Noise temperature limit of a superconducting hot-electron bolometer mixer B. S. Karasika) and A. I. Elantiev Department of Physics, Moscow State Pedagogical University, Moscow 119435, Russia
~Received 18 July 1995; accepted for publication 28 November 1995! A theoretical analysis for the noise temperature of a low-temperature hot-electron superconducting mixer has been presented. The contribution of both Johnson noise and electron temperature fluctuations has been evaluated. The theoretical limit of the single-side band noise temperature of the device due to the intrinsic noise mechanisms has been estimated to be as low as 20– 40 K, depending on the film material. The corresponding noise bandwidth can be as large as 2–3 GHz, for a device with the electron-phonon cooling mechanism. An improvement of the sensitivity is not necessarily followed by a decrease of the mixer conversion gain bandwidth. © 1996 American Institute of Physics. @S0003-6951~96!02406-8#
A rapid progress in the development of low noise submillimeter heterodyne receivers has made researchers turn to the problem of terahertz frequencies, where commonly used superconductor-insulator-superconductor ~SIS! mixers, so far, have not been demonstrated to perform well. Today, the most promising device which can compete with the SIS mixer at terahertz frequencies is a superconducting hotelectron bolometer ~HEB! mixer.1 A new approach has been proposed2 for the HEB mixer device development where a diffusion cooling rather than electron-phonon relaxation should limit the bandwidth. Currently, hot-electron mixer devices exploring both electron-phonon relaxation in Nb ~Ref. 3! and NbN ~Ref. 4! as well as diffusion cooling in Nb ~Ref. 5! are being developed. The double-side band ~DSB! noise temperature of a HEB mixer have been measured to be 80– 150 K at 20 GHz,3 about 1000 K at 100 GHz,4 and 560 K at 533 GHz.5 The improvement of the HEB mixer sensitivity is raising the interest in the limiting parameters of such devices. A theory of a HEB mixer conversion gain was given for the first time for an InSb mixer device6 and is applicable for a superconducting HEB device as well.3 The noise temperature of the mixer device was estimated using the numerical modeling.1 As low as 40 K the single-side band ~SSB! mixer noise temperature, along with a 40 MHz bandwidth, has been predicted for a Nb device. Since then, NbN has been used for the development of HEB mixers. Here, we present an analytic theory for the noise performance of the superconducting HEB mixer, applicable for different materials. The HEB mixer device is a thin superconducting strip on a dielectric substrate, cooled below its critical temperature T c . Different external factors, e.g., magnetic field, transport current, or local oscillator ~LO! power may partially destroy the superconductivity and create a resistive state where the film resistance depends on the electron temperature u. In the low-temperature limit, when the electron specific heat, c e , is much larger than the phonon specific heat c p , the electron temperature relaxation is governed by a single time constant, t u . 7 For a relatively long film strip the time constant is t u 5 t e-ph1 t esc e /c p , where t e-ph is the electron-phonon ena!
Present address: Jet Propulsion Laboratory, Pasadena, CA 91109-8099. Electronic mail:
[email protected]
Appl. Phys. Lett. 68 (6), 5 February 1996
ergy relaxation time, t es is the phonon escape time. When the film is very thin ~d t e-ph . For short bridges,2 with the length L! AD t e-ph, the value for t u is t diff >(L/4) 2 /D (D is the electron diffusivity!. For the lumped bolometer the change of the resistance DR caused by the radiation power can be expressed through the change of the electron temperature Du using the phenomenological parameter ]R/]u: DR5
S D
]R Du. ]u
~1!
The particular physical mechanism responsible for the relationship given by Eq. ~1! depends on the origin of the resistive state. If the resistivity is caused by a vortex flow when u is close to T c or when an external magnetic field is applied,8 then ]R/]u does not seem to demonstrate any time dependence. For a bolometric device Johnson noise and noise caused by the thermal fluctuations of the electron temperature are the two main noise sources. If a Fermi-like electron distribution function can be introduced for the nonequilibrium resistive state, then the intrinsic noise sources are given by the ordinary formulas for the equilibrium state with the temperature substituted by u. For example, in such a case the Johnson noise spectral density agrees with that obtained from kinetic theory9 within 2% accuracy. A simple way to derive the expressions for the noise temperature of a HEB mixer is to use the characteristics of a corresponding HEB detector. The mixer conversion efficiency, h~v!, when the device is terminated by the load R L is1,2,6
h ~ v ! 52 a S 2 ~ v ! P LOR 21 L ,
~2!
where P LO is the incident LO power, a is the coupling factor of radiation to the mixer, S~v! is the voltage responsivity of the HEB detector. The noise equivalent power ~NEP! is NEP~v!5e~v!/S~v!, where e~v! is the noise voltage across the load. Then the mixer SSB noise temperature is T M~ v !5
e 2 ~ v ! NEP2 ~ v ! 1 5 , k Bh~ v ! R L 2 a k B P LO
~3!
where k B is the Boltzman’s constant. The noise mechanisms in superconducting bolometers have been thoroughly studied during many years ~see, e.g.,
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© 1996 American Institute of Physics
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Ref. 10! and a rigorous theory of the bolometer sensitivity was given.11 For Johnson noise the NEP is given by NEPJ 5 a 21 A4k B u P dc•
U
U
Z ~ 0 ! 1R • u 11 j v t u u . Z ~ 0 ! 2R
~4!
Here, P dc is the Joule power dissipated in the device, Z~0! is the dc differential resistance, and R is the dc resistance. The difference between Z~0! and R arises because of self-heating of the bolometer. Assuming that Z( v ) / v →` 5R one can express Z~0! in terms of the self-heating parameter C [I 20 ( ] R/ ] u )/G( u ) 8,11,12 @G~u! is the thermal conductivity between the electrons and phonons, I 0 is the bias current#: Z ~ 0 ! 5R @~ 11C ! / ~ 12C !# .
~5!
From Eqs. ~3!–~5! the Johnson noise contribution is
FIG. 1. A broken-line model of the superconducting transition.
T JM ~ v ! 52 v C 22 a 22 ~ P dc / P LO!~ 11 v 2 t 2u ! .
~6!
If u'T the NEP due to thermal fluctuation noise is
10,11
2 NEPTF 54k B u 2 G ~ u ! a 21 .
~7!
In the nonequilibrium state when the HEB mixer is pumped with an intense LO power and u 2 @T 2 the fluctuations of the heat sink temperature T can be neglected and the NEP is 2 52k B u 2 G ~ u ! a 21 . NEPTF
~8!
The following spectrum of the electron temperature fluctuations can be derived using a Langevin technique.13 ~9! ^ D u 2 & v 52k B u 2 G ~ u ! G * 22 ~ u !~ 11 j v t * ! 21 . Here, G*~u!5G~u!~12C! is the effective thermal conductivity and t * u 5 t u /(12C) is the apparent time constant
(R L →`). In combination with Eq. ~1! and the bolometer 21 21 voltage responsivity S( v )5 a I 21 (11 j v t * 0 C(12C) u) it gives Eq. ~8!. Consequently, the thermal fluctuations contribution is 2 2 T TF M 5G ~ u ! u / ~ a P LO ! .
~10!
Equations ~6! and ~10! suggest that whereas the T JM value increases at intermediate frequencies ~IF! larger than (2 p t u ) 21 the T TF M value does not depend on the IF. Thus the equivalent noise bandwidth of the mixer device is given by J D f 5 ~ 2 p t u ! 21 A11T TF M /T M ~ 0 ! .
~11!
temperature point u * 5T c 2DT c /2, just below the superconducting transition edge ~see Fig. 1!. The LO power is then given by P LO5 a 21 AV(T nc 2T n 2nT n21 DT c ). The current– c voltage ~IU! characteristic for this case is shown in Fig. 2. Finally, we obtain T JM 5
a 21 nT nc DT c , n T c 2T n 2nT n21 DT c c
T TF M5
a 21 nT n11 c . n n T c 2T 2nT n21 DT c c
~13!
The lowest noise temperatures can be obtained if T nc @T n (T JM 5nDT c / a , T TF M 5nT c / a ). Then the noise bandwidth of the device is given by D f 5 ~ 2 p t u ! 21 A11T c /DT c .
~14!
T TF M
The minimum value does not depend on the superconducting transition width, if DT c is very small. For a particular case a P LO5 P dc5G(T c )DT c /4 Eq. ~10! gives T TF M 54 a 21 T 2c /DT c , which is similar to the corresponding formula of Ref. 2. The conversion gain can be estimated using a formula3,6
h ~ v ! 52 a 2 C 2 3
P LO RR L 1 2 P dc ~ R1R L ! @ 11C ~ R2R L ! / ~ R1R L !# 2
1 , 11 ~ v t * ! 2
~15!
which gives for the ‘‘broken-line transition’’ model
For further analysis it is useful to assume that the R(T) characteristic of the bolometer has a broken-line shape2 ~see Fig. 1!. With help of a steady-state heat balance equation AV ~ u n 2T n ! 5 P dc1 a P LO
~12!
we can obtain G( u )5nAV u n21 , where V is the film volume, A and n are the material dependent constants, e.g., for Nb A '104 W cm 23 K24 and n54.8,14 Since u 'T c and R 'R n /2 then C52 P dc /(nAVT n21 DT c ), where DT c is the c superconducting transition width. An increase of both P LO and C @Eqs. ~6! and ~10!# leads to a decrease of the noise temperature. At the same time P dc decreases since both u and T are fixed @see Eq. ~12!#. The most favorable regime of operation is when C51, which corresponds to P dc5nAVT n21 DT c /2. To ensure this condic tion the applied LO power has to heat the device up to the 854
Appl. Phys. Lett., Vol. 68, No. 6, 5 February 1996
FIG. 2. Unpumped and pumped @C51, Z~0!5`# IU characteristics. B. S. Karasik and A. I. Elantiev
TABLE I. Theoretical limits for the noise performance of HEB mixers.
Material
Tc @K#
DT c @K#
n
T JM @K#
T TF M @K#
tu @ns#
Df @GHz#
Nb NbN
5 9
0.1 0.5
4 3.6
0.4 2.2
22 41
0.5a 0.2b
2.3 3.5
a
These data are taken from Ref. 14 assuming the electron diffusivity to be 1 cm2/s. b We use the experimental mixing data ~Refs. 4 and 15!.
h ~ 0 ! 5 ~ a /n !~ T c /DT c !~ R L /R ! .
~16!
The apparent time constant t* for the conversion gain differs from t u due to the self-heating caused by the IF current flowing through the load:3,8
S
t * 5 t u 11C
R2R L R1R L
D
21
.
~17!
An increase of the ratio R L /R can ensure a very large conversion gain, h~0!.1, but along with this the gain bandwidth can degrade significantly. However, a gain as large as h~0!>1 can be obtained even if R L 5R. In this case t * 5 t u . This is important if one wants to increase the gain bandwidth and, hereby reduce the noise contribution of an IF circuit at higher intermediate frequencies. The contribution to the noise temperature from the mixer itself does not depend on the IF load. The estimations from Eqs. ~13! and ~14! for materials used for fabrication of hot-electron mixers and a51 are presented in Table I. One can see that an excellent noise performance can be expected for both Nb and NbN superconducting HEB mixers. In practice, however, there is a physical phenomenon ~normal domain formation! preventing such a good mixer performance. As a result, the necessary LO power is determined by the smaller effective thermal conductivity for the slower process as given by Eq. ~12!. The fast process, responsible for mixing at the higher IF frequency, is thereby ‘‘underpumped.’’ For instance, for a NbN HEB mixer made of ;20 nm thick film, when pumping was done with 300–350 GHz radiation, a uniform resistive state appeared only when T'7.5– 8 K whereas T c >9 K. 15 However, even in this case the thermal fluctuation noise dominated, giving about 400 K SSB contribution. The Johnson noise contributed as little as 40 K. The influence of the domains can be hopefully reduced at terahertz frequencies
Appl. Phys. Lett., Vol. 68, No. 6, 5 February 1996
where the absorption of radiation is more uniform. In conclusion, we have presented a theoretical analysis of the HEB mixer noise performance. It has been shown that the mixer noise temperature is dominated by the thermal fluctuation noise, which in turn can be very low for an optimized mixer device. The effective noise bandwidth of the mixer can be much wider than the conversion gain bandwidth, determined by the inverse electron temperature relaxation time. The latter circumstance can relax the narrow bandwidth problem of the devices, where the relaxation is governed by the electron-phonon interaction, and make them more competitive. The authors thank D. E. Prober, K. S. Yngvesson, E. Kollberg, and H. Ekstro¨m for stimulating discussions. This research was made possible in part by Grant No. NAF000 and Grant No. NAF300 from the International Science Foundation and Russian Government. The support from the Russian Program on Condensed Matter ~Superconductivity Division! under Grant Nos. 93169 and 94043 is also acknowledged.
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