Single-Shot Readout Performance of Two Heterojunction-Bipolar-Transistor Amplification Circuits at Millikelvin Temperatures

High-fidelity single-shot readout of spin qubits requires distinguishing states much faster than the T1 time of the spin state. One approach to improving readout fidelity and bandwidth (BW) is cryogenic amplification, where the signal from the qubit is amplified before noise sources are introduced and room-temperature amplifiers can operate at lower gain and higher BW. We compare the performance of two cryogenic amplification circuits: a current-biased heterojunction bipolar transistor circuit (CB-HBT), and an AC-coupled HBT circuit (AC-HBT). Both circuits are mounted on the mixing-chamber stage of a dilution refrigerator and are connected to silicon metal oxide semiconductor (Si-MOS) quantum dot devices on a printed circuit board (PCB). The power dissipated by the CB-HBT ranges from 0.1 to 1 μW whereas the power of the AC-HBT ranges from 1 to 20 μW. Referred to the input, the noise spectral density is low for both circuits, in the 15 to 30 fA/\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\bf{Hz}}}$$\end{document}Hz range. The charge sensitivity for the CB-HBT and AC-HBT is 330 μe/\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\bf{Hz}}}$$\end{document}Hz and 400 μe/\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sqrt{{\bf{Hz}}}$$\end{document}Hz, respectively. For the single-shot readout performed, less than 10 μs is required for both circuits to achieve bit error rates below 10−3, which is a putative threshold for quantum error correction.


I. SET GEOMETRIES AND DETAILS
The SET connected to the AC-HBT uses a single layer doped poly-Si electrode structure on 50 nm thick SiO 2 , providing a mobility of 19,500 cm 2 /Vs at 4 K. The poly-Si gate layer is etch-defined into electrodes that control the formation of the SET (upper left in Figure 1(a) SEM image) and two quantum dots (under gates RD and LD). Regions of electron enhancement are indicated by the highlighted regions.
The Si-MOS device in the CB-HBT circuit is similar to the Si-MOS device in the AC-HBT circuit with the exception that the SiO 2 layer is 35 nm thick and the bottom layer is isotopically purified silicon (500 ppm 29 Si). The 28 Si isotope has no net nuclear spin, therefore it is ideal for qubits to be formed in because decoherence due to magnetic noise is highly suppressed. Phosphorous ( 31 P) donor atoms are imbedded in the 28 Si layer using ion implantation near where the quantum dot is intended to be formed (red dot in Figure 2(a) SEM image). The CB-HBT and AC-HBT were characterized using different Si-MOS devices possessing different electrostatic gate layouts ( Figure S1). The geometry of the gate layout affects the mutual capacitance between the SET and the quantum dot. More capacitive coupling results in larger changes in the electrochemical potential of the chargesensor for a given quantum dot charging event 1 . Since changes in electrochemical potential of the charge-sensor result in changes in current through the charge-sensor, larger changes result in larger signal. Therefore, more mutual capacitance leads to larger readout signals, faster readout times, and higher readout fidelity.
The gate geometry used in the Si-MOS device connected to the CB-HBT had the SET 33% closer to the quantum dot than in the Si-MOS device connected to the AC-HBT. The closer SET proximity in the CB-HBT resulted in an increase in sensitivity of approximately 34%. We compare the sensitivity of both circuits by dividing the voltage shift of the dot occupancy transition by the charge-sensor Coulomb blockade peak period. For the CB-HBT, the voltage shift was 18 mV and the chargesensor period was 337 mV (5.34% change). For the AC-HBT, the voltage shift was 12 mV and the charge-sensor period was 350 mV (4% change). Therefore, the SET in the CB-HBT was around 34% more sensitive to charging events than the AC-HBT.

II. CURRENT-BIASING EFFECT OF CB-HBT CIRCUIT
Since the node that connects the SET source to the HBT base is floating, the bias across the SET cannot be set to a fixed voltage in the CB-HBT circuit. Verilog-A models were created to simulate the behavior of the circuit when biasing the SET through multiple regions of Coulomb blockade via an electrostatic gate. As the SET resistance changes due to Coulomb blockade, the sourcedrain bias across the SET changes to allow current to flow into the base of the HBT ( Figure S3(b)). In order for this to happen, the HBT trades base-emitter voltage for minimal impact to operation. Although the trade in voltage results in a relatively small change in HBT collector current during, for example, a single-shot readout event, this signal is approximately 100 larger than the SET source-drain signal without an HBT (e.g. ΔI C = 10 nA vs. ΔI SET = 100 pA).
The Verilog-A model estimates the small signal resistances as: r set = 200 kΩ and r π = 10 MΩ (where r π is the small signal resistance of the base-emitter junction). Most of the emitter bias voltage is across the baseemitter junction at all times (since r set < < r π ), therefore the CB-HBT is a current-biasing circuit. The currentbiasing behavior is highlighted in Figure S4(a), where three Coulomb blockade peaks are plotted. For comparison, three Coulomb blockade peaks are plotted for the AC-HBT case ( Figure S4(b)). The CB-HBT amplified peaks are broadened by the current-biasing effect and the blockade region never reaches zero current as it would with a smaller constant voltage bias. The AC-HBT amplified peaks are much narrower and minimally broaden due to having a constant, small voltage bias regardless of HBT power. Comparable sensitivities can be achieved for either circuit around 10 µA/V.

III. ELECTRON TEMPERATURE MEASUREMENT
Heating of electrons in the quantum dot due to the operation of the connected HBT is a concern, therefore we examined the dependence of electron temperature on HBT amplifier bias (Figure 3(e)). For the CB-HBT, the electron temperature of the QD was measured by extracting the width of a Coulomb blockade peak of the QD (not SET) as a function of fridge temperature. The QD was tuned to a transport regime where the QD was approximately equally tunnel-coupled to both reservoirs and there were around 10 electrons in the QD. The sourcedrain bias was reduced to 5 µV rms to avoid bias heating. A Coulomb peak was chosen where a minimum width was observed in Coulomb diamond measurements. After extracting the lever-arm of the gate used to measure the broadening (13 µeV/mV), we find that the minimum linewidth yields an electron temperature around 150 mK. Heating of the QD begins where the CB-HBT is operating with over 100 gain, therefore the CB-HBT circuit can amplify well while heating the electrons to 160-200 mK.
For the AC-HBT setup, the base electron temperature was around 120 mK. This is confirmed by the measurements of the electron temperature when measuring the SET signal directly through the shunt resistor (R S in There is a slight difference in the two curves, however the performance at 60 mK is enough to efficiently amplify and perform single-shot readout. ( Figure S5). For powers above this threshold, the electron temperature increases approximately linearly with power. This might be due to local heating of the PCB and wires, which increase the temperature of the nearby device 2 . No effort has been made to heat sink the AC-HBT in this experiment, so further tests with various heat sinking options will be performed to minimize the increase in electron temperature. Nonetheless, an electron temperature of 200 mK is achieved for the bias condition that provides the minimum amplifier noise.

IV. HBT CHARACTERIZATION
Before being used in either amplification circuit, HBTs are initially characterized in liquid helium at 4 K using PCBs with eight HBTs mounted on them. We find that HBT performance at 4 K-particularly current gain vs. base current-changes minimally when HBTs are cooled down to 20-60 mK in a dilution refrigerator ( Figure S6(b)). This is most likely due to the charge-carrier transport mechanism changing from a drift-diffusion regime (temperature dependent) to a tunneling regime (barrier dependent) at around 30 K 3 .
In to the HBT base and collector terminals. A power supply (emitter bias) is connected to the HBT emitter terminal and used to bias the HBT to different operating regimes. The emitter bias has to reach approximately -1 V for the HBT to begin operating in an amplifying regime. As the emitter bias is changed from -1.00 V to around -1.07 V, the collector and base current begin to increase exponentially. The current gain, defined by dividing the collector current by the base current, also increases exponentially as emitter bias changes.
Previous measurements without HBT amplification circuits indicate that the SET current should be below several hundred pA in order to avoid QD electron heating. For the CB-HBT, we select HBTs based on their current gain at low base currents. Around 20% of HBTs characterized will have current gain > 100 at base current < 200 pA ( Figure S6(a)). For the AC-HBT, the transconductance (g m ) is the only metric required for selection. Since the HBTs were fabricated with g m as a primary metric, > 80% of HBTs are usable for the AC-HBT circuit even at low temperatures. However, g m does not scale ideally in these HBTs at cryogenic temperatures. For a given HBT, g m ∝ I n C , where n = 1 in normal conditions. In the HBTs used in this work, n ≈ 0.8, which leads to suboptimal SNR at higher power. CB-HBT effective circuit model. The shot noise current source, i b-shot , is in parallel with rset and rπ. Most of the shot noise does not enter the base of the HBT because rset < < rπ. The signal, iset, is also shown, which is amplified according to Equation S1.

V. CB-HBT SMALL SIGNAL GAIN
The gain of the CB-HBT is calculated using a standard BJT small-signal model. A small voltage fluctuation at the base node is usually converted to a large current fluctuation at the collector node by the transconductance, g m = dic dv be . This voltage fluctuation is usually the smallsignal base-emitter junction resistance, r π , multiplied by the base current. However, in the case of the CB-HBT, r set || r π , therefore the parallel combination of the two resistances is required to calculate gain:

VI. NOISE MODELS
Sources of noise in the HBT amplification circuits include: shot noise, Johnson noise, triboelectric noise associated with the coaxial lines coupled to fridge vibration 4 , room temperature amplifier noise, and other instrumental noise. At relatively low power operation regimes (< 1 µW for the AC-HBT and < 200 nW for the CB-HBT), the noise due to vibrations in the fridge dominates at around 1 pA/ √ Hz. The input noise spectral density of the room temperature amplifier is relatively low (100-500 fA/ √ Hz), therefore we focus on noise sources much more dominant. When either circuit is operating in a regime appropriate for single-shot readout, the base shot noise is greater than the collector shot noise (Figures 1(e) and 2(e)). For the SET shot noise in either case, we do not consider a Fano factor, which would reduce the noise for a given power 5,6 . The total noise for either circuit is calculated by assuming noise sources are independent processes and adding noise sources in quadrature.
Noise modeling for the CB-HBT circuit is nontrivial because of current division at the HBT base node since r set r π . The SET and base current are reduced to a Norton equivalent circuit, and the HBT is reduced to r π connected to a current source which takes voltage fluctuations (v be ) across r π and converts them to collector current via the transconductance, g m . For the CB-HBT, the noise model is a shot noise current source (i b−shot = √ 2 e I B ∆f , where I B is the DC base current, and ∆f is the bandwidth centered on frequency f ) in parallel with r set and r π ( Figure S8(b)). Since r set r π , most of the shot noise current goes through the SET to ground, and a much smaller amount enters the HBT base and is amplified. The amplified base shot noise is shown in Equation S2: This amplified base shot noise is estimated in Figure 2(e) as the orange curve where g m and r π are calculated from Gummel plots of the HBT and r set is assumed to be 3 MΩ, which was verified in later measurements with the HBT disconnected from the Si-MOS device.
The noise model for the AC-HBT is similar to the CB-HBT with r S and r B added in parallel to r set and r π . The coupling capacitor, C, is considered a short at the frequencies appropriate to model noise in the AC-HBT. The Johnson noise of R S in the AC-HBT circuit is v s−jn = √ 4 k B T R S ∆f (where T is the temperature) and does not contribute significantly in the single-shot operation regime. Since the AC-HBT has a separate current to bias the base-emitter junction, I SET = I B , therefore the base shot noise and SET shot noise are considered separately. However, I SET < I B , so the base shot noise is always dominant in amplifying regimes.

VII. AC-HBT BIAS TEE PARAMETERS
The bias tee parameters for the AC-HBT were chosen to be R S = 100 kΩ and C = 10 nF, which sets a high pass filter at 160 Hz. Operating the circuit at frequencies higher than 160 Hz aids in avoiding higher noise levels at lower frequency due to 1/f-like noise behavior in the system.
The shunt resistance value is chosen to be less than r set (100s of kΩ) so that most of the SET bias voltage drops across the SET.