Calculation and design of GaAs quantum dot devices where the vibrational modes can be frozen out at cryogenic temperatures

We calculate how the frequencies of the vibrational modes in a free-standing GaAs bar are changed as a function of the bar’s geometrical features such as length, thickness and shape. After understanding the effect of the physical characteristics we add finger gates that will be used to define quantum dots on the bar and study the system as a function of the length of the suspended finger gates, and their material properties. Finally, we strengthen the bridges in order that the first vibrational modes occur at a temperature of 100 mK or more, so that all modes can be frozen out when operated in a dilution refrigerator.


Introduction
Micro/nano electromechanical systems (MEMS/NEMS) have attracted a lot of attention over the past 30 years, as these devices can show very high fundamental resonance frequencies, have negligible mass and can exhibit very high quality factors resonances [1][2][3][4][5][6][7][8][9][10][11][12].The MEMS/NEMS systems, combine mechanical and electrical components and can ranges from a few tens of nanometres to hundreds of micrometres.They have the ability to react to but also sense and measure properties of their environment and control a systems' behaviour in the macro-scale.
The above properties of the micro/nano electromechanical oscillators have been utilised in both simulations and experimentally in organic and inorganic structures [13][14][15][16][17][18][19][20][21].Currently there is a growing interest in trying to observe many body localisation in solids state systems (MBL) [21].MBL can result in non-ergodic behaviour where disorder combined with electron-electron interactions can lead to regions of a device where the electrons states remain out of equilibrium with the rest of the device.In order to observe this we need to make structures where energy loss mechanisms such as electronphonon emission are greatly reduced.The aim of this work is to push up the low-energy phonon vibrational modes of the device so that they are greater than the temperature of operation to assist in the observation of the MBL state.
As the suspended semiconductor structures are very light compared to their mechanical strength, their main factor for collapsing (or not), is the stress or differential stress that they experience once they are released from the substrate or the adhesion force between the double clamped cantilever and the substrate versus the force from the spring constant of the double clamped cantilever.A cantilever like that, shown in figure 1, has both its ends connected to the heat-bath, whilst it is suspended along its long axis.When the different semiconductor layers, that are forming the structure, are grown on top of each other, there is always a small mismatch in their lattice constants that can cause stress effects [22].By removing part of the structures lithographically to create airbridges, the remaining beams will then be under tensile or compressive force which in turn will result in them forming an arc with a peak or a trough if the bridge is under compressive stress.If this is combined with differential stress the beam will either bend up or down depending on the differential stress direction.After a critical length for the suspended part, the Euler buckling instability phenomenon will have to be taken into account [22].The latter means that if the energy stored in the cantilever is not bigger what is required to overcome the adhesion force, then the beam will bend out of shape.
In the literature, groups have suspended both metals, like AuPd [19] and semiconductors like Si [23], Si 3 N 4 [24,25], GaMnAs [26], InGaAs [21] and GaAs/AlGaAs [22,27] and GaAs [28] just to mention a few.The fundamental frequencies of the double clamped cantilevers in some of the studied structures above, are in the range of less than a MHz, up to a few tens of MHz.If this frequency is then matched to the temperature at which it appears, it should be in the range of a few µK to a few mK, making it hard for experiments to study them in the phonon freeze-out regime.
The localisation of the phonons has been a heated part in the nanoscale physics community, as the heat transfer through phonons at these length-scales, can affect the devices' properties by increasing the lattice temperature well above the intended one.One way that this problem can be resolved, is by employing smart designs or geometries.Two very characteristic examples of this are shown in the experiments of Heron et al [29] and Blanc et al [5] and by Monte Carlo simulations by Moore et al [30].In [29], a phonon blocking effect at low temperatures was achieved by fabricating both ends of a silicon nanowire to be serpentine-like.They measured a reduction in the thermal conductance of between 20%-40% compared to that of a straight wire, supported by a theoretical model that took into account the frequency dependence of phonon transport.In [5], they compared the thermal conductance of straight and corrugated Si nanowires at temperatures below 5 K, and found out that there is a significant reduction to the phonons mean free path, related to multiple reflections caused by the shape of the wire.
We will experimentally study MBL effects using a NEMS system and specifically electrostatically defined quantum dots (QDs) using metal gates operating in depletion mode on a 2 dimensional electron gas (2DEG) that can also be used as thermometers or in refrigeration [27,31,32].The QDs will later be used to observe the MBL state as described by Nico-Katz et al [33].One key signature of many-body localisation is the observation of regions of non-ergodic behaviour, but From top to bottom are the first three phonon modes that will be seen in the system for a double clamped cantilever.The above simulated phonon modes have been produced for a 300 nm wide, 120 nm thick, 1000 nm long GaAs bar with a Au bar with the same length but 60 nm wide and 30 nm thick on top, using the mechanical package of ANSYS.The fundamental frequencies that these modes occur, are at about 462 MHz, 1139 MHz and 1996 MHz, that correspond to 22 mK, 55 mK and 96 mK respectively.electron-phonon scattering can impact this with hot electrons emitting phonons.To increase the lifetime of the non-ergodic state we need to engineer a reduced electron-phonon scattering rate by controlling the phonon density of states.In practice, we can quantize the phonon states in the z direction by growing a quantum well (QW) heterostructure with a sacrificial layer at a depth of 100 nm below the surface.This can be etched away to make a free-standing slab, but we still have phonon modes in the x and y direction.To remove the modes in the y direction we narrow the channel to make a 1D free standing wire where the modes are now frozen out in the z and y direction.Next we need to quantize the phonons in the x direction which requires making the wires short and making a large phonon mismatch between the ends of the beam and the phonon paths that we make electrical contact too.
In this paper, we focus on modelling the designs required to achieve phonon freeze-out at 100 mK to predict the geometrical features needed for the experimental observation of MBL, using the simulation software ANSYS and in the last part, on the cleanroom fabrication of the devices described throughout.

System description
The heterostructure consists of a 20 nm thick GaAs QW, 50 nm beneath the surface.The structure is symmetric on either side of the QW, that is sandwiched between two AlGaAs spacers and two Si-doped AlGaAs layers.The important part is the 500 nm thick Al-rich Al x Ga x−1 As layer with a composition x = 66%, below the 120 nm thick GaAs/AlGaAs heterostructure.The latter is a sacrificial layer that will be removed using a 10% hydrofloric acid (HF) solution, to suspend the active part of the device, but more will be discussed in the III C part.
The flexural motion of a 1D beam with a width of w, thickness t and length l, is a very well studied system and can be described by the Euler-Bernoulli equation, as it has already been shown [34][35][36], Equation ( 1) is valid for isotropic materials and Y(x,t) is the transverse displacement of the beam centre as described in [36].The boundary conditions of the equation are chosen because the ends of the bar are clamped at x = 0 and x = L.
, where ω is the angular momentum, κ is the wavevector, E is the material's Young's modulus and ρ is material's density, I is the bending moment of inertia I = wt 3 / 12 and A = wt its cross-sectional area.The mth eigenfunction of the system can be calculated as: To convert the eigenfrequencies to temperature so that we know the minimum temperature we may need see what temperature would activate the corresponding phonon modes of the suspended double clamped cantilever, we worked the following way.
From the Einstein relation E = mc 2 = hc/λ = hf, where E is the energy, c is the speed of light in vacuum, h is Planck's constant of 6.63 * 10 −34 Js, λ is the wavelength and f is the frequency, we can calculate what the energy is for a certain vibrational mode in Joules.The energy is then related to the temperature through E = k B T, with k B the Boltzmann's constant and T the temperature.Given that k B is 1.381 * 10 −23 J K −1 [37], the final outcome is the temperature at which the vibrational modes occurs in o K.
Applying the above to equation ( 2), a new one that describes the temperature that each of these fundamental frequencies occur at, can be formed as: Equation ( 3), is analogous to equation ( 2) multiplied by a prefactor that equals the ratio of Planck's constant to Boltzmann's constant.The first three modes of a double clamped cantilever can be seen in figure 1, as calculated using ANSYS Mechanical R1 2020.The system consists of a Au metal wire on top of the GaAs bar.The thickness of the GaAs (Au) bar was 120 nm (30 nm), its width 300 nm (60 nm) and its length 1000 nm (1000 nm).Running the simulation, the calculated fundamental frequencies for these three modes were about 462 MHz, 1139 MHz and 1996 MHz, that correspond to 22 mK, 55 mK and 96 mK respectively.The shape of the bar for these frequencies is exactly what would be expected for a harmonic oscillator [38].
It is worth pointing out that since the frequencies of the vibrational modes depend on material parameters, the same physics that can describe the vibrations of a bridge, will describe those of a nano-bridge as well.This of course will change in the case of a narrow beam as it will be under tension on the top surface of an upward curving surface and under compression at the bottom.This will cause a change in its shape and hence a change in the moment of inertia.To help the aim of the study, all the data resulting from the simulations will be plotted as 'activation temperature', that describes the temperature the system needs to be at, so that the corresponding vibrational mode can materialise.

Initial characterisation
For our first characterisation, we have assumed for simplicity that the suspended bar is GaAs, rather than GaAs/AlGaAs heterostructure, and has a stripe consisting of a Au top gate that will be used as the central gate that will assist in electrostatically defining the QDs in the actual devices.The bar, as described in section 2 is clamped on either side, and there is no thermal bath underneath or on its sides, connected to the system.For the GaAs and Au we used respectively 86.2 GPa [39] and 79.0 GPa [40] for the Young's modulus E, 0.32 [39,41] and 0.42 [42] for the Poisson ratio v, and 5360 kg m −3 [43] and 19 300 kg m −3 [44] for the material density, ρ.As the values for the materials' Young's modulus, Poisson ratio and density, have an almost negligible temperature dependence, we used them for the low temperature predictions for our devices.
To understand how the first phonon mode depends on the geometrical features of the suspended bar, our first simulation tests for the devices, shown in figure 2(b), has a fixed length of 1500 nm, width of 300 nm and varying thickness from 30 nm to 2000 nm, followed by a second test, in figure 2(a), where the thickness is fixed to 120 nm, width at 300 nm and varying length from 250 nm to 3500 nm.The Au bar in both examples For the length dependence test shown in (a), we kept the thickness of the device fixed to 120 nm and varied its length from 250 nm up to 3500 nm.A T ∝ 1 / l 2 trend characterises the system for L ⩾ 500 nm, as the Euler-Bernoulli equation describes it for L / t > 1.This agrees with equation (3) where the activation temperature is proportional to the 1/l 2 , shown by the red straight line fitting, as long as the length is bigger that the device's width.In the thickness dependence test, shown in (b), the length was fixed to 1500 nm and the thickness varied from 30 nm to 2000 nm.When the thickness matches the width, any increase to it up to 2000 nm has a negligible effect to the activation temperature.In the inset is the device under test, the width of which was kept fixed to 300 nm and the straight line fitting shows the linear response of the activation temperature to the device's width.
was fixed to a width of 60 nm and thickness of 30 nm-same as in the real device-and varying length to match the length of the GaAs bar.The width of 300 nm was selected because when fabricating a real device, there will be a lateral depletion from the etching and the above number that represents the electronic device width, is the narrowest a GaAs device can be in order to have conduction through it [6].
In the first test, there is linear dependence in the bar's thickness up to approximately 300 nm, shown by the straight line fitting of figure 2(b) and the geometry used is included as an inset of the figure.As the bar gets thicker, the temperature required to activate the first phonon mode of the bar increases from about 2 mK at 30 nm to about 20 mK at 300 nm.The tenfold change in temperature (and frequency) is what should be expected for a tenfold change in the thickness, following equations ( 2) and ( 3), as T ∝ d.After that critical thickness, the frequency of the first phonon mode almost saturates; this is a turning point as it only mildly depends on any further change.After the thickness of the device matches its width, the first mode is almost unaffected by any further increase in thickness, up to 2000 nm.
As far as the length dependence is concerned, the temperature presents a linear trend to the 1/l 2 for lengths of 500 nm or more, as expected from equation ( 2), presented in a linear fitting in figure 2(a).From the same equation and for a given width, the longer the bar the lower the temperature that the phonons freeze-out.In both cases, there is a deviation from the linear fit for small enough lengths as according to the Euler-Bernoulli theory the L/t ratio is required to be a lot bigger than 1 [45] to work.

Full device characterisation
The temperature that the first vibrational mode will appear, depends strongly on the geometrical characteristics, as shown in the initial characterisation (IIIA), the material parameters and hence the material itself, as predicted from equation (2).
As mentioned in the Introduction (I), this is a study for a system where QDs will be defined electrostatically, using finger gates in depletion mode in a 2DEG.The cross-sectional area of the fingers are 30 nm by 30 nm and the spacing between each finger is 30 nm.As shown in figure 3(a), there is insignificant change to the temperature where the phonon freeze-out takes place no matter if the system is just a bar (figure 3(b), black points on figure 3(a)) or has the finger gates as well (figure 3(c), red points on figure 3(a)), for a gap of 100 nm (gap at which the finger gates are suspended).The first three modes for both of them, as shown in figures 1(a)-(c), are at approximately the same temperature with a slight suppression in the device with the gates.Interestingly after that, for the device with the finger gates 'plateaus' appear in the activation temperature, with the first one between 70 mK-80 mK, for the next almost 30 modes.This indicates that the system becomes a lot more complicated, with a lot more extra phonon modes entering the system from the finger gates with a very small increase in the device's temperature or at a very short range of frequencies.What is happening, is that the vibration modes deriving from the finger gates, include frequencies for when all of them are vibrating coupled together as a system, and also each and every finger gate vibrating on its own.
As mentioned earlier, both the geometrical features and the material itself will have an effect on the phonon modes and of course in the heated part of the phonon freeze-out temperature.
Commonly for electrostatically defined QDs, both Au and Al can be used [27,46,47].The simulated system is a bar with a width of 500 nm, length of 1000 nm and a varying gap.It is important to point out that for GaAs/AlGaAs devices, the width should be no less that 300 nm for the case (a) By adding the finger gates (points in red, using geometry in (c) and in the red frame) to the bar (points in black, using geometry in (b) and in the black frame), the first phonon modes appear at the same temperature, within a small margin.The increase in the number of modes, though, changes massively on the device with the finger gates.The system gets a lot more complicated as apart from the bar's vibrational modes, the ones from 14 finger gates are added to it.As these can vibrate as a collaborate system and each one individually at different frequencies with a small difference between them, their activation temperature will vary with those frequencies.In (b) and (c) are the simulated systems mentioned above.
where electrical conduction is needed, as the electronic width is reduced due to lateral depletion from the sample surface [6].The summary for the two materials and our system is shown in figure 4. Depending on the material and the gap over which each finger gate is suspended, the first phonon mode of the system will be either from the bar shown in figures 4(a) and (b) as Au (b) and Al (b) of from the finger gates shown in figures 4(a) and (c) as Au (f) and Al (f).Irrespective of the finger gates gap up to 700 nm, the first phonon mode of the bar is at (19.4 ± 1.1) mK for Au and (22.5 ± 0.5) mK for Al, presenting a small change of about 3.5 mK from one material to the other.As the gap increases, the first phonon mode of the system as a total, shifts from that of the bar to that of the finger gates.This happens at a lower gap for the Au (400 nm) compared to the Al (600 nm) and the change in the system with the Au is a lot more dramatic, as the first mode drops to less than 6 mK compared to less than 14 mK at a gap of 700 nm for Al.
There are two ways to strengthen the system, in case a longer gap is needed for an actual device.This could be by In (a), there is small increase in the phonon freeze-out temperature coming from the bar (geometry and deformation used, is shown in (b)) when using Al (b) rather than Au (b) gates by about 3.5 mK, as their Young's modulus are quite similar, but the densities are quite different.The effect of the material and first phonon mode is more interestingly seen for longer finger gates' (geometry and deformation used, is shown in (c)) gaps for Al (f) and Au (f).For both materials, the increase of the gap that the finger gates are suspended for, has an opposite effect to the temperature at which the first mode appears, as the source shifts from the bar to the finger gates.In (b) is the deformation of the system when a mode derives from the bar and in (c) when a mode derived from the finger gates.
depositing material on top of the finger gates on their own or by including the bar as well.In the first case, this can be made by either depositing an extra layer using the same PMMA layer used to deposit the finger gates metal, or by adding an extra step where only the finger gates are exposed for the extra deposition and hence there will be material both on the finger gates and in-between them.The data from the simulations for the above suggestion are illustrated in figure 5.
In figure 5(a), depositing material on the finger gates and in-between has a better response irrespective of the material of the gate.For the Au and Al cases, 30 nm of PMMA, SiO 2 and AlO x had their response simulated independently, the most successful being the latter, where the phonon freeze-out temperature was increased by almost a factor of two in both metal cases.
This result is a lot easier to visualise, by plotting the number of modes that propagate, as per the figure 5(b).There is a huge suppression in the temperature of the phonon modes, as for example, the first 50 modes in the reference device appear by ∼150 mK, while for the corresponding system with 30 nm of AlO x deposited fully on the finger gates this happens at ∼250 mK, according to the simulations, as shown in the inset of the same figure .A much better and easier way to help the device's performance, is to deposit the strengthening material fully on the bar and gates.In our simulated case this is in a cross-like shape, where the AlO x is covering both the bar and the finger gates.The geometry is illustrated in the inset of figure 6(a).We test the response for thickness of the insulator from 0 nm up to 450 nm.For a bar with a width of 500 nm, length of 1000 nm and a gap of 100 nm, we find that there is a linear increase of the activation temperature of the first phonon mode.After To strengthen the finger gates we can deposit material through a PMMA layer only on them (fingers only, red), or on and in-between them (fingers full and gaps, black).Both cases lead to an increase in the temperature of the first mode as shown in (a), with the latter being the most efficient.Apart from the material of the gates on their own, three more materials were tested for their response, PMMA, SiO 2 and AlOx, all were 30 nm thick.The activation temperature both for Au and Al gates almost doubled to that of the uncoated gates.In (b) is the number of modes as a function of activation temperature for the material with the best performance, AlOx, for the first 10 modes and up to 375 modes in the inset.
the thickness of 200 nm though, the slope is still linear but different to the initial one.As shown before in figure 4, by increasing the gap, the phonon modes appear at a lower temperature.
At this point it is important to emphasise the fact that the AlO x layers needs to stop at the end of either side of the suspended part, as the mismatch in the phonon speed of sound is needed to assist with the phonon localisation.
Using the extreme case of the 450 nm thick AlO x that presents an activation temperature of more than 140 mK, as presented in figure 6(a), we run the previous scenario of the gap increase, shown in figure 6(b).Increasing the gap, the first mode changes rapidly and is slightly below 100 mK once the gaps reach 500 nm.If we need to go for a device with a bigger gap-up to 1500 nm-the activation temperature for the first mode drops down to about 40 mK which is approximately 5 mK for every 100 nm increase on the gap.In that case, a smarter geometry for example a fishbone-like structure, as in the inset of figure 6(b), could help with the phonon mode temperature.The two red points indicate an addition of two more arms per side-far right column, cartoon in the middle.For the 1000 nm gap, there is an increase of 18 mK from 67 mK to 85 mK.For the 700 nm gap, the corresponding increases slightly smaller-14 mK-from 84 mK to 98 mK.In the case of a further addition of two more arms per side, indicated with a star in figure 6(b), there is a further increase of 8 mK, from 98 mK to 106 mK.
In practice, employing atomic layer deposition (ALD) to deposit the AlO x layer, it will conformally cover the sample's surface, and in addition, the amount of material that will be deposited in the trenches and underneath the bar will be the same as that deposited on the top surface [48,49].It worth pointing out that the amount of material to be deposited on the top of the bridge using this technique, is limited by the gap between the bridge and the heat bath.Given the nature of the ALD, the same amount of material will get deposited at the bottom of the bridge and on the top of the substrates.That leads to the conclusion that the maximum AlO x thickness (t AlOx ) to be deposited, should be less than half of the distance between the bottom of the bridge and the top of the substrate (d bb-hb ), or t AlOx < 1 2 d bb-hb , so that these two remain disconnected.
In inset of figure 6(c) is the model used for this simulation and the outcome as a comparison to the test device with a single layer on top is the main figure of 6(c).Four more models were created with a thickness of the insulating layer being 60 nm, 120 nm, 150 nm and 200 nm, with the same amount at the bottom of the bar as well.With the extra layer of AlO x at the bottom of the bar and finger gates, the device is gaining about 10 mK more on the suppression of the first phonon mode, without any extra cost to the fabrication.Instead of depositing at least 200 nm of the insulating material-if it were a single layer, the same outcome can be achieved with 150 nm (95 mK) or for a slightly lower temperature (86 mK), with 120 nm.

Cleanroom fabrication
Multiple steps were needed to fabricate the devices with standard optical lithography ones for the Mesa etching and AuGeNi Ohmic contacts steps.The MBE grown wafer structure used, can be seen in the supplementary information.The second mesa that defines the double clamped cantilevers, was patterned using electron beam lithography on a PMMA 950 K A11 in MIBK (1:5) resist, approximately 80 nm thick.The resist is developed using an IPA based fast feature developer, IPA:MEK:MIBK (15:5:1), for approximately 8 s and then cleaned in IPA.The cantilevers were defined using an etching solution which is the same as before, and etching down to 120 nm so as to fully expose the surface of the sacrificial layer.
Finally, there followed another optical lithography step which exposed only the part of the chip that the sacrificial layer would be removed from, in order to release the bridges.The sample fabrication was completed by dipping the devices in a solution of 10% concentration of HF, to remove the aforementioned sacrificial layer underneath the Hall bars.The challenges of fabricating and measuring sub-micron wide freestanding semiconductor devices has been reported earlier [19,28,50,51] and more recently [21].
After the device is written on the PMMA layer using electron beam lithography, to remove the broken polymer chains the sample needs to be developed in a electron beam lithography-compatible solution so that its shape can be formed on the absence of resist-initially-and on the GaAs/AlGaS heterostructure through wet etching-in the end.The nanofabricated device modes can be altered as a results of a few extra seconds of development, because this can change the bar's size and also the gap over which the finger gates will be suspended for, by a lot.Some of the cleanroom fabrication steps, can be seen in figures 7(a) and (b).The top part is after depositing the gate electrodes on the GaAs/AlGaAs heterostructure, where the top and bottom row of finger gates are the plungers and barriers, while the electrode in the middle is the central gate.After that, an electron-beam lithography defined mesa etching step follows, to form the transport channel, exposing the Al rich AlGaAs sacrificial layer.An HF etching is then used to remove the latter.In figure 7(b) we fabricated arrays of free-standing devices where the different geometrical features are explored, in order to form a reliable fabrication recipe.The SEM images of figure 7 are missing the deposition of the AlO x layer that was discussed earlier.

Conclusion
Using the simulation software ANSYS (Package: Ansys mechanical R1, 2020, ANSYS, Inc), we studied how to create phonon localisation conditions on suspended electrostatically defined QD devices in a 2DEG with gates on depletion mode.We show that the first phonon mode depends strongly on the material used for the finger gates and also the material used to cover those and the bar.The best candidate is AlO x , a material that can commonly be deposited either using ALD, oxygen plasma sputtering or ebeam evaporation that almost doubles the first phonon mode temperature.
Covering the system fully, improves the goal even further and increasing the thickness to about 200 nm can set the first phonon mode temperature to more than 100 mKa safe temperature that can routinely be achieved on a dilution refrigerator system.Working towards a more experimentally realistic system, the extra AlO x layer at the bottom of the bar further suppresses the first phonon mode.For a system of 1000 nm long, 120 nm thick, 500 nm wide with finger gates suspended for 100 nm and a crosssectional area of 30 nm × 30 nm, the temperature of the first phonon mode can be pushed almost five times from its fundamental frequency, around 20 mK to that of 100 mK.

Figure 1 .
Figure 1.From top to bottom are the first three phonon modes that will be seen in the system for a double clamped cantilever.The above simulated phonon modes have been produced for a 300 nm wide, 120 nm thick, 1000 nm long GaAs bar with a Au bar with the same length but 60 nm wide and 30 nm thick on top, using the mechanical package of ANSYS.The fundamental frequencies that these modes occur, are at about 462 MHz, 1139 MHz and 1996 MHz, that correspond to 22 mK, 55 mK and 96 mK respectively.

Figure 2 .
Figure 2.For the length dependence test shown in (a), we kept the thickness of the device fixed to 120 nm and varied its length from 250 nm up to 3500 nm.A T ∝ 1 / l 2 trend characterises the system for L ⩾ 500 nm, as the Euler-Bernoulli equation describes it for L / t > 1.This agrees with equation (3) where the activation

Figure 3 .
Figure 3.(a) By adding the finger gates (points in red, using geometry in (c) and in the red frame) to the bar (points in black, using geometry in (b) and in the black frame), the first phonon modes appear at the same temperature, within a small margin.The increase in the number of modes, though, changes massively on the device with the finger gates.The system gets a lot more complicated as apart from the bar's vibrational modes, the ones from 14 finger gates are added to it.As these can vibrate as a collaborate system and each one individually at different frequencies with a small difference between them, their activation temperature will vary with those frequencies.In (b) and (c) are the simulated systems mentioned above.

Figure 4 .
Figure 4.In (a), there is small increase in the phonon freeze-out temperature coming from the bar (geometry and deformation used, is shown in (b)) when using Al (b) rather than Au (b) gates by about 3.5 mK, as their Young's modulus are quite similar, but the densities are quite different.The effect of the material and first phonon mode is more interestingly seen for longer finger gates' (geometry and deformation used, is shown in (c)) gaps for Al (f) and Au (f).For both materials, the increase of the gap that the finger gates are suspended for, has an opposite effect to the temperature at which the first mode appears, as the source shifts from the bar to the finger gates.In (b) is the deformation of the system when a mode derives from the bar and in (c) when a mode derived from the finger gates.

Figure 5 .
Figure 5.To strengthen the finger gates we can deposit material through a PMMA layer only on them (fingers only, red), or on and in-between them (fingers full and gaps, black).Both cases lead to an increase in the temperature of the first mode as shown in (a), with the latter being the most efficient.Apart from the material of the gates on their own, three more materials were tested for their response, PMMA, SiO 2 and AlOx, all were 30 nm thick.The activation temperature both for Au and Al gates almost doubled to that of the uncoated gates.In (b) is the number of modes as a function of activation temperature for the material with the best performance, AlOx, for the first 10 modes and up to 375 modes in the inset.

Figure 6 .
Figure 6.On a bar with a width of 500 nm and length 1000 nm, three test were conducted and shown in (a).The AlOx used was either covering 150 nm of the width, 300 nm or 450 nm symmetrically from the centre, the geometry of which, is shown as an inset.The thicker the insulating layer the more the temperature of the first phonon mode is increased, initially in a linear way and then the slope is changing.For the thickest AlOx layer, a gap size test (shown in (b)) with an increase from 100 nm to 1500 nm increases the activation temperature by 100 mK.Adding an extra set of side-arms can push it up by almost 20 mK (bottom right inset) and for 700 nm gap another set pushes it to almost 110 mK (top right inset).In (c) is presented a comparison of the temperature of the first mode as a function of AlOx thickness that has been grown on top of the bridge (in black) or by using ALD (in red).That means that the insulator will be grown both on top and bottom of the bridge, as shown in the inset.

Figure 7 .
Figure 7.Both the top and bottom SEM images are test devices and arrays of devices, in order to create a reliable fabrication recipe.The top figure has a fully suspended device held only by the metal electrodes and the bottom is an array of the simulated device, where the different geometrical features are explored.