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Optoelectronic Devices and Properties
to the critical current density based on quasiparticle tunneling by Ambegaokar et al.
(Ambegaokar & Baratoff, 1963)-(Ambegaokar & Baratoff, 1963). We have found out that
the normal state resistivity of the nanowire is about 200 μΩ-cm, which is the range of
granular tunneling range at 20 K regime as discussed in (Kampwirth & Gray, 1981)-(Gray
et al., 1985).
2 Microwave nonlinear surface resistance Zs: NbN has been long employed in microwave
applications, primarily useful for making transmission lines and resonators. The power
handling capability of such devices has been a topic for many decades (Chin et al.,
1992)-(Boin & Safa, 1991). The nonlinear dependency of the surface resistance has been
attributed to the granular nature of NbN thin films (? ), (Chin et al., 1992)-(Boin & Safa,
1991), that has been prepared by the RF sputtering technique. The dependency has been
modeled on the basis of tangential magnetic field that is parallel with the superconducting
surface. The model involves magnetic field penetration depth. This does not seem
applicable to our case directly, because the DUT acts only as a lumped load; but the
dissipation of microwave power by granular boundary remains the same mechanism to
account for the nonlinearity.
Additionally, we also calculated the phase-slip induced I-V relationship at the transition from
the superconducting state to the normal state when the temperature increases across the
transition temperature Tc. The exponential increase of resistance indicates the pronounced
existence of phase-slip centers in the nanowire meander-line. We suspect that these phase
slip centers stem from the granular boundaries during the formation of thin film. It is well
known about the process of sputtering thin films. Some seeds of metallic grains expand
and gradually merge into one continuous film. Thus, we conclude that the superconducting
nanowire is patterned on such thin film and its weaklink effect has to be considered in almost
every testing, such as dc I-V and RF/microwave measurements.
Thus, we heuristically conclude that the strong-weaklink network (? ), or inhomogeneities
of the order parameter in one dimension superconducting nanowire, is responsible for both
nonlinear kinetic inductance and resistance. The amplitude of complex order parameter has
been reduced on average, allowing the G-L equation to be valid in a wider temperature range,
e.g. at 0.5 Tc.
We have developed a relation of current bias ratio with respect to kinetic inductance Lk for a
clean, uniform bulk superconducting 1-D wire, with the absence of magnetic field, in (Yan et
al., 2007), and the analytical solution reads
Lk( β) =
Lk 0
√
(5)
1− β 2
1 − 4 cos2
π − 1
3
3 arctan
β
where Lk 0 is the zero bias kinetic inductance. It is device dependent, and can be measured via
the method in this paper when the setting dc bias to zero. β = Ib IGL is the ratio of dc bias
c
current Ib to that of bulk critical current IGL
c
. Since the order parameter has been reduced due
to the strong weaklink network, the (5) is valid for temperature at 4.2 K. However, attention
has to be paid to the fact that IGL
c
Ic, where Ic is defined as the critical current in the context
of SNSPD. In terms of thin film quality, or QE of a given SNSPD in the literature, we substitute
β with βb = γβ, in which γ is a figure of merit of the thin film quality, defined as γ = Ic , and IGL
c
βb = Ib .
Ic
Investigating Optoelectronic Properties of the NbN
Superconducting Nanowire Single Photon Detector
275
Parameter SNSPD # 11 SNSPD # 22
Jc ( A/ m 2)
6.2 × 1010
3.7 × 1010
Lk (nH) 3
200
230
ρ ( μΩ · cm)
195
214
λL (0) (nm)
383
411
λGL (0) (nm)
400.7
407.9
λ (0) (nm)
490.6
499
ξ 0 (nm)
6.18
5.91
ltr (nm)
3.97
4.00
κGL
70.88
73.45
1Device QE is 2%
2Device QE is 0.2%
3Extracted at T = 4.2 K with Tc = 10.4 K for all calculations
Table 2. Computed superconducting material parameters
Note that technically IGL
c
can not be measured directly, but can be deduced through the ratio
of Lk( βb) when β
L
b
1 based on the relationship of (5). Evidently, the highest γ can be close
k 0
to 0.6 at temperatures of 4.2 K . It would become relatively higher when the temperature is
decreased, but we anticipate that γ will saturate at some point due to the possibility that the
weaklink will limit Ic from going close to IGL
c
. For two of our DUTs, the higher quality one
has γ = 0.5; and γ = 0.4 for the lower one.
Finally, the effort to account for nonlinear resistance is more complicated, because it involves
combinations of material, processing, electrical and thermal physics. The difficulty is added
because of the quasiparticles are involved in the presence of RF field. The strong fluctuations
of Rs result not only from the quasiparticle generation and recombination process, but also
from their tunneling through the weaklinks. We think that detailed discussions are much
beyond the scope of this thesis, and we will address this topic in the future works.
5. NbN superconducting thin film properties
We treat superconducting nanowire devices as a nonlinear load of a 50 Ω transmission line.
The incident RF/microwave power (ac current) is weakly applied. The reflected power is
measured and analyzed as discussed in the preceding sections. Series of experimental results
to measure kinetic inductance Lk to study the nonlinearities stemming from dc bias by one
port S-parameter measurement. The weak ac current was used (less than 1 μ A) as a probe for
all measurements.
By knowing the kinetic inductance of the superconducting nanowire via the method described
in the preceding section, we are able to obtain some insights to the material properties
of the superconducting parameters that is prepared for making NbN SNSPD. After post
processing the measured S-parameter, we are able to retrieve London penetration depth,
coherence length, and very importantly the first and second critical magnetic field Hc 1 and
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Optoelectronic Devices and Properties
Hc 2. However, the second critical magnetic field intensity can not be simply computed for
NbN thin film, especially for the ultra thin film (e.g. 4 nm).
The extraction of values of the penetration depth λ is based on the relations in Ref. (Orlando
et al., 1979). The normal state electron specific heat γ 0 = 2.1 × 10−4 J · cm−3 · K−2 is from (Johnson et al., 1996). All of the input parameters, including the kinetic inductance Lk in
conjunction with the critical current density Jc and the normal state resistivity ρn at 20 K are
measured parameters. We have also evaluated BCS coherent length ξ 0 and electronic mean
free path ltr. Interestingly, we have found that they are comparable. Consequently, all values
of different penetration depth are becoming comparable accordingly, which is significantly
different from the bulk properties of NbN. This exceptionally long mean free path seems to
arise from the two magnitude lower electron gas density at Fermi surface than that of bulk
superconductors. Table 2 lists all of input parameters and output results for this evaluation.
Oates et al. (Oates et al., 1991) reported the penetration depth λ(0) obtained from their surface
impedance measurement in the microwave range. The comparison shows that our penetration
depth value is in good agreement with their results provided that thin film modifies the values
of BCS coherent length ξ 0 and mean free path ltr. As a result, their London penetration depth
λL(0) becomes considerably lower than λ(0) due to the high ratio of ξ 0 to ltr.
6. DC I-V measurement in the absence of optical power
The dc I-V measurement of the NbN nanowire shows four distinct regimes.
The first
regime is the superconducting regime and the slope of I-V curve is primarily determined
by the cryogenic cable resistance; The second is the switching state (Lindgren et al., 1996),
in which spontaneous thermo-electrical relaxation oscillations is dominantly observed; The
third regime is the formation and expansion of a stable hotspot, which is the starting point of
a self-heating effect. The hotspot grows up with the increment of dc bias to occupy one entire
segment of the nanowire. After this point, the fourth regime is entered when one segment of
the nanowire becomes a hotwire and starts to heat up the adjacent segments of nanowires.
6.1 Superconducting and switched states
For the first two regimes as demonstrated in Fig. 10, the figure depicts the forward I-V curve
obtained via two scan speeds: fast computerized scan and slow manual scan, in which the
inset panel displays a typical computerized scan in both forward and backward directions.
The manual mode is persistently useful for the small signal S-parameter measurements. In
both manual and computerized scan modes, the return current, which manifests the plateau
of a hotspot formation, shows the same value. However, the occurrence of this point in the
manual scan mode is different from the computerized fast scan mode. For the former mode,
it occurs at higher voltage than does the latter, but the latching current (approximately 7 μ A)
is almost identical. We attribute this to three possible mechanisms: scanning speed, scanning
direction and electrical grounding. The manual mode always has slower variation of bias
voltage. The voltage relaxation induced by quasiparticle excitations results in a metastable
states, which is very susceptible to external perturbations, such as a sudden bias voltage
variation, and then stabilizes itself at the latching current.
In addition, the scanning direction depends on whether it starts from a higher bias voltage
down to zero (backward), or vice versa (forward). Within the switched states as defined in
(Lindgren et al., 1996), the nanowire will exhibit a hysteresis I-V curve due to the thermal
healing of the hotspot, i.e. the voltage scans of the forward and backward mode show the
hysteresis behavior; and the measured current remains at the level of latching current until
Investigating Optoelectronic Properties of the NbN
Superconducting Nanowire Single Photon Detector
277
30
3 0
2 5
F orw ard
Backw ard
25
2 0
1 5
μA)I (
1 0
20
5
0
0
1
2
3
4
5
15
B ia s V o lta g e (m V)
rrent (μA)
Cu 10
5
Manual Scan
Computerized Scan
0
0
2
4
6
8
10
Bias Voltage (mV)
Fig. 10. Experimental voltage and current relationship measured via both manual and
computerized scan modes under forward voltage bias scheme; inset panel is a computerized
scan curve in both directions.
the bias voltage decreases below the forward latching voltage point. The detail theory has
been discussed by Tinkham et al. (Tinkham et al., 2003). However, the theory can not account
for the different hysteresis loops in two scan modes. We suspect that the electrically shared
ground of the analog and digital parts in the source meter is responsible for the difference.
In the computerized scan mode, communication between the source meter and computer can
induce extra ground voltage fluctuations. Consequently, a smaller hysteresis I-V curve results
as demonstrated in the inset panel of Fig. 10.
√
The dc I-V curve measurement also indicates that when Ib is greater than 2 Iret, it is no longer
a single value function of bias voltage. On the other hand, for any given bias voltage, there
exists a single-value current. Lindgren et al. used to bias an HTS bridge in the switched
state to measure photoresponses (Lindgren et al., 1996). The values of Rs by small signal
RF/microwave measurement results in less than 20 Ω, even when Ib is 0.5 μ A less than Ic
(28.5 μ A in this case), which is much lower than the stable resistance of a hotspot by Joule
heating.
6.2 Temperature dependent Ic
The critical current Ic can be experimentally determined for a given SNSPD by inspecting the
maximum current in Fig. 10. 2 The temperature dependence of Ic can be experimentally
found by inspecting the maximum current in the dc I-V measurement at all temperatures. Fig.
11 illustrates the measured zero maximum current for the DUT with higher QE. In the figure,
the dot line is a linear fitting curve, whose slope is -5.88 μA/ K; the zero temperature Ic is projected to be 50.74 μ A. The red square in the plot is the DUT provider measured Ic at 2 K,
which is below the linear fitting trend line. When the temperature ranges from 7.5 K to 9.5 K,
approximately a second slope can be observed. Higher 9.5 K, we can only measure submicron
ampere current, which is hard to identify it as a maximum current. The linear relation of Ic
with the decreased temperature is not consistent with what Ginsburg-Landau theory predicts
in the bulk superconductors, from which we expect a relation like Ic ∝ T−3/2 curve. We
2 when the bias current Ib is close to Ic, the superconducting nanowire will exhibit a considerable dc resistance primarily due to voltage spikes arising from dark count. Consequently, this will affect the
accuracy of the Ic measurement, as well as latching problem in the experiment.
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Optoelectronic Devices and Properties
Fig. 11. The zero voltage maximum current, or critical current Ic vs temperature dependence
for the SNSPD of higher QE.
attribute the linear relationship of Ic( T) particulary exhibited in the SNSPDs to the NbN thin
film, in which the superconducting weaklinks formed during its fabrication process.
The NbN thin film is sputtered on sapphire substrates; and is studied in (Kampwirth &
Gray, 1981)-(Gray et al., 1985). The weaklink inherently presents during the ultra thin film
sputtering, due to the formation of granular grain boundaries. Kampwirth et al. (Kampwirth
& Gray, 1981) approximated the critical current based on the quasiparticle tunneling, which is
studied by Ambegaokar (Ambegaokar & Baratoff, 1963)-(Ambegaokar & Baratoff, 1963). We
can relate Ic to the granular size of the NbN thin film a 0
Δ ( T)
a 0 = πAΔ( T) tanh
(6)
2 Icqeρ 20 K
2 kBT
where kB is the Boltzmann constant; qe is the electron charge; Δ( T) is the temperature
dependent BCS energy gap; A is the cross section area of the nanowire, and ρ 20 K is the
resistivity and will be addressed later in this section. The value of a 0 provides a quantitatively
commencement point in the self-heating model for the SNSPD nanowire that will discussed
next.
6.3 Modeling of superconducting and switched state
The I-V curve shown in Fig.
5 (a) prior and after the critical current Ic point, is the
superconducting and switched state (Lindgren et al., 1996), respectively. For the latter, it is
characterized by a spontaneous relaxation oscillation at about 800 KHz counting rate, when
we use the method to perform photon/dark count characterization in (Yan et al., 2009).
Hadfield et al. (Hadfield et al., 2005) suspected about 5000 Josephson Junction-like behavior
exhibited in their SNSPD, which has the identical geometrical size as that of our DUT. It is
actually in consistency with what we calculated based on what Kampwirth (Kampwirth &
Gray, 1981) estimated about the single granular size of 100 nm. We further developed a lump
5000-stacked Josephson Junction circuit model to simulate to the total current driven by a
voltage source. The voltage source spans from 0 to 10 mV. The simulation results is plotted
in a solid curve in Fig. 12. The comparison to the experimental result in the figure indicates
that the simulation does not display a relatively flat current in the switched state. This is
Investigating Optoelectronic Properties of the NbN
Superconducting Nanowire Single Photon Detector
279
Fig. 12. The simulated dc I-V scan by voltage bias in superconducting and switched states
Name
Return Current Ambient Temperature Heat Transfer Coefficient Resistivity at 20K
Description
Ir ( μ A)
Tb( K)
α( W/ m 2 · K)
ρ 20 K(Ω · m)
DUT 1
5.66
4.2
2 × 104
2.05 × 10−6
DUT 2
3.96
4.5
2 × 104
2.14 × 10−6
Table 3. The self-heating hot spot model parameters
primarily due to the fact that the spontaneous voltage oscillation at about 800 KHz has not
been considered in the simulation. The strong voltage pulses can act as another resistance,
which is beyond our stacked Josephson Junction model.
6.4 Modeling 1-D self-heating
We are not able to observe any relaxation oscillations when the device is latching. We believe
that when the bias voltage is higher than the latching voltage point, the nanowire formulates
a non-reversible hotspot that can not be thermally healed by the heat bath. Then the nanowire
is believed to enter the third regime dominated by self-heating effect.
Carefully examining the I-V curve in Fig. 13 (a) and (b), we notice that there exist a series
of zigzag current fluctuations with the increment of bias voltage. We anticipate that the
self-heating hotspot expansion along one segment of the nanowire is quantized as resistance
staircases. We suspect that those resistive steps are associated with granular nature of the ultra
thin NbN thin films.
With the increment of dc voltage bias, a stable hotspot will develop at a certain section of the
nanowire. The voltage width of the switched state depends also on the scanning direction and
mode. Usually manually increasing the vo