superconducting microwire
carrying large oscillating currents
at low temperatures
THESIS
submitted in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE in PHYSICS Author : K.M. Bastiaans Student ID : 0947962 Supervisor : MSc. J.J.T. Wagenaar
Prof. Dr. Ir. T.H. Oosterkamp
2ndcorrector : Prof. Dr. J. Aarts
superconducting microwire
carrying large oscillating currents
at low temperatures
K.M. Bastiaans
Huygens-Kamerlingh Onnes Laboratory, Leiden University P.O. Box 9500, 2300 RA Leiden, The Netherlands
February 17, 2015
Abstract
The working temperature is a limiting factor for improving the sen-sitivity of Magnetic Resonance Force Microscopy towards imaging of a single nuclear spin. In this report we take a step in lower-ing the worklower-ing temperature by reduclower-ing the energy loss of the radio frequent source, using a superconducting NbTiN microwire. A cryogenic calorimeter with 100 nW resolution at 4 Kelvin is de-veloped to investigate the power dissipation of the detection chip, giving new insights on the current design. The use of a NbTiN mi-crowire enables to keep the mixing chamber of a dilution refriger-ator at 10mK. The presence of flux vortices penetrating the NbTiN material seems to play an active role in the energy losses. Improve-ments are proposed on the design of the detection chip to reduce the power dissipation further for future experiments.
1 Introduction 7
2 Theory 9
2.1 Quasiparticles 9
2.1.1 Two-fluid model 9
2.1.2 Quasiparticle density of states 12
2.1.3 Calculations for a NbTiN wire 15
2.2 Vortices 16
2.2.1 Calculations for a NbTiN wire 16
2.2.2 Flux flow 17
2.2.3 Flux pinning and oscillations 18
2.3 Dielectric losses 19
3 Materials and Methods 21
3.1 Detection chip 21
3.2 Measurement setup 25
3.2.1 Calorimeter 25
3.2.2 Measurement procedure 29
3.2.3 Equipment and Electronics 29
3.2.4 Resolution 30
4 Results and Discussion 31
4.1 4 Kelvin 31
4.1.1 Discussion 31
4.2 100 milliKelvin 34
5 Conclusions and Outlook 37
5.1 Outlook 37
Chapter
1
Introduction
One important limiting factor for Magnetic Resonance Force Microscopy (MRFM) is the working temperature. Lowering it would improve the sen-sitivity of MRFM towards the goal of single spin detection. A considerable limitation to this temperature is the radiofrequent heating of the radiofre-quent source. Using a copper wire as radiofreradiofre-quent source for flipping the nuclear spins provides for 350 µW of dissipated power, permitting a working temperature down to 300 mK [1]. This allowed the Rugar group in 2009 to obtain a 5 nm resolution image of a tobacco virus [2]. In this thesis we will report on the power dissipation of a NbTiN wire, a type II superconductor, which we want to use as radiofrequent source. We will investigate whether the use of a NbTiN microwire will enable us to work down to tens of milliKelvins, improving the sensitivity of MRFM.
Chapter
2
Theory
In order to investigate energy losses in a superconducting microwire car-rying an alternating current, we first need to discuss the relevant theory. Where the static superconductor is treated as an entirely lossless conduc-tion channel, dynamic -carrying an alternating current- superconductors will always show finite dissipation. Therefore we will review three mech-anisms for energy losses in our radiofrequent wire. First we will argue whether the existence of quasiparticles in the superconducting conden-sate enhances the energy losses. Second we will consider the effect of flux lines penetrating the superconductor and their response to an alternating electromagnetic field. The third mechanism includes the dielectric losses.
2.1
Quasiparticles
2.1.1
Two-fluid model
In this section we will introduce the oversimplified two-fluid model as a standard working model for understanding energy losses in a super-conductor. [3] The idea is the following: model the superconducting con-densate as a superposition of two co-excisting fluids. One so-called ”su-perconducting” fluid contains the superconducting electrons. These are the states of opposite momentum and spin that via a net attraction form the paired states, the Cooper pairs. [4] The other ”normal” fluid consists of ”normal” electrons. These are electron-like elementary excitations of the groundstate. These quasiparticles are broken Cooper pairs, meaning that one of the two states is occupied and one is empty. This breaking into quasiparticles can occur by thermal excitation or by mechanisms that
change the pairing itself.
Now the reason why one always would expect a non-zero dissipation for a superconductor carrying an alternating current can be shown by a simple argument of Tinkham. [3] Consider the London equation:
∂Js
∂t = nse2
m E (2.1)
where ns is the density of superconducting electrons, e the charge and m the mass. An alternating supercurrent Js requires an electric field E to ac-celerate the superconducting electrons. But at the same time this electric field also accelerates the quasiparticles in the normal fluid, which shows normal Ohmic dissipation. Ofcourse this is a crudely simple argument, but it shows that the response of a superconductor to an alternating cur-rent is not the same as for the lossless static case.
Back to the two-fluid model, we have assumed that the total electron density n can be modeled as a superposition of two parts: n = ns+nn, where ns is the density of superconducting electrons and nn is the density of quasiparticles. In the Drude model
mdv
dt =eE− mv
τ (2.2)
they both have different scattering times τs = ∞, since we assume that the superconducting electrons don’t scatter, and τn. For the superconduct-ing electrons we find the London equation 2.1 when we plug τs = ∞ into equation 2.2. Likewise the quasiparticles form a parallel ohmic channel. Now if we are only interested in the linear response of the superconduct-ing condensate to an alternatsuperconduct-ing electromagnetic field, we can write the conductivity of the condensate as a complex conductivity of the two par-allel channels
σ(ω) = σ1(ω) −iσ2(ω) (2.3) where σ1is the conductivity of the quasiparticle channel and σ2of the su-perconducting one. Since this two-fluid approximation only holds for fre-quencies below the energy-gap (above the gap other loss mechanisms set in), we can assume that frequencies are low enough such that ωτn 1. This enables to express the conductivity of the two fluids as [3]:
σ1(ω) = nne2τn/m (2.4a)
Figure 2.1: Circuit analogy of the two-fluid model. The superconducting fluid is represented as an inductor with conductance σ2 = 1/iωL. The resistor reflects
the quasiparticle fluid where σ1 =1/R.
These expressions for the complex conductivity of the superconductor make clear that for any non-zero frequency the conductivity has a real part and an imaginary part.
We can model this very nicely in a circuit analogy, as shown in figure 2.1. Represent the quasiparticle channel with a resistor, σ1=1/R, and the superconducting channel as an inductor, σ2 = 1/iωL. The voltage drop over the superconducting inductive channel (V = I·iωL) will accelerate quasiparticles in the parallel channel, resulting in ohmic energy losses.
To express the ratio of currents in the two channels we make use of the complex conductivity, since j =σE:.
Js Jn = nse 2/mω nne2τn/m = ns nnωτn (2.5)
From this equation becomes clear that for every non-zero frequency rent will flow in the dissipative ”normal” fluid channel. For a certain cur-rent density J, the power dissipated per unit volume is P = Re(1/σ)|J|2. Where the conductivity σ is the complex conductivity from equation 2.3 and 2.4. If we assume that the conductivity in the ”normal” channel is much lower than in the superconducting channel (σ1 σ2), we can ap-proximate the dissipated power:
P = nnτnm
n2 se2
ω2J2 (2.6)
Although this is a simplified model, we still can make some correct con-clusions from this approximation. [3] First the energy losses due to quasi-particles in the superconducting condensate are quadratic in frequency
(P ∼ ω2) and current (P ∼ J2). Second the dissipated power depends on the density and lifetime of the quasiparticles (P ∼ nnτn). In order to give an estimate for the power we should expect for our radiofrequent wire, we need to give an estimate for the product
nnτn =
Z
dED(E) ·e
−E
kBT ·τ(E) (2.7)
where D(E)is the quasiparticle density of states.
2.1.2
Quasiparticle density of states
As described in the previous section, the Cooper pairs are formed in pairs of states with opposite momentum and spin. The Fermi spheres where these states belong to are symmetric; lifting this symmetry weakens the superconductivity. Like we said earlier, pair breaking can occur by ther-mal excitation or by other pair breaking mechanisms, such as the influence of an external magnetic field or the presence of a supercurrent. The mod-ern basis of understanding for describing superconductivity modified by these pair breaking mechanisms are the Usadel equations. [5] They ap-ply in the diffusive limit for superconductors (meaning that the electron mean-free-path is short compared to the BCS coherence length ξ0) and al-low us to describe these modifications by a single parameter, the depairing energyΓ. Anthore et al. (2003) have shown that the density of the broken pair states, the quasiparticle density of states, within a superconducting microwire varies with the supercurrent. Their results (see figure 2.2) are very well described using the Usadel theory. [6]
In the Usadel theory, interactions between the states of opposite mo-mentum and spin are described by the pairing angle θ(r, E), a complex function, which obeys the following diffusive differential equation:
¯hD 2 ∇
2
θ+ [iE−Γ cos θ]sin θ+∆ cos θ =0 (2.8) Here D is the diffusion coefficient,Γ again the depairing energy and ∆ the energy of the BCS gap. This equation is called the Usadel equation. It bears a diffusive term, but also a term accounting for the pair breaking (hence theΓ). By solving equation 2.8 one obtains an expression for the depairing angle θ(r, E), which in his turn enables to obtain the local density of states D(r, E) = D(0)Re[cos θ(r, E)] (2.9) where D(0)is the density of states at the Fermi level of the material. Un-fortunately, being a non-linear ordinary differential equation, the Usadel
Figure 2.2:By Anthore et al. (2003). In this figure the normalized differential con-ductance dI/dV(V)of a tunnel junction probing a superconducting microwire is shown (this corresponds with the local density of states). The DOS varies as func-tion of the supercurrent, which can be described very well by the Usadel theory.
equation is this form is hard to solve exactly. But for samples where the current density is homogeneous, we can assume that θ is spatially inde-pendent (this can be justified when the transverse dimensions of the wire are smaller than the coherence length ξ0, a characteristic length for the variations of θ. This is not the case for our NbTiN wire, but will anyhow use this approximation since it broadly describes the main phenomena). In this approximation we can eliminate the diffusion term [6], and it follows
E ∆ +i Γ ∆cos θ−i cos θ sin θ =0 (2.10)
where we also divided the terms by∆ sin θ. This is a far more manageable equation to solve. Using the solutions of equation 2.10 we calculated the local density of states (equation 2.9), as shown in figure 2.3.
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Egap gamma gap 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -6-4-20246
Figure 2.3: Contourplot of the local density of states calculated from equations
2.10 and 2.9. On the horizontal axes we have E/∆ and on the vertical axes Γ/∆. The blue region corresponds with D(r, E) = 0 and the dark red with D(r, E) =
D(0). The regions in between represent D(r, E) = x·D(0), where x can be read of in the legend on the right.
2.1.3
Calculations for a NbTiN wire
Let’s take a moment to realise what we have achieved by doing these cal-culations. What we eventually are after is an estimation for the dissipated power in our NbTiN radiofrequent wire due to the presence of quasiparti-cles. In order to do so we need to know the quasiparticle density of states. The Usadel theory gives us a framework to achieve this. Estimate the key parameter in the Usadel theory, the depairing energy:
Γ ∆ = ∆ US(Γ) Is IΓ 2 + B BΓ 2 (2.11) This equation consists of two terms, both pair breaking mechanisms: the effect of the supercurrent and the external magnetic field. We have in-troduced the depairing current IΓ =
√
2∆/eR(ξ0) (where R(ξ0) is the resistance of the wire with length ξ0) and the depairing magnetic field BΓ =
√
6(¯h/e)/(wξ0)(w is the width of the wire). Here we can clearly see how these depairing mechanisms come into the Usadel theory. We will fo-cus only on a wire carrying a supercurrent, so no external magnetic field. Some material properties of NbTiN (we will discuss more on this in chap-ter 3): the energy gap of NbTiN is∆ ≈ 1.76kTC ≈ 2.2meV, the coherence length ξ0 ≈ 4nm. From the resistivity of our film (92µΩcm) we evaluate that R(ξ0) = 6.1mΩ. This results in a depairing current of IΓ = 510mA. For kBT ∆, that is for kBT 26K, one can approximate
US(Γ) ∆ ≈ π 2 −1.8 Γ ∆− Γ ∆ 2 (2.12) Plugging this in equation 2.11, together with IΓ = 510mA, we find for a supercurrent of Is =10mA a depairing energy ofΓ/∆≈0.65. From figure 2.3 we can see that for such a depairing energy there is a region where the quasiparticle density of states is zero up to E = 0.15∆. This would
im-ply that for our NbTiN radiofrequent wire carrying a 10 mA supercurrent there would be no quasiparticles below E=0.15∆, which is equivalent to
3.8K. This can be viewed as an onsetpoint for a finite quasiparticle den-sity of states to appear in our NbTiN microwire. What we learn from this is that the power dissipation due to the presence of quasiparticles should clearly be temperature dependent.
2.2
Vortices
The second mechanism for energy losses in a superconductor is related to flux vortices penetrating the material. The presence of magnetic flux can come from a present magnetic field, like the earth magnetic field, or from a field induced by the wire itself. When a superconductor carries a current it will induce a magnetic field. For type II superconductors, it will become energetically favorable for flux to penetrate if this self induced field is larger than the first critical field
Hc1 = Φ0
4πµ0λ2ln λ ξ0
(2.13) whereΦ0 = h/2e is the flux quantum, λ the penetration depth and ξ0 the coherence length of the superconductor. One could include temperature by [3] Hc1(T) ' Hc1 1− T 2 T2 c (2.14) The penetrating flux vortices are distributed in a periodic lattice with a nearest neighbour distance of roughly ∼ √Φ0/B. Eventually when the current increases such that the induced field exceeds Hc2, the vortices will overlap and the superconducting state will be destroyed.
2.2.1
Calculations for a NbTiN wire
If we calculate this for our NbTiN microwire (where λ ≈ 300nm and ξ0 ≈ 4nm), we find that we would need a field of Hc1 ≈ 6282mA, which corresponds to a magnetic field of Bc1 = µ0Hc1 ≈ 8mT, for the first flux vortex to enter. The distance between the vortices are then∼ 510nm. Us-ing the results of the Ginzburg-Landau theory, in the approximation of a round wire and assuming that the current density is homogeneous, the current corresponding to a field Hc1 is Ic1 ≈ 3.3mA for our NbTiN mi-crowire. Hence, Ic1 = 4 3√6 2πrBc1 µ0 ≈3.3mA (2.15)
where r is the diameter of the wire (equation adapted from Tinkham [3], the prefactor is a result of the more exact Ginzburg-Landau theory). Exper-imental values will differ from this estimation since our wire is rectangular instead of round and its dimensions are large compared to ξ0, therefore the current density will not necessarily be uniform. As a concequence Ic1 will usually be less than 2.15, due to effects like current crowding at edges of
the wire. As a solution one could make the film thinner than λ such that the product of width and height is smaller than λ2. In this way the current will truly be uniform over the wire even if the width is larger than the pen-etration depth. [3, 7] Also one could think of the width of the wire. Stan et al. observed that the critical field for vortices to enter is proportional to Hc1 ∼ Φ0/W2, where W is the width of the wire. [8] This implies that superconducting channels should be designed narrower if one would not want vortices to enter.
2.2.2
Flux flow
We will concentrate on the regime where flux vortices will penetrate the superconductor. They will form a periodic lattice, but what will happen if we would also include the effect of thermal fluctuations. Absent in the mean-field approximation of Abrikosov, the thermal fluctuations will show to be a cause of energy losses in the vortex state of type II super-conductors. Basically when the flux vortices penetrate the material they will pin to impurities in the crystal lattice, preventing the vortex to move. Forget for a moment about pinning, current flowing trough the supercon-ductor makes a Lorentz force act on the vortices FL =j×B, where j is the current density. This will set the vortices in motion and make them move from their pinning sides, inducing an electric field E= B×v, causing the energy loss. In return, the movement in counteracted by a friction force
Fη = −ηv, where v = η1j×B is the velocity of the vortex. [9] The total
energy dissipated due to the movement of vortices equals P= (j×B)
2
η (2.16)
and is strongly dependent on the friction coefficient η. In the Bardeen-Stephen model this η is determined by making the approximation that the core of the vortex, radius ξ0, is fully normal. Dissipation occurs by normal resistive processes in this core. [10] We need to stress that these energy losses do not arise because of current flowing through the normal cores, currents would just flow around these cores, but the movement of the vor-tices is essential. In this model the energy dissipation can be expressed by
P=ρnj2 B
Hc2 (2.17)
with ρn the normal state resistivity of the sample. Notice that the energy losses in the superconductor due to flux (vortex) flow, without pinning, is a factor B/Hc2 of the dissipation in the normal state.
2.2.3
Flux pinning and oscillations
Now also consider the aspect of flux vortex pinning. Competing with the Lorentz force there is a pinning force Fpin that keeps the vortices at their pinningsides. Only when the current density j exceeds jc = Fpin/B the Lorentz force is larger than the pinning force and flux vortices are de-pinned from their sides. If depinning occurs we are in the region of flux flow. Now we will consider the case where the pinning force is strong enough to keep the flux lines pinned. The pinning potential can very well be modelled as a generic Lorentzian function. [11] The vortices oscillate inside this potential well as a response to the alternating electromagnetic field, causing energy loss. [12]
Figure 2.4: Schematic representation of a flux vortex oscillating in its pinning
potential as a response to the alternating electromagnetic field. The pinning po-tential has width ξ due to its Lorentzian shape U(r) = −U0/(1+ (r/ξ)2), where
U0is the pinning energy and r the distance to the pinning defect. [11]
The observed power dissipation due to these oscillations should be lin-ear with the frequency of the alternating field. Once the pinning force is exceeded the flux abandons its potential well, enabling flux flow, and dis-sipation should be quadratic with frequency.
Figure 2.5:From Barends et al. [13] Noise spectra of the normalized frequency of a NbTiN coplanar waveguide resonator, with and without coverage of a dielectric layer. Black: bare NbTiN sample. Yellow, Orange, Red: covered with SiOx. Blue: covered with Ta
2.3
Dielectric losses
As a third mechanism for dissipation we will consider the dielectric losses. Due to the interaction with the alternating electromagnetic field, dipoles in the substrate will move as the polarization of the field switches. This movement is a origin of energy loss. To express this for our NbTiN mi-crowire we will use the paper of Barends et al. where they investigated the contribution of the dielectrics to dissipation in a NbTiN superconducting resonator. [13] In figure 2.5 the effect of coverage of a dielectric layer on top of a NbTiN sample is depicted. Compared to the bare NbTiN sample a change in the normalized frequency noise spectra is visible when a dielec-tric layer is present. It clearly shows that the noise increases significantly when the sample is covered with a layer of SiOx, and is independent of the layer thickness. The authors suggest that the increase of noise is re-lated to quasiparticle trapping and release at the interface. Model this Silicon / NbTiN interface as a LC resonator circuit. From the noise spec-tra one can determine the noise in the phase response of the resonator δθ = 2.5mrad/
√
Hz. For a standing wave amplitude in the LC resonat-ing circuit of Vrms = 14mV the noise in the phase response translates to a voltage jitter of δV = Vδθ = 35µV/√Hz in phase with the current. We hypothesize that the power dissipation is P ' IδV√BW, where BW is the bandwidth.
Chapter
3
Materials and Methods
In this chapter we will discuss two things. First the sample under con-sideration: the NbTiN radiofrequent wire on the detection chip planned to use for MRFM experiments. Second the measurement setup for investi-gating the power dissipation of the sample. We build a so-called cryogenic double compensated calorimeter, by the idea of Kajastie et al. [14], to de-termine the losses.
3.1
Detection chip
For the fabrication of the detection chip we work closely together with the group of Teun Klapwijk from TU Delft. Being experts in the field of sputtering NbTiN, David Thoen and Akira Endo provided us with the NbTiN wafers. The wafer characteristics can be found in the table below
Thickness (nm) 378
Resistivity (µΩcm) 92
Transition temperature (K) 14.9
Penetration depth (nm) 280
Patterning (using e-beam) the superconducting structures and plasma etch-ing (together with David Thoen) have been done by Jelmer Wagenaar and Arthur den Haan. During this proces the thickness of the NbTiN film could have been reduced due to overetching and/or the use of resist. Fig-ure 3.1 shows two SEM images of the structFig-ures on the detection chip, with the one investigated in this thesis being the radiofrequent wire. Contact can be made by connecting external leads to the contactpads via wirebond-ing.
Figure 3.1: Optical image of the detection chip. Three different components are visible in this image. Vertically we see the radiofrequent wire (indicated with RF-wire). The length of this wire is 100 micrometer between the two narrowings. Width 2 micrometer and height approximately 300 nanometer. The two leads after the broadning go to two big (2 by 2 millimeter) contactpads. The second component is the pick up coil for the SQUID detection (indicated with pick up coil). Third, above the pick up coil, we have a copper pad. This is used for copper saturation experiments to verify the Korringa relations (not used in this thesis). Below a close up of the structure.
Since we purely want to investigate the power dissipation of the ra-diofrequent wire, it is necessary for all other contacts to be non-dissipative. Therefore we tested to kind of wire bonding materials: 1) so-called Tanaka (an alloy of Pb76In20Au4) and 2) Al (Aluminium) wirebonds.
Tanaka Pb76In20Au4 is a bonding wire previously produced by the Tanaka Denshi Kogyo company in Japan. ∗ Bonding can be done using a standard wire bonder, on NbTiN settings: 2.75 power and 3.00 time, for a wire thickness of 25µm. Annealing is not necessary because it should attach easily thanks to the softness of the material. The alloy is known to be highly reliable for superconducting wire bonding and has a transition temperature around Tc ≈7K. [15] This sounds like a suitable candidate for contacting the NbTiN device, without generating to much heat itself. Un-fortunately we are unable to determine the antiquity of our Tanaka wire supply, therefore it is possible that the material has degraded over time. Figure 3.2 shows a IV-characteristic of the NbTiN device contacted with Tanaka wire bonds, measured inside a 4 Kelvin pulsetube cryostat system. One can clearly see the superconducting transition of NbTiN to the nor-mal state. The concerning part is observed below this transition. Here a remaining 15Ω resistance is measured. Most likely the Tanaka wire bonds are not superconducting; heating the NbTiN device, eventually destroying superconductivity, which can explain the very low critical current. There-fore it is not usefull to use the Tanaka wire bonds.
2 . 0 2 . 2 2 . 4 2 . 6 2 . 8 3 . 0 0 . 0 2 . 5 5 . 0 7 . 5 1 0 . 0 1 2 . 5 V o lt a g e ( V ) C u r r e n t ( m A ) R = 1 5 Ω R = 4 0 9 0 Ω IC = 2 . 5 m A
Figure 3.2:IV-characteristic of the NbTiN device contacted by Tanaka wire bonds.
∗Tanaka Japan has stopped production of this product, currently we are looking for a
Aluminiumis a widely used wire bonding material. The wire bonding settings for bonding a 25µm Al wire on NbTiN are: 3.60 power and 4.10 time. The material becomes superconducting at a temperature of ∼ 1K, therefore it would not form a superconducting connection to the NbTiN device inside the 4 Kelvin system, but will do so in a dilution refrigerator with a working temperature around 10mK. Figure 3.3 shows the measured IV-characteristics of the NbTiN device contacted by Al wire bonds in the 4 Kelvin system. Like for the Tanaka wire bonds, the superconduting tran-sition of the NbTiN is again clearly visible. Below the trantran-sition there is also a remaining resistance, but now expected since Al is not yet super-conducting at 4 K. The resistance of the Al wire bonds is 6mΩ, generating little heat, allowing a much higher critical current. This will be even bet-ter when working at 10mK. Therefore we decide to use Aluminium wire bonds for contacting the NbTiN device.
1 7 . 9 1 8 . 0 1 8 . 1 1 8 . 2 1 8 . 3 0 5 1 0 1 5 2 0 2 5 3 0 3 5 V o lt a g e ( V ) C u r r e n t ( m A ) R = 0 . 0 0 6 W T C = 1 8 . 1 m A
Figure 3.3: IV-characteristic of the NbTiN device contacted by Aluminium wire
bonds. Note: The IV curve above the superconducting transition is flat because the limit of our digital multimeter is reached.
Figure 3.4: Schemetic representation of a cryogenic dual compensated calorime-ter, by Kajastie et al. [14] The temperature of the heat baths is controlled by a temperature control loop, containing a heater ˙qF1/2and a thermometer T1/2.
3.2
Measurement setup
To determine the losses of our NbTiN device we developed a cryogenic calorimeter as measurement setup, by the idea of Kajastie et al. [14] In this section we will discuss the design of the calorimeter and the needed equipment to operate it.
3.2.1
Calorimeter
The measurement setup we constructed is a so-called cryogenic dual com-pensated calorimeter. The method of dual compensation relies on the abil-ity to keep both the target bath (containing the sample under test) and the reference bath at stable temperatures in cryogenic environments. [14] The operation of the calorimeter is shown in figure 3.4. The target heat bath is heated above the reference bath, making heat flow from the target to the reference via a primary heat link RT1. In his turn, the reference bath is heated above the base temperature, making the heat flow from reference to the 4 Kelvin plate via RT2. The principle of dual compensation relies on the fact that RT2 RT1, the thermal time constant of the reference bath is much smaller than of the target, maken both feedback systems
indepen-dent. The measured quantity is the power used by the heater of the target bath ˙qF1. The target heater power balances the heat flows
˙qF1= ˙qT1+ ˙qP1− ˙qs (3.1) where ˙qT1 is the heat flow through the primary heat link, ˙qP1 are the par-asitic heat losses and ˙qs are the losses generated by the sample under test. Now comes the working principle of the calorimeter. As long as both heat baths T1and T2are kept at a constant temperature, the heat flow through the primary heat link ˙qT1will remain constant. This enables us to measure the losses ˙qs of the sample (the detection chip) directly by measuring ˙qF1.
The calorimeter is designed to measure an energy loss of approximately ˙qs ≈100nW. If we estimate the parasitic losses to be ˙qP1 =σT4 ≈10−13W and demand that the heater power is only about 25% greater than the predicted losses, we can estimate the primary heat flow ˙qT1 ≈ 10−6W. This is the information needed for the design of our setup. Anticipating a temperature difference Tsample −Tre f erence = 10 mK, the thermal resis-tance of the primary heat link needs to be RT1 ≈ 104K/W. The heat link connecting the reference bath to the 4 Kelvin plate is designed to be ( RT1) RT2 ≈ 125K/W. By chosing to construct both heat links of stain-less steel the ratio RT1/RT2 can be tuned purely by the dimensions of the links, without having to concern that the thermal conductivity changes with temperature. The dimensions of the heat links can be determined via
RT = l
A·κ (3.2)
where RTis the thermal resistance, l the length of the heat link, A the area and κ the thermal conductivity. For our setup we decided to use stainless steel to construct the heat links, κ = 0.1 W / m K. The primary heat link from target to reference bath consists of 4 pillars with a height of 12.5 mm and a radius of 1 mm, resulting in a thermal resistance of RT =104K / W. The link connecting the reference bath to the cryostat plate is constructed of a stainless steel ring, height 18.5 mm and radius 30 mm. In this ring a compartment is made to accomodate the reference heater and thermome-ter (40.5 by 33 mm). The thermal resistance of the ring contact is RT =125 K / W. In this way the requirement RT2 RT1is fulfilled (RT1/RT2 =80). Figures 3.5 and 3.6 give an impression on the final design of the calorime-ter.
Figure 3.5: Design of the calorimeter. On top there is the target bath, harboring the sample and a printed circuit board for interfacing the sample. The primary heat links are the four stainless steel pillars, anchored to the reference bath. The reference bath is a brass housing coated with gold. It can be closed off on top with a cover. Connection to the 4 Kelvin plate is made by the stainless steel ring on the bottom.
Figure 3.6: Top:Photo of the calorimeter mounted on the 4 Kelvin plate.
3.2.2
Measurement procedure
How to operate the developed calorimeter? The idea is the following: First heat the target heat bath to a set temperature, without passing a current through the sample detection chip. Measure how much power the target heater needs to reach the set temperature (P1). Next, after cooling back down to the stable reference temperature, start to apply a current through the sample. Then the target heat bath is heated again to the set temperature and measure how much power the heater needs (P2). The total dissipated power of the sample is Ps =P1−P2, as illustrated is figure 3.7. An
impor-Figure 3.7:Temperature profile to demonstrate the measurement procedure.
tant factor for the duration of this measurement procedure is the thermal RC time of the system. This is of the order of∼100s. For each step in the procedure you have to wait 5 times the thermal time to assure a stabilized temperature. For future experiments we believe there are ways to improve on this.
3.2.3
Equipment and Electronics
The measurement setup can be mounted on the 4 Kelvin plate of a pulse-tube refrigerator for experiments at 4 Kelvin. In the same way it can be mounted in a dilution refrigerator to work at 10 milliKelvin. One only need to think of what thermometers to use. For the 4 Kelvin experiments we use RuO2 thermometers, which can be read out using a Keithley digi-tal multimeter. For the 10 milliKelvin experiments HDL designed special thermometers to be read out using a AVS resistance bridge. The heaters in the temperature control loops are custom made heaters, made out of twisted pair Constantan wire (38Ω/m). The heaters can be controlled using a stable Delta Elektronika UCS50B current supply. The whole tem-perature control loop is controlled via Labview with a custom made PID program.
In order to apply an (alternating) current through the sample an ar-bitrary waveform function generator is used. The current amplitude is controlled by controlling the voltage drop over an known resistor in series with the sample (outside of the measurement setup). The voltage drop is measured using a lock-in amplifier and is used as feedback to set the function generator. In this way one can apply a stable alternating current through the NbTiN wire.
3.2.4
Resolution
Using the developed calorimeter, we calibrated the system by measuring a heater as sample. Figure 3.8 shows the dissipated power measured by the calorimetric setup versus the electrical 4-point measured power of the sample heater. This demostrates that we can obtain a 100nW resolution using our setup, according to the design.
0 . 0 2 0 0 . 0 n 4 0 0 . 0 n 6 0 0 . 0 n 8 0 0 . 0 n 1 . 0 µ 1 . 2 µ 0 . 0 2 0 0 . 0 n 4 0 0 . 0 n 6 0 0 . 0 n 8 0 0 . 0 n 1 . 0 µ 1 . 2 µ Pm e a s u re d ( W ) P s a m p l e ( W ) T 2- T b a s e= 0 . 1 K T 1- T 2= 1 0 m K
Figure 3.8: The calotimetricly measured dissipated power of a sample heater
versus 4-point measured power. The inset on the top left shows the tempera-ture differences between the heat baths. The red line is a linear fit with slope 1.00002±0.04. A 100 nW resolution is obtained.
Chapter
4
Results and Discussion
In this chapter it will show that the developed calorimetric setup can be used to measure the energy losses of the NbTiN detection chip. The main results of this project will be discussed for two types of measurements: at 4 Kelvin and at 10 milliKelvin. We will see that at both temperatures the current detection chips show a non-negligible power dissipation.
4.1
4 Kelvin
At 4 Kelvin the energy loss of the NbTiN detection chip, when carrying al-ternating supercurrents, is measured using the calorimetric measurement setup. Figure 4.1 shows the dissipated power versus the frequency of the alternating current for various current strengths. We observe a significant amount of dissipated power. The energy loss increases with frequency and current strenght. Figure 4.2 displays the current dependence of the mea-sured dissipated power at a frequency of 3 MHz. The data is fitted with allometric function P = a·Ib, where P is the dissipated power and I the current. Fitting the data reveals that the power dissipation is quadratic in the current (coefficient b, see inset of the figure). Coefficient a corresponds with an effective resistance of the sample and is plotted in figure 4.3. One can distinguish an increasing resistance with frequency.
4.1.1
Discussion
The first thing that stands out is the significant amount of energy loss of these NbTiN detection chips. It largely exceeds the amount of dissipated power one would expect for the 6mΩ Aluminium wire bonds, therefore
0 1 M 2 M 3 M 4 M 5 M 6 M 0 5 0 µ 1 0 0 µ 1 5 0 µ 2 0 0 µ 2 5 0 µ 3 0 0 µ 3 5 0 µ 4 0 0 µ 4 5 0 µ 7 m A 6 m A 5 m A 4 m A 2 m A D is s ip a te d P o w e r (W ) F r e q u e n c y ( H z )
Figure 4.1:Measured power dissipation of the NbTiN detection chip at 4 Kelvin.
The energy loss is measured at various frequencies for the alternating current at various current strengths.
0 2 4 6 8 0 . 0 5 0 . 0 µ 1 0 0 . 0 µ 1 5 0 . 0 µ 2 0 0 . 0 µ 0 1 2 3 4 5 6 1 . 5 2 . 0 2 . 5 D is s ip a te d P o w e r (W ) C u r r e n t ( m A ) F r e q u e n c y 3 M H z F r e q u e n c y ( M H z )
Figure 4.2: Current dependence for the measured dissipated power at 3MHz.
Red line is a fit of the data with function P= a·Ib. The inset shows the value of
0 1 2 3 4 5 6 0 2 4 6 8 1 0 1 2 R e s i s t a n c e f r o m d a t a f i t t i n g R e s is ta n c e ( W ) F r e q u e n c y ( M H z )
Figure 4.3: Resistance of the NbTiN detection chip when carrying an alternating
current, calculated from fitting of the data (coefficient a). An increase with fre-quency is observed. Two types of cruves are fitted to the data. Blue: Linear Red: Quadratic.
the main contribution to the measured energy loss comes from the NbTiN detection chip. Clearly the response of the superconductor to an alternat-ing electromagnetic field is different than for the totally lossless case con-cerning a direct current. The amount of power dissipated increases with the amplitude and frequency of the current (figure 4.1). The dependence of the frequency is hard to determine with this dataset, due to the behav-ior around 2 and 4 MHz in figure 4.1, which could be assigned to overseen resonances. Fitting of the data (figure 4.2) reveals that the current depen-dency of the energy loss is quadratic for all measured frequencies (inset of figure 4.2). This is what you would expect if the energy loss is dominated by quasiparticles or flow/oscillations of vortices. Therefore the effect of dielectric losses can be discarded due to the linear behavior of the mech-anism. Fitting the data also enables us to calculate an effective resistance of the NbTiN sample (figure 4.3). The character of the increased resistance with frequency is hard to determine exactly with this dataset, but a fre-quency dependence is clearly visible. This is fitted in figure 4.3 with a linear (blue) and quadratic (red) curve, the quadratic relation again hint-ing on presence of quasiparticles or vortices. To investigate the effect of temperature on these observation further experiments were done at 100 milliKelvin.
4.2
100 milliKelvin
Since a non-negligible power dissipation is measured at 4 Kelvin, we de-cided to investigate the effect of temperature on these observations. The next serie of measurements are done in a dilution refridgerator with a working temperature around 10 milliKelvin. The NbTiN detection chip is mounted inside a MRFM setup, therefore these measurements are not performed in the calorimetric measurement setup. Instead the present heater and thermometer on the sample stage of the MRFM setup are used to mimic the calorimeter experiment. The mass connected to the MRFM setup can be thermally controlled using a temperature control loop (PID heater and thermometer), therefore acting as a reference bath.
0 2 0 k 4 0 k 6 0 k 8 0 k 1 0 0 k 0 2 5 n 5 0 n 7 5 n 1 0 0 n 1 2 5 n 1 5 0 n D is s ip a te d P o w e r (W ) F r e q u e n c y ( H z ) I = 2 m A T = 1 0 0 m K
Figure 4.4:Measured dissipated power of the NbTiN detection chip, carrying an
alternating current of 2 mA at 100 mK, for various frequencies. An increase with frequency is observed. Note that the black line indicates zero. The red line gives an indication to the error.
Simply no dissipation (not even a small rise in temperature) was mea-sured when applying a direct current up to 15 mA. When applying a 2 mA alternating current a finite dissipated power was measured at 100 mil-liKelvin for various frequencies (figure 4.4). A lower frequency range is investigated than for the measurements at 4 Kelvin, since this is a inter-esting frequency range for first MRFM experiments. During the measure-ments the mixing chamber maintained a working temperature around 10
milliKelvin. The total amount of dissipation at 90 kHz permits a sample-holder temperature just below 100 milliKelvin.
4.2.1
Discussion
Like for the 4 Kelvin measurements an energy loss increasing with fre-quency is observed. Since the working temperature is much lower, one could question what causes the dissipation at such temperatures. Accord-ing the calculations in section 2.1, the quasiparticle density of states for the NbTiN microwire should be zero for IAC =2mA at T =100mK. More-over, if one would calculate the AC sheet resistance for the NbTiN wire in the Mattis-Bardeen model, one would find Rs ∼ 4·10−13Ω, far too low to explain the measured energy loss. [16, 17] Also for quasiparticles to be te main loss mechanism one would expect a temperature dependency. To inspect whether the dissipation mechanism is the same for both measure-ments, and to see whether temperature has an effect, we can compare the 100 mK data and the 4 K data. Both are plotted in figure 4.5.
1 E + 0 4 1 E + 0 5 1 E + 0 6 1 E + 0 7 1 E - 0 9 1 E - 0 8 1 E - 0 7 1 E - 0 6 1 E - 0 5 1 E - 0 4 f1 . 6 4 P o w e r (W ) F r e q u e n c y ( H z ) 1 0 0 m K 4 K ~ f1 ~ f2
Figure 4.5: Combination of the 100 milliKelvin (low left) and 4 Kelvin (up right)
data for IAC = 2mA. The measured dissipated power is plotted versus the
fre-quency of the alternating current in a log-log fashion. The red line is a data fit P= const·f1.64. The dashed lines P∼ f and P∼ f2act as guidelines for the eye, representing respectively flux oscillations and flux flow.
With this figure the two different frequency regimes can be compared. The 100 mK data is shown at the low left corner and the 4 K at the top
right in the log-log plot. The 4 K data seems to be in line with extrapola-tion of the 100 mK data. Figure 4.5 is a strong evidence that the dissipaextrapola-tion mechanism for both temperatures is of the same origin. For quasiparticle dissipation the energy loss should exponentially increase with tempera-ture, but there is certainly no effect of temperature visible. Therefore we believe that the measured dissipation in the NbTiN microwire is caused due to the presence of flux vortices in the superconductor. Energy losses can occur due to the oscillation or flow of flux, both represented by the dashed lines in figure 4.5. To confirm this hypothesis more measurements needs to be done, but we can use this hypothesis to further improve the design of the detection chip.
Chapter
5
Conclusions and Outlook
In this thesis we investigated the use of a superconducting NbTiN ra-dio frequent wire for future MRFM experiments. We measured a non-negligible energy loss both at 4 Kelvin and 100 milliKelvin, but low enough to keep the mixing chamber around 10 milliKelvin. We believe that this energy loss is due to flux vortices in the superconductor, since other loss mechanisms (quasiparticles and dielectric losses) can be excluded. The power dissipation can occur due to oscillations or flow of the flux vortices. With this hypothesis we give the following recommendations for further improvements on the detection chip design.
5.1
Outlook
The first recommendation for reducing the energy loss of the detection chip is to shorten the length of the supply lines, running from the contact pads to the radio frequent wire. We propose a lay-out as in figure 5.1
Figure 5.1: Proposed design. Shorten the length of the supply lines. From left to
right: Contact pad, supply line, rf wire, supply line and contact pad.
Second, optimize the length and width of the radio frequent wire itself. According to Stan et al. the amount of vortices penetrating the wire can be
controlled by the width. [8]. One should find an optimum for the width of the wire. Reducing the width lowers the number of vortices, but increases the current density. If the Lorentz force acting on the vortices (due to an in-creasing current density) exceeds the pinning force we enter the regime of flux flow. Preferably we would like to stay in the flux oscillations regime, where P∼ f , since in the flux flow regime P∼ f2.
One could also think of ways to transport the generated heat away from the radio frequent wire, for instance by covering the NbTiN wire with a gold layer. The power dissipation in the gold layer due to Eddy currents should be sufficiently low enough
P= π
2B2
peakd2f2
6ρD (5.1)
where P is the power dissipation per unit mass, Bpeakis the peak magnetic field, d the thickness of the layer, f the frequency, ρ the resistivity of the material and D the density. [18] This is P ∼ 3500W/kg for a gold layer with Bpeak =10 mT, f =10 MHz and d = 300 nm. This corresponds to a total power dissipation due to Eddy currents to P∼4 nW for a gold layer of 100µm long, 2µm wide and 300 nm high.
Figure 5.2:Radio frequent wire covered with a gold layer to transport the
Chapter
6
Acknowledgements
The work reported in this thesis could not have been done without the hard work of a lot of people. I would like to thank Gert Koning and Fred Schenkel of the Fine Mechanical Department for the development of the calorimeter and support on the cryostats. Co Koning helped me with all the electronical aspects. David Thoen, Akira Endo and Teun Klapwijk pro-vided the NbTiN wafers and helped with etching of the detection chip. I am also very thankfull to Peter Kes for the valuable discussion on vortex dissipation. Last but not least I would like to thank all the members of the Oosterkamp group for all the help and support. Especially Jelmer and Tjerk for your supervision and our valuable disscussions. I really enjoyed being part of this team effort!
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