Heterostructure growth<\/p>\n
The Ge\/SiGe heterostructure material is grown using reduced-pressure chemical vapour deposition in an ASMI Epsilon 2000 reactor. Starting from a Ge wafer, a 2.5\u2009\u03bcm strain-relaxed Si(1\u2013x)Gex buffer is grown at a temperature of 800\u2009\u00b0C with a final Ge concentration of x\u2009=\u20090.83 using three grading steps (1\u2009\u2212\u2009x\u2009=\u20090.07, 0.13, 0.17). We lower the growth temperature to 500\u2009\u00b0C for the growth of the final 200\u2009nm of the SiGe buffer layer, the 16\u2009nm Ge quantum well, the 55\u2009nm SiGe barrier layer and the sacrificial passivated Si cap layer. As measured in ref. 21<\/a>, this heterostructure supports a 2D hole gas with a high maximum mobility of 3.4(1)\u2009\u00d7\u2009106\u2009cm2\u2009V\u20131\u2009s\u20131 and a low percolation density of 1.22(3)\u2009\u00d7\u20091010\u2009cm\u22122. The threading dislocation density is 6(1)\u2009\u00d7\u2009105\u2009cm\u22122, nearly an order of magnitude lower than for the growth of Ge quantum wells with similar strain starting from a Si wafer.<\/p>\n Device fabrication<\/p>\n All devices are fabricated on the same Ge\/SiGe heterostructure on a Ge wafer as that measured in ref. 21<\/a>. The two linear array quantum dot devices are fabricated with multiple layers of Ti\/Pd and platinum\u2013germanosilicide (PtSiGe) ohmic contacts. These are defined by electron-beam lithography and created by thermally diffusing Pt at a temperature of 400\u2009\u00b0C. A first layer of Ti\/Pd (3\/17\u2009nm) barrier gates is separated from the heterostructure by a 7\u2009nm Al2O3 insulating oxide grown using atomic layer deposition. A second layer of Ti\/Pd (3\/37\u2009nm) plunger gates is created and separated by 5\u2009nm of Al2O3 from the first layer of barrier gates. Details for the fabrication of the 3\u20134\u20133 device (2D quantum dot array) can be found in ref. 6<\/a>.<\/p>\n Flank method electrical characterization of quantum dots<\/p>\n We cool multi-hole quantum dots defined underneath the charge sensor or plunger gates of a device in a Leiden cryogenic dilution refrigerator operating at a mixing chamber base temperature of 70\u2009mK. To measure the charge noise using the flank method, we first set a source\u2013drain bias (Vsd) of 0.1\u2009mV across the device and subsequently tune the surrounding gates until we measure a current of 1\u2009nA through the device. We define a multi-hole quantum dot underneath a sensor or plunger gate by fine tuning the barrier gates surrounding the gate of interest until we observe a spectrum of Coulomb peaks. We then measure the source\u2013drain current Isd on the left flank of the Coulomb peak where the slope \u2223dIsd\/dVsd\u2223 of the Coulomb peak is the largest at a rate of 2\u2009kHz for a duration of 100\u2009s using a Keithley DMM6500 multimeter. To calculate the current power spectral density SI, we first split each 100\u2009s current trace into ten segments of 10\u2009s each, and subsequently average over the current power spectral densities (\\({S}_{{{I}}}=1\/N\\mathop{\\sum }\\nolimits_{i = 1}^{N}{S}_{{{I}}}^{i}\\)) that we evaluate from each segment. For each Coulomb peak that we analyse, we convert the current power spectral density SI into charge noise power spectral density S\u03f5 using15<\/a><\/p>\n $${S}_{\\epsilon }=\\frac{{a}^{2}{S}_{{{I}}}}{| {\\rm{d}}{I}_{{\\rm{sd}}}\/{\\rm{d}}{V}_{{\\rm{S}}}{| }^{2}}\\,,$$<\/p>\n \n (1)\n <\/p>\n where VS is the sensor voltage and a is the lever arm extracted from the analysis of the corresponding Coulomb diamond.<\/p>\n Coulomb peak tracking<\/p>\n On sensor 1 of device 2, we perform an ~18\u2009h CPT experiment tracking the current Isd through a quantum dot while continuously sweeping across the quantum dot using its plunger gate. For the four charge sensors of the 3\u20134\u20133 device, we track the reflected signal of each charge sensor for ~2\u2009h when sweeping the sensor plunger gate voltage. We extract the position of the Coulomb peak by fitting each Coulomb peak to a hyperbolic secant function<\/p>\n $$y=\\frac{a}{{\\cosh }^{2}\\left(b\\,(x-{x}_{{\\rm{0}}})\\right)}+c\\,,$$<\/p>\n \n (2)\n <\/p>\n where y reflects the measured signal, x is the sensor\u2019s plunger gate voltage, x0 is the position for which the Coulomb peak is maximum and a, b and c are free fitting parameters. To calculate the voltage power spectral density SV from the Coulomb peak fluctuations, we split them into ten equal segments and for each segment we calculate the voltage power spectral density from a Fourier transformation. We find the charge noise power spectral density S\u03f5 (Fig. 3c<\/a>) using the evaluated lever arm of each sensor and using<\/p>\n $${S}_{\\epsilon }={a}^{2}{S}_{{{V}}}.$$<\/p>\n \n (3)\n <\/p>\n Voltage noise estimation on single-hole quantum dots<\/p>\n To estimate the voltage noise of the quantum dots where we know the exact hole occupancy, we repeatedly load a hole into a dot by continuously sweeping the plunger gate voltage under which the quantum dot is defined. We keep track of the voltage for which a hole loads into a dot by fitting each voltage sweep to a sigmoid function given by<\/p>\n $$y=\\frac{a}{1+\\exp \\left(\\frac{x-{x}_{{\\rm{0}}}}{\\tau }\\right)}+b\\,,$$<\/p>\n \n (4)\n <\/p>\n where y reflects the measured signal, x0 is the voltage for which a hole loads into a dot and a, b and \u03c4 are free fitting parameters. We split the voltage fluctuations into ten equal segments and calculate the voltage power spectral density SV from a Fourier transformation. The final voltage power spectral density is calculated from the average of the ten segments. Because of the complexity in determining with accuracy the lever arm of the plunger gates to the quantum dots in this regime, we maintain the metric of the charge noise in voltage, rather than in energy.<\/p>\n Estimation of the relative out-of-plane angle between magnetic field and substrate<\/p>\n We consider the average effective g-factor geff\u2009=\u20090.58 of the ten qubits49<\/a> and assume in-plane principal g-tensor components of gx\u2009=\u2009\u2212gy\u2009=\u20090.04 (ref. 6<\/a>) and an out-of-plane component of gz\u2009=\u200911 (ref. 26<\/a>). Assuming the g-tensor axes follow the crystal directions, the effective g-factor can be written in terms of the out-of-plane angle \u03b8 and the principal components:<\/p>\n $${g}_{{\\rm{eff}}}=\\sqrt{{g}_{x}^{2}\\cos {\\theta }^{2}+{g}_{z}^{2}\\sin {\\theta }^{2}}.$$<\/p>\n \n (5)\n <\/p>\n By inverting the equation, we estimate the most plausible misalignment angle of \u03b8\u2009=\u20093\u00b0.<\/p>\n Extraction of Rabi and Larmor frequency<\/p>\n We extract the Larmor frequency fL and Rabi frequency fR by fitting the data with the following functions. For the Rabi frequency we use<\/p>\n $$P_{\\uparrow}=A\\sin (2\\uppi {f}_{{\\rm{R}}}t+\\phi )\\exp (-{t}^{2}\/{\\tau }^{2})+C\\,,$$<\/p>\n \n (6)\n <\/p>\n where P\u2191 represents the measured up probability and t is the duration of the microwave burst. The Rabi frequency fR, decay time \u03c4, phase \u03d5, amplitude A and offset C are free fitting parameters. The extraction of the Larmor frequency is done using<\/p>\n $$P_{\\uparrow}=A\\frac{{f}_{{\\rm{R}}}^{\\,2}}{{f}_{{\\rm{R}}}^{\\,2}+{\\it{\\varDelta} }^{2}}{\\sin }^{2}\\left(0.5t\\sqrt{{f}_{{\\rm{R}}}^{\\,2}+{{\\it{\\Delta}} }^{2}}\\right)+C\\,.$$<\/p>\n \n (7)\n <\/p>\n Here \u0394\u2009=\u2009f\u2009\u2212\u2009fL with f the probed frequency range during measurement; t, A, fR, C and fL are free fitting parameters.<\/p>\n Qubit noise model<\/p>\n Low-frequency and hyperfine noise affecting the qubits hosted in the natural Ge\/SiGe heterostructure is modelled by26<\/a><\/p>\n $${S}_{{f}_{{\\rm{L}}}}=\\frac{{S}_{{\\rm{0}}}}{f}+{S}_{0,{\\rm{hf}}}\\exp \\left(\\frac{-{(\\,f-{f}_{{\\rm{Ge}}-73})}^{2}}{2{\\sigma }_{{\\rm{Ge}}-73}^{2}}\\right)\\,.$$<\/p>\n \n (8)\n <\/p>\n Here, S0 represents the low-frequency noise component at 1\u2009Hz and S0,hf represents the effective strength of the hyperfine noise acting on the qubit; fGe\u221273\u2009=\u2009\u03b3Ge\u221273B is the precession frequency of 73Ge determined by its gyromagnetic ratio \u03b3Ge\u221273\u2009=\u20091.48\u2009MHz\u2009T\u20131 and the magnetic field B; and \u03c3Ge\u221273 represents the spread of the 73Ge precession frequencies. We follow the same fitting procedure as outlined in the methods of ref. 26<\/a> to extract S0, S0,hf and \u03c3Ge\u221273.<\/p>\n For the fitting of the data containing influence of the 29Si nuclear spin, we expand equation (8<\/a>) with a second Gaussian peak:<\/p>\n $$\\begin{array}{ll}{S}_{{f}_{{\\rm{L}}}}=\\frac{{S}_{{\\rm{0}}}}{f}&+{S}_{0,{\\rm{hf}}}^{{\\rm{Ge}}-73}\\exp \\left(\\frac{-{(\\,f-{f}_{{\\rm{Ge}}-73})}^{2}}{2{\\sigma }_{{\\rm{Ge}}-73}^{2}}\\right)\\\\ &+{S}_{0,{\\rm{hf}}}^{{\\rm{Si}}-29}\\exp \\left(\\frac{-{(\\,f-{f}_{{\\rm{Si}}-29})}^{2}}{2{\\sigma }_{{\\rm{Si}}-29}^{2}}\\right)\\,.\\end{array}$$<\/p>\n \n (9)\n <\/p>\n The fitting procedure is split into a two-stage process, where we first use equation (8<\/a>) to find \\({S}_{0,{\\rm{hf}}}^{{\\rm{Ge}}-73}\\) and \u03c3Ge\u221273. We fix these parameters and then use equation (9<\/a>) to find S0, \\({S}_{0,{\\rm{hf}}}^{{\\rm{Si}}-29}\\) and \u03c3Si\u221229. The precession frequencies of fGe\u221273 and fSi\u221229 are also fixed.<\/p>\n Estimation of the hyperfine noise contribution<\/p>\n Using the average extracted parameters of \\({S}_{0,{\\rm{hf}}}^{{\\rm{Ge}}-73}=1.1(3)\\times 1{0}^{6}\\,{{{\\rm{Hz}}}^{2}} \\, {{\\rm{Hz}}^{-1}}\\) and \u03c3Ge\u221273\u2009=\u200912(1)\u2009kHz that describe the modelled Gaussian peak in the power spectral density at the frequency fGe\u221273, we estimate an integrated noise of the Larmor frequency fluctuations of<\/p>\n $${\\sigma }_{f}=\\sqrt{\\sqrt{2\\uppi } \\times {S}_{0,{\\rm{hf}}}^{{\\rm{Ge}}-73}{\\sigma }_{{\\rm{Ge}}-73}}=180(8)\\,{\\rm{kHz}}.$$<\/p>\n \n (10)\n <\/p>\n This sets an approximate boundary for the hyperfine-limited dephasing time of \\({T}_{2}^{\\,* }={(\\uppi \\sqrt{2}{\\sigma }_{f})}^{-1}=1.25(5)\\,\\upmu {\\rm{s}}\\), qualitatively similar to what is measured experimentally, on the order of 1\u2009\u03bcs to 2\u2009\u03bcs (ref. 49<\/a>).<\/p>\n We use the same procedure to assess the influence of the interaction with the 29Si nuclei in the barrier to the Ge hole spin qubits. We consider the average parameters extracted from the fits in Fig. 4<\/a>, \\({S}_{0,{\\rm{hf}}}^{{\\rm{Si}}-29}=9.0(8)\\times 1{0}^{3}\\,{{{\\rm{Hz}}}^{2}} \\, {{\\rm{Hz}}^{-1}}\\) and \u03c3Si\u221229\u2009=\u200999(20)\u2009kHz, that lead to an integrated noise of \u03c3f\u2009=\u200947(5)\u2009kHz and a 29Si-limited \\({T}_{2}^{\\,* }=4.8(4)\\,\\upmu {\\rm{s}}\\).<\/p>\n Effective voltage noise and charge-noise-limited \\({{\\boldsymbol{T}}_{\\mathbf{2}}}^{\\mathbf{*}}\\)<\/p>\n We assume a spatially homogeneous distribution of uncorrelated fluctuators, and exploit knowledge of the g-factor susceptibility to voltage variations in all the surrounding gates49<\/a>. Considering traps under the gates the most dominant, uncorrelated noise sources, we can estimate the resulting overall g-factor susceptibility of the hole spin qubits as \\(\\frac{\\Delta g}{\\Delta V}=\\sqrt{{\\sum }_{i}{(\\frac{\\updelta g}{\\updelta {V}_{i}})}^{2}}\\approx 6.7\\times \\,1{0}^{-4}\\,{{\\rm{mV}}}^{-1}\\), with \u03b4g\/\u03b4Vi the susceptibility of the g-factor g to a voltage variation on each gate i (Vi) of the twelve barrier and ten plunger gates controlling the array. The associated Larmor qubit fluctuations result then into \\(\\frac{\\Delta {f}_{{\\rm{L}}}}{\\Delta V}=\\frac{1}{h}\\,\\frac{\\Delta g}{\\Delta V}\\,{\\mu }_{{\\mathrm{B}}}B\\), where h is the Planck constant and \u03bcB is the Bohr magneton. For B\u2009=\u2009117.5\u2009mT the value of the fluctuations is ~1.1\u2009MHz\u2009mV\u22121. We can then derive an effective voltage power spectral density value at 1\u2009Hz of<\/p>\n $${S}_{{{V}}}=\\frac{{S}_{{\\rm{0}}}}{{\\left(\\frac{\\Delta {f}_{{\\rm{L}}}}{\\Delta V}\\right)}^{2}}=140(30)\\,\\upmu {{{\\rm{V}}}^{2}} \\, {{\\rm{Hz}}^{-1}}$$<\/p>\n \n (11)\n <\/p>\n that results in an effective voltage noise of \\({S}_{{{V}}}^{1\/2}=12(1)\\,\\upmu {\\rm{V}} \\, {\\rm{Hz}}^{-1\/2}\\).<\/p>\n We evaluate the charge-noise-limited \\({T}_{2}^{\\,* }\\) using the extracted voltage noise amplitude of \\({S}_{{{V}}}^{1\/2}=12(1)\\,\\upmu {\\rm{V}} \\, {\\rm{Hz}}^{-1\/2}\\) at 1\u2009Hz and the effective g-factor susceptibility of \\(\\frac{\\Delta g}{\\Delta V}=6.7\\times 1{0}^{-4}\\,{{\\rm{mV}}}^{-1}\\). The dephasing time \\({T}_{2}^{\\,* }\\) can be approximated in a quasi-static configuration by6<\/a>,15<\/a>,50<\/a><\/p>\n $${T}_{2}^{\\,* }=\\frac{1}{\\sqrt{2}\\uppi \\Delta f},$$<\/p>\n \n (12)\n <\/p>\n with \u0394f the amplitude of the qubit frequency fluctuations due to a voltage noise with root mean square amplitude of \u0394Vr.m.s.<\/p>\n $$\\Delta f=\\frac{\\Delta g}{\\Delta V}\\Delta {V}_{{\\rm{r.m.s.}}}.$$<\/p>\n \n (13)\n <\/p>\n We compute \u0394Vr.m.s. over a period of time T assuming a power spectral density of SV\/f, and integrating the noise from low-frequency and high-frequency cut-off values, fL\u2009=\u2009T\u22121 and fH, respectively:<\/p>\n $$\\Delta {V}_{{\\rm{r.m.s.}}}=\\sqrt{\\mathop{\\int}\\nolimits_{{f}_{{\\mathrm{L}}}}^{{f}_{{\\mathrm{H}}}}\\frac{{S}_{{{V}}}}{f}\\,{\\mathrm{d}}f}=\\sqrt{{S}_{{{V}}}\\ln \\frac{{f}_{{\\mathrm{H}}}}{{f}_{{\\mathrm{L}}}}}.$$<\/p>\n \n (14)\n <\/p>\n By considering realistic values of fL\u2009=\u20091\u2009mHz and fH\u2009=\u20091\u2009MHz, we determine \\(\\Delta {V}_{\\rm{r.m.s.}}=54(5)\\,\\upmu {\\rm{V}} \\, {\\rm{Hz}}^{-1\/2}\\). We then distinguish two cases. For B\u2009=\u2009117.5\u2009mT, the resulting fluctuations lead to \u0394f\u2009=\u200960(5)\u2009kHz and a charge-noise-limited dephasing time of \\({T}_{2}^{\\,* }=3.7(3)\\,\\upmu {\\rm{s}}\\). For a low magnetic field of 10\u2009mT, we obtain an increased value of \\({T}_{2}^{\\,* }=44(4)\\,\\upmu {\\rm{s}}\\) due to the more than tenfold reduction in \u0394f.<\/p>\n","protected":false},"excerpt":{"rendered":"Heterostructure growth The Ge\/SiGe heterostructure material is grown using reduced-pressure chemical vapour deposition in an ASMI Epsilon 2000…\n","protected":false},"author":3,"featured_media":44397,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25],"tags":[26985,2270,834,19533,26986,26984,492,14882,13632,159,67,132,68],"class_list":{"0":"post-44396","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-biomaterials","9":"tag-condensed-matter-physics","10":"tag-general","11":"tag-materials-science","12":"tag-nanotechnology","13":"tag-optical-and-electronic-materials","14":"tag-physics","15":"tag-quantum-dots","16":"tag-quantum-information","17":"tag-science","18":"tag-united-states","19":"tag-unitedstates","20":"tag-us"},"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@us\/114808557936249672","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/44396","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/comments?post=44396"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/44396\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media\/44397"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media?parent=44396"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/categories?post=44396"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/tags?post=44396"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}