by the respective parameter p, the presence of 1/f noise reduces the signal to noise ratio by less than 10%. For lower values of the dynamical resistances R d (higher p) the tolerable values of c become smaller. The marked point is discussed in the text.
462
J. Jochum et at I Nucl. Instr. and Meth . in Phys. Res. 338 (1994) 458-466
tunneling time Tl of quasiparticles and is inversely proportional to the tunnel area A, Hence, the dynamical resistance of a tunnel junction is R j a d'7t /A t . To increase the signal charge Q, the tunneling time T, has to be minimized (cf. Eq. (4)). To optimize the signal to noise ratio, a high quality factor a is essential .
3 . Series and parallel connected tunnel junctions We shall now compare the various possibilities to enlarge the sensitive area of detectors: trapping detectors, parallel junction detectors, and serial junction detectors. An alternative would be to use tunnel junctions of large area with consequently large detector capacitances. Being equivalent to many tunnel junctions of small area ("normal sized" junctions) connected in parallel, we shall refer to large tunnel junctions as a parallel junction detector . To scrutinize the performance of the various detector types we shall now compare their equivalent noise charges . As a reference we quote the equivalent noise charge produced by a single "normal sized" tunnel junction (index j) . Exploiting quasiparticle trapping [4], the sensitive detector area can be enhanced significantly employing only a small number of normal sized tunnel junctions . The equivalent noise charge of a single junction detector is thus a measure for the equivalent noise charge of a trapping detector. According to Eq . (8) the optimal signal to noise ratio Msn and the equivalent noise charge Qn , neglecting 1/f noise, are given by 2 Qc
,rrK 2 b a f~ (1 + C ) 2TS)(1 + 1 Qt K Qn
1/2
d
c) 2TP
1/2
2ba (TS +'r,) }
'
(l0a)
= K~26a (TS + rp) =K
4ab a Ci + 2b aT p .
(10b)
The first term of Qn under the square root (2b aT 5 ) can be interpreted as the noise charge collected during the optimal shaping time r, . The second term (26 aT p ) is associated with the additional noise charge generated by the effective current noise during the time Tp required for the collection of the signal charge . Both contributions depend on the effective parallel noise ba . The first term arises due to the FET series noise a and the second term due to the nonvanishing pulse length
,r p . Therefore, we shall call the first term the FET noise term and the second one the pulse noise term . To assess the noise charges Q n for different detector types, the dependence of the values a, ba , c, Ci and TP on the detector type and on its parameters has to be determined : Pulse length T P : TP depends on quasiparticle lifetime and tunneling time and thus on the purity of the films of the tunnel junctions and on the quality of the tunnel barrier . Both are independent of detector type, implying equal pulse lengths for single, serial, and parallel junction detectors . White series noise constant a : For a JFET a is given by a=
2kT gm
0 .65 .
(11)
T, k, and g m are the temperature of the FET, the Boltzmann constant, and the FET forward transconductance, respectively [ll] . gn, is proportional to the gate capacitance C g , whence the FET noise term to the equivalent noise charge Qn (Eq . (10), with b a = b) is 1 4abC ;a f(Cd +C g ) .
(12)
In order to minimize this noise term, the gate capacitance of an FET has to be equal to the detector capacitance . For a properly matched FET the series noise a then becomes a a 1/Cd .
(14)
Minimizing the FET noise term in the equivalent noise charge, Eq . (12), the contribution of a/R2 to the effective parallel noise was neglected . If this is taken into account, the FET gate capacitance C g has to be chosen somewhat larger than the detector capacitance Cd . The changes, however, which arise from this modified FET matching, are of less importance, especially for the values of interest here . Input capacitance Ci : Ci = Cd + Cg . With an FET matched to the detector one obtains Ci = 2Cd . 1 /f noise constant c : The constant c depends on the gate size and thus on the gate capacitance in the same way as constant a [12] . c a 1/Cg .
(15a)
For the case of CR-RC shapers having equal time constants r, for differentiation and integration, 1/f noise contributes to Q n with a term proportional to cCi [13] . This contribution is likewise minimized by the condition C g = Cd , yielding c a 1/Cd .
(15b)
46 3
J. Jochum et al. I Nucl. Instr. and Meth. in Phys. Res. 338 (1994) 458-466
The parallel noise constant b : The parallel noise consists of the thermal noise of the feedback and bias resistors Rf and R b as well as on the shot noise of the detector bias current Ib and the FET gate leakage current Ig. b = 2kT(Rf'+Rb')+e(Ib +Ig ) .
(16)
By lowering the operating temperature T of the feedback resistor, the bias resistor, and the input FET, these contributions to parallel noise can be reduced substantially. The parallel noise is then dominated by the bias current contribution . Provided tunnel junctions are equal, each of them requires the same bias current Ib. Hence, a detector consisting of M tunnel junctions connected in parallel requires a bias current MIb. Consequently, the parallel noise constant of a parallel junction detector is M times larger than for a single junction detector . M series connected tunnel junctions have to be biased with the current Ib of a single tunnel junction . Thus, each tunnel junction exhibits a shot noise b = eIb. The noise current source is parallel to the respective tunnel junction in the same way as the signal current is in Fig. lb. The corresponding noise power density at the preamplifier output is proportional to bIMZ (cf. Eq . (6)). As there are a number of M such noise current sources, this leads to a noise power density proportional to b/M. Compared to a single tunnel junction, the parallel noise constant of M series connected tunnel junctions is therefore reduced by a factor M. To give an example for typical noise contributions we refer to an aluminum tunnel junction as used in ref. 4, with a tunnel area of At = 1.6 X 10 -4 cm', exhibiting a capacitance of Ci = 0.8 nF (by parallel plate model) and a pulse length of rp = 50 ws . A properly matched FET for this tunnel junction is the NJ 3600 exhibiting a forward transconductance of gn, = 200 mfZ- ' (at a drain current of Ids = 20 mA). The result-20 V2 s. ing series noise (cf. Eq . (11)) is a = 2.6 X 10 noise constant of this FET is approximately The 1/f c = 5 X 10 -16 V2. For tunnel junctions of this area A t a dynamical resistance in the range of Ri >_ 50 kfl can easily be achieved . If such a tunnel junction is biased with a typical voltage of Vb = A/4 e, the bias current is about Ib = 1 nA (Fig . 5), causing a shot noise of b = 2 X 10 -28 AZ s, Eq . (17) . Thus, -r s = 19 ws and the two noise contributions to the equivalent noise charge Qn are: FET noise term 2bjs = 543 electrons (rms) and pulse noise term 2b a,p = 906 electrons (rms). We shall now evaluate the equivalent noise charges Qn for the different detector types. The quantities ail bj, bail cj, Tsi, and pj, specified by the index j, denote the values of a single "normal sized" tunnel junction
Fig. 5. Current-voltage characteristics of a tunnel junction with a dynamical resistance of Rj = 50 kf and a tunnel area of At = 1 .6 x 10 -4 cm2. The bias point has to be chosen at the largest possible dynamical resistance. A bias current in the range of Ib = 1 nA is typical for our aluminum tunnel junctions of this size . detector as defined in Eq . (8). Cj , Rj are the capacitance and the dynamical resistance of a normal sized tunnel junction, respectively . The corresponding values for the parallel connection or the series connection of M such tunnel junctions are compiled in Table 1. With the help of Eq . (10) and the values of Table 1, the minimal equivalent noise charges for the different detector types are compared : 4a jbaj 2Cj + 2b ai Tp = Qnj, single junction, Qn =
4a~b at 2MC, + 2Mbaj T p
= V1~~
Qnj,
= Y1~~
Qnj ,
parallel connection, M 4a ibat 2 V M + 2 M 7-1, series connection .
(18)
Since the equivalent noise charges Qt, are calculated under optimal conditions, they represent the lowest possible limits . The lowest equivalent noise charge Qn is obtained for a single junction detector . The parallel and series connections of M tunnel junctions exhibit noise charges which are in both cases larger by a factor of ~-M _. The best signal to noise ratio is obtained in the case of a single junction detector, in a situation where the amount of energy deposited is independent of detector size, e.g . in the case of X-ray detection. By contrast a different situation arises, if the amount of energy detected increases in proportion to the sensitive area of a tunnel junction detector, e.g. in the case of phonon detection [14,15]. In this situation the signal charge Qt
independent 1/f Dependence parallel adetector advantage independent energy noise detector quantities single the term can be noise junction 1noise Mi to connection to tunnel of to therefore estimate ratio junction connection however, charge conditions, parameters M ratio detector detector), than noise area noise matched easily with area film equivalent Tp, atunnel noise especially charge the affected tunnel or junction times charges nonvanishing absorbed type isAn p,junction series exhibits determining materials of for abut required taken ratios ratio, of ofsingle visualize of injunctions r,ratio the in the detector conclude of smaller (serial of the may junction aand increase employing the signal larger toas such by single aconnections detector this ifnoise of energy the the detector, detector into are the sensitive tunnel given well low ajunction detector increases the the be the parallel higher gap that junction, ato factor enhanced best the for comparison by detector account the turns junction equivalent of charges FET 1/f by detector ratio detector that, as element gaps noise aratio by junction equivalent Jochum quasiparticle normal parallel the A best afactor on asignal type detector detector signal [4] VMi/M noise, matching Eq out factor and detector ifof parallel of of in relations the +ratio the parallel reduces capacitance, do detector /M detector noise only A M tunnel type to Cg et(8) (cf proportion size serial Thus, parameters The or detector noise to trapping Mi/M not to albe tunnel qualitatively we energy noise VM, area has -'~25n by Table Inoise series With By or with has constants is for difficult Nucl change, trapping, junction consider influence between junction, junction vFM charge the in then serial for whence to charge, contrast, The and aan on compared junctions absorbed the terms depends lM compete serial junction detector an ratio Instr phonon 1)the ratio initially relative equivatounder tunnel equivK2ba of juncsame even FET compared of The and c/a the the pathe dethe has and an or or of A of isa /Mfor if i affects Phys the noise On determined = reducing FET =equivalent tunneling limited have tunnel to bias determined between (20-80) our length three term -QO/1/2ba(7s tothe be area noise reduce Q Res pulse /M signal etc noise of detector parallel ratio to +the bias noise both to noise aor resistors mainly ofTtThe aluminum different 338 initial contrary by If as junction baTp chance immediately will the optimal by to tunneling the length signal 7p ws, current time can an to the noise the effective ba term signal "back (1994) by influence ratio quasiparticle isthe be noise noise determined excess tunnel by At operating r,isexcess and signal the electronic be /M 7tshorter, detector and to =given charge elucidated of 7p charge the if signal tunnel 458-466 +shaping, parameters, present back is charge 19 illustrated tunneling" quasiparticle (cf tunnel ba, 7p) time thus quasiparticles ratio isQn parallel then density back the lifetime once charge quasiparticles, ws area aby of r,Eq tunneling might [16] types critical to junctions, (eqs temperatures involving pulse rt, see the increases noise, input by Qt our the tunneling, dominated trapping repeatedly they by in(16)) noise isThis given noise of sections the by signal and (10), of this eventually ismeans large The contribution energy like length parameter, quasiparticles whence have lifetime FET determined considering the shot situation ratio By would section equivalent both by ba (18)) the (cf pulse tunneling compared [41, Eq to quasiparticles tunneled, (cf 2the and by the reducing on resolution of noise pulse of Eq and itof noise quasipartilifetime 7, (4) Under dominate the leave the aEq tunneling length isthis feedback effective (8)) because isaand of 3) proper imporand of by length "back signal single often pulse (19)) noise stays time, ratio electo The way and the the the the the by 7, isT,is
464
J.
.
.
.
Meth. n
.
.
Table The Single Parallel Series
may detector alent tectors, signal to sensitive the single using lent to However, created signal with absorber To effect signal properly rameters are only One different if We is optimal obtained the sensitive tion trapping detection, parallel
K 1 1 M
Cd 2Ci 2MCi 2Ci
a ai ai Mai
.
. . . . . .
. .
.
.
.
. . .
4. junction The
on trapping individual
Rd Ri Ri MRi
.
b bi Mbi bi
ba bdi Mb,i bai
c ci ci/M Mci
the lifetime . The ratio fraction the condition tronic junction 7 sn Tp In 7p, quasiparticle (Tp signal noise still tant By and parallel would detector effective the The it charge . . to tunneling". confined geometry cles amplifies referred then T, . tunnel pulse and 7,7t 7, 7,,
P Pi Pi Pi
Tsi T,i 'r'i
.
.M,a .
.
.
.
(19)
.
; . .
.
.
.
.
.
.
.
.
.
.
: without with
.
(20)
J. Jochum et al. I Nucl. Instr. and Meth. in Phys. Res. 338 (1994) 458-466
Back tunneling was considered as an amplification process for tunnel junction detectors increasing the signal charge Qt by a gain factor gbt 71 gbt =1 + - . Tt
(21)
This gain 9bt, however, is due to a longer pulse length, Eq . (20), rather than a higher current, giving at the same time rise to a higher noise charge . The gain G bt associated with the signal to noise ratio is somewhat smaller : G bt
~
7",
(1 +
7'
SI T t ) (1 + Tt
gbt( 1 +
7t
(22a)
with Ts ,=TS T1/(Ts +T,) . We thus obtain gbt > Gbt >
gbt
(22b)
If the pulse noise term b aTP of the equivalent noise charge dominates, the noise charge is considerably increased by a longer pulse length . In this case, only the square root of the signal charge gain gt,t enters in the signal to noise ratio G bt = gbt . Back tunneling on the one hand maximizes the signal to noise ratio, while on the other hand it limits the maximum counting rate . We will nevertheless assume in the following the presence of back tunneling with TP =T1. At the usual low operating temperatures, the bias current of our tunnel junction detectors is caused by imperfections of the tunnel barrier and of the film quality . The bias current is then inversely proportional to the dynamical resistance R~ 1 of a tunnel junction (Fig . 5). Consequently we obtain for the effective parallel noise ba aA t /~f7t,
(23)
where Eq . (9) was employed. The series noise a for a properly matched FET is inversely proportional to the capacitance C j of the tunnel junction and hence to the tunnel area A t . a a 1/At .
(24)
Eventually, four parameters are decisive for the contribution of electronic noise to the signal to noise ratio of a tunnel junction detector : - the quasiparticle lifetime - the tunneling time Tt , - the quality factor of the tunnel junction &I = Rj/R , - the tunnel area A t . For the signal to noise ratio `Wsn (cf . Eq . (19)) we
465
obtain Qt
2bT P a if pulse noise term dominates, rP >> Ts Qt
,r2--b,.
a T,
V
A, 7, Tt
if FET noise term dominates, TP << 7s . (25) In our typical experimental situation, the pulse noise term dominates and T, enters into both, the noise and the signal charges (cf. Eq . (19) rP = r,), whence the signal to noise ratio becomes proportional to T, . If experimental progress permits a lower effective parallel noise ba or a shorter pulse length rP , the FET noise term might dominate. In this case -~Fsn is directly proportional to the quasiparticle lifetime r,, because T, enter only in the signal charge . The effective parallel noise ba enters differently into the FET noise term bars and the pulse noise term baTP , thus the quality factor & becomes more important in case the pulse noise term dominates . If the tunnel area A t is enlarged, the detector capacitance and consequently the FET noise term increases . Simultaneously the dynamical resistance reduces requiring a higher bias current, and thus leading to a higher effective parallel noise. In both cases the signal to noise ratio becomes therefore proportional to 1/ A t , necessitating a small tunnel area . On the contrary there exist lower limits for the tunnel junction size, due to the quasiparticle pair recombination, which would lead to a nonlinear energy response of the detector [5] .
5. Summary The signal to noise ratio for tunnel junction detectors was calculated depending on detector type (single, serial, or parallel junction detectors). The influence of 1/f noise was shown to depend upon pulse length and in particular on the dynamical resistance of the tunnel junctions at the bias point . The dynamical resistances required to suppress 1/f noise are technically feasible . The signal to noise ratio, which can be reached under optimal conditions such as bias and feedback resistors operated at low temperatures, an input FET matched to the detector, and optimal pulse shaping, was com-, pared for the cases of a single tunnel junction and a parallel or a series connection of M tunnel junctions . For a fixed signal charge, as prevailing in the case of X-ray detection, a single junction detector yields the
466
J. Jochum et al. /Nucl. Instr. and Meth . in Phys. Res. 338 (1994) 458-466
best signal to noise ratio. The ratio becomes worse by a factor rM for a parallel or series connection of M tunnel junctions as compared to a single tunnel junction. If the signal charge increases with the sensitive area, as in the case of phonon detection, a parallel or serial junction detector or a trapping detector would be the better choices.
Back tunneling increases the signal charge but at the same time increases also the noise charge to some extent . Consequently, the gain in signal to noise ratio by implementation of back tunneling is smaller than anticipated by the gain in signal charge alone. To maximize the signal to noise ratio, a long quasiparticle lifetime rl, a short quasiparticle tunneling time
Tc , a high quality factor ci, and a small tunnel area A t
are essential. Under the conditions prevailing in our experiments, the signal to noise ratio depends on the square root of all these parameters . If the quality
factor is very large and the parallel noise is small due to a low bias current, the FET series noise matters and thus matching of the FET to the detector is crucial. Acknowledgement This work has been supported by the Bundesministerium für Forschung und Technologie. References [1] L. Brogiato, D.V. Camin and E. Fiorini (eds.), Proc. Low Temperature Detectors for Neutrinos and Dark Matter 111, Gran Sasso, L'Aquila, 1989 (Editions Frontières, Gif- sur-Yvette, 1990). [2] A. Barone, R. Cristiano and S. Pagano (eds .), Proc . X-ray Detection by Superconducting Tunnel Junctions, Naples, 1990 (World Scientific, Singapore, 1991).
[3] N.E . Booth and G.L . Salmon (eds.), Proc. Low Temperature Detectors for Neutrinos and Dark Matter IV, Oxford, 1991 (Editions Frontières, Gif- sur-Yvette, 1992). [4] H . Kraus, F. v. Feilitzsch, J. Jochum, R.L. M6ssbauer, Th . Peterreins and F. Prèbst, Phys . Lett . B 231 (1989) 195. [5] J. Jochum, H. Kraus, M. Gutsche, B. Kemmather, F. v. Feilitzsch and R.L. M6ssbauer, Ann. Phys . 7 (1993) 611. [6] D.J . Goldie, N.E. Booth and G.L . Salmon, Proc. Low Temperature Detectors for Neutrinos and Dark Matter IV, Oxford, 1991, eds. N.E . Booth and G.L . Salmon (Editions Frontières, Gif- sur-Yvette, 1992) p. 245. [7] M. Kurakado, A. Matsumura and T. Takahashi, Proc . X-ray Detection by Superconducting Tunnel Junctions, Naples, 1990, eds. A. Barone, R. Cristiano and S. Pagano (World Scientific, Singapore, 1991) p. 77. [8] J.P . Maneval, S. Labov, R.W . Bland, S.C . Dickson, K. Laws, R.T. Johnson, J. Lockhart, R. Martin, M.W . Simon, D.A . Stricker and R.M . Watson, Proc . Low Temperature Detectors for Neutrinos and Dark Matter IV, Oxford, 1991, eds. N.E . Booth and G.L . Salmon (Editions Frontières, Gif- sur-Yvette, 1992) p. 291 . [9] E. Gatti and P.F . Manfredi, La Rivista del Nuovo Cimento 9(3) (1986) . [10] M . Tsukuda, Nucl . Instr. and Meth . 14 (1961) 241. [11] R.S .C. Cobbold, Theory and applications of field effect transistors (Wiley/Interscience, New York, 1970). [12] P.O . Lauritzen, Solid-State Electrons 8 (1965) 41 . [13] E. Kowalski, Nuclear Electronics (Springer-Verlag, Berlin, Heidelberg, New York, 1970). [14] Th . Peterreins, J. Jochum, F. Prèbst, F. v. Feilitzsch, H. Kraus and R.L. M6ssbauer, J. Appl. Phys . 69(4) (1991) 1791 . [15] D.J . Goldie, Proc. X-ray Detection by Superconducting Tunnel Junctions, Naples, 1990, eds. A. Barone, R. Cristiano and S. Pagano (World Scientific, Singapore, 1991) P. 98 . [16] F. Prèbst, H. Kraus, Th . Peterreins and F. v. Feilitzsch, Nucl . Instr. and Meth . A 280 (1989) 251 .