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the cold - electron bolometer is a broadband bolometric detector where the only fundamental limit to the frequency response comes from the spectral transmission of the absorber . in order to tune the detector to respond to a certain frequency band ,
the absorber is coupled to an antenna structure which defines the frequency response.the original , and still most common , cold - electron bolometer design involved a normal - metal absorber with tunnelling contacts to superconducting leads ; see , for example , the work of kuzmin.@xcite this structure has also been used in the so - called _ microrefrigerator _ devices to cool electrons to below the lattice temperature .
savin et al.@xcite showed that if the normal - metal absorber is replaced by a degenerately doped , electrons can be more efficiently cooled due to the reduced electron - phonon coupling in such a material compared to a metal.@xcite it has been shown that introducing strain into the silicon lattice can further reduce this coupling , allowing microrefrigerator - type devices to achieve lower electron temperatures compared to unstrained silicon.@xcite    based on this lower electron temperature achieved using strained silicon it is expected that the responsivity of cold - electron bolometer based on strained silicon should be higher than a comparable detector utilising unstrained silicon . a common advantage that both strained- and unstrained - silicon cold - electron bolometers share , compared to those using a normal - metal absorber , is that they do not require the fabrication of an insulating layer , since the naturally forming schottky barrier may be used instead .
here we explore the performance of cold - electron bolometers utilising degenerately - doped silicon as the absorbing element .
we take two devices : the _ control _ device using doped silicon , and a device where the doped absorber is strained by a layer of .
we compare these devices by characterising them in a dark environment and also when illuminated .
for both device the doping concentration was @xmath3 . in order to measure the noise of the devices themselves
we present a novel concept of cross - correlating the outputs of two matched jfet amplifiers to remove the uncorrelated amplifier noise leaving only the device noise .
the cold - electron bolometer consists of an absorbing island with tunnelling contacts to superconducting leads either side of the absorber . in the absence of any bias ,
the fermi energy is the same in the absorber and the superconducting contacts and thus carriers can not tunnel between the two .
however , in the presence of an external bias , applied across the structure , the fermi level in the superconductor can be moved relative to that of the absorber , allowing tunnelling from the absorber to the vacant states above the superconductor s energy gap .
this structure is described to greater detail by gulobev and kuzmin , and brien.@xcite    the current , @xmath4 , flowing through each of the symmetric junctions is given by : @xmath5\,\mathrm{d}e\ , ,      \label{eqn : iv}\end{aligned}\ ] ] where @xmath6 is the normal - state resistance , due to tunnelling , of an individual tunnelling barrier , @xmath7 is half the superconducting bandgap , @xmath8 is the voltage across both junctions due to tunnelling ( i.e. it does not include voltages due to the resistance of the absorber or any leakage across the junction ) , and @xmath9 is the fermi distribution of electrons at temperature @xmath10 . this tunnelling current either increases or decreases the average energy of the electrons in the absorber.this change in energy can be expressed as a tunnelling power , @xmath11 , given by : @xmath12\,\mathrm{d}e\ , ,      \label{eqn : ptun}\end{aligned}\ ] ] where @xmath13 is the temperature of the electrons in the absorber and @xmath14 is the temperature of the normal - state electrons in the superconductor .
a negative value of @xmath11 represents a reduction in the energy of the electrons in the absorber and thus corresponds to electron cooling . in a cold - electron bolometer
it is this tunnelling power which cools the absorber .
the temperature of the electrons in the absorber is also affected by the weak thermal link to the phonons .
this thermal power , @xmath15 , is given by : @xmath16 where @xmath17 is the area of the absorber , @xmath18 is the phonon temperature , @xmath19 is a material constant that has been measured to be @xmath20 and @xmath21 for the control and strained silicon materials,@xcite and the power @xmath22 have been found@xcite to be @xmath23 .
the responsivity , when current biased , @xmath24 , of such a detector has been derived by golubev and kuzmin@xcite and is given by : @xmath25    the dark noise - equivalent power is comprised of several terms , corresponding to various thermal and electronic process , and has been derived , for a current - biased device , by golubev and kuzmin@xcite to be : @xmath26 where @xmath27 is the noise voltage introduced by the readout , @xmath28 is boltzmann
s constant , @xmath29 is the heat - flow noise , and @xmath30 is the current - shot noise . in the presence of optical power ,
the noise - equivalent power can be further degraded by noise introduced by the absorption of photons .
this photon - noise limit to the noise - equivalent power is given by the well - known equation:@xcite @xmath31 where @xmath32 is planck s constant , @xmath33 and @xmath34 are the frequency and power of the radiation incident upon the detector , and @xmath35 is the optical bandwidth .
as is often the case , the internal noise of device under test here was lower than the input referred noise of the amplifier used ( @xmath36 in this case ) . to combat this
we have used a novel solution of feeding the output of the device to two matched jfet amplifiers . the output of these two amplifiers
was then fed to a data - acquisition system and cross - correlated in software .
this treatment has the effect of removing the amplifier , which is uncorrelated , and leaving only noise present in the system prior to the splitting of the signal wiring .
this approach is illustrated in fig .
[ fig : crosscol_flow ] .
since the noise of each amplifier is random there is a finite chance of correlations between the two for any particular acquisition , due to this , in order to sufficiently reduce the amplifier noise in the final spectrum it is necessary to average multiple cross - correlations .    in reality there will always be a weak level of correlated amplifier noise in the final measurement due to stray capacitances in the system which provide a route for the amplifier noise to couple to the device .
this low level of correlated amplifier noise presents itself as a noise floor below which further averaging offers little improvement .
both the control and strained devices have been made and fabricated to a common design and process .
for both devices the absorber is a @xmath37 tall mesa formed from etching n^++^-doped silicon ( @xmath38 ) and the contacts are sputtered aluminium . for the strained device
the doped silicon is strained by a buffer layer between the absorber and the silicon substrate .
the sputtered aluminium is etched to form a twin - slot antenna , designed to respond to 160-ghz radiation .
both devices have been tested in two scenarios .
firstly , dark ( looking at a blank at @xmath18 ) tests were performed to verify the formation of a schottky contact and to baseline device performance .
the devices were then illuminated by radiation from external black - body sources ( eccosorb ) with temperatures of @xmath39 and @xmath40 .
radiation was coupled to the detector via a pair of back - to - back horns and a hemispherical silicon lens which focussed the radiation on to the twin - slot antenna .
the power incident on the detector was limited by both a low - pass filter , with an edge frequency of @xmath41 , and by a @xmath42 limiting aperture .
the detector was biased via a pair of @xmath43 resistors and readout via cross - correlated jfet amplifiers ( as discussed above ) .
all optical measurements were performed at a bath temperature ( @xmath44 ) of @xmath45 .
+    to compare the strained- and unstrained - silicon cold - electron bolometers , both current - voltage ( @xmath4-@xmath8 ) characteristics as well as noise spectra of the devices have been measured , these measurements have been performed dark and in the presence of optical power .
@xmath4-@xmath8 curves are presented in fig .
[ fig : iv ] . for both devices
there is a clear voltage response to increasing the source temperature from @xmath46 ( dashed lines ) to @xmath40 ( dotted lines ) and this can be seen to a greater extent when comparing these curves to the dark data ( solid lines ) . from fig .
[ fig : rderiv ] it can be also seen that the unstrained device ( blue curves ) exhibits a lower normal - state ( @xmath47 ) resistance along with greater sub - gap leakage .
this is attributed to lower - quality schottky contacts in this device along with the absorber resistance being lower in this material ( @xmath48 compared to @xmath49 in the strained detector ) .
+    the electron temperature for these data has been computed by fitting the data to eq .
[ eqn : iv ] . using the electron temperature at zero bias , along with eq .  [ eqn : pe - ph ] , it is possible to calculate the power absorbed within the silicon in each scenario . for the control device the absorbed power was @xmath50 for the 77-kelvin source and @xmath51 for the 300-kelvin source . for the strained detector
the absorbed power was @xmath52 for the 77-kelvin source and @xmath53 for the 300-kelvin source .
the difference between these powers for the two detectors is due to alterations in the optical setup and in both measurements we believe there to be a substantial background power due to the cryostat optics and stray light ; these values for the absorbed power are supported by our noise modelling , an example of which is shown in fig .
[ fig : noise ] . from the computed electron temperature it is also possible to calculate the responsivity using eq .
[ eqn : responsivity ] , the results of this are shown in fig .  [ fig : responsivity ] .    fig .
[ fig : responsivity ] shows that when the incident power is increased there is an overall drop in the responsivity of the detector ( the difference between the dashed and dotted lines ) , this is attributed to an overall increase in the electron temperature for the higher incident power . if the two curves for the 77-kelvin source are examined ( dashed lines in fig .
[ fig : responsivity ] ) , then the difference in peak responsivity is a factor of @xmath54 .
this value is very close to the difference in the @xmath19 parameter for the two materials ( @xmath55 ) .    from the responsivity , and by using eq .
[ eqn : darknep ] and eq .
[ eqn : photonnep ] , it is possible to calculate the noise - equivalent power for the two detectors .
this has been found to be @xmath56 for the control device and @xmath57 for the strained device , both of these values are found when observing the 77-kelvin source . in the case of the control device the limiting noise source , across the entire measurement , was the amplifier noise remaining after cross correlation ( increased by the poorer responsivity ) .
for the strained device the noise - equivalent power was photon - noise limited for biases corresponding to voltages in the range @xmath58 ( as shown , for the 77-kelvin source , in fig .
[ fig : noise ] ) , outside of this range the noise - equivalent power was limited by a combination of tunnelling and the amplifier noise .
the introduction of a straining layer to a silicon cold - electron bolometer has increased the detector responsivity by a factor of @xmath59 .
this increased responsivity can be attributed to the weaker coupling between the electrons and the phonons in the strained material . in terms of sensitivity
the limiting noise source for the unstrained device was the readout , whereas in the strained detector the sensitivity was photon noise limited , afforded by the improved responsivity lowering the amplifier noise contribution . taking the strained - silicon device we achieve a minimum noise - equivalent power of @xmath2 at optimum bias .
this work has been financially supported by the stfc through grant st / k000926/1 , the epsrc through grant numbers ep / f040784/1 and ep / j001074/1 , and the academy of finland through grant 252598 .
+ information on how to access all data supporting the results in this article can found at cardiff university data catalogue at doi : http://dx.doi.org/10.17035/d.2016.0008254680[10.17035/d.2016.0008254680 ] .
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we describe the optical characterisation of two silicon cold - electron bolometers each consisting of a small ( @xmath0 ) island of degenerately - doped silicon with superconducting aluminium contacts . 
 radiation is coupled into the silicon absorber with a twin slot antenna designed to couple to 160-ghz radiation through a silicon lens . 
 the first device has a highly - doped silicon absorber , the second has a highly - doped strained - silicon absorber . using a novel method of cross - correlating the outputs from two parallel amplifiers we measure noise - equivalent powers of @xmath1 and @xmath2 for the control and strained device respectively when observing radiation from a 77-kelvin source . in the case of the strained device , 
 the noise - equivalent power is limited by the photon noise .