The deviations of cryogenic collector current–voltage characteristics of SiGe heterojunction bipolar transistors (HBTs) from ideal drift-diffusion theory have been a topic of investigation for many years. Recent work indicates that direct tunneling across the base contributes to the non-ideal current in highly scaled devices. However, cryogenic discrepancies have been observed even in older-generation devices for which direct tunneling is negligible, suggesting that another mechanism may also contribute. Although similar non-ideal current–voltage characteristics have been observed in Schottky junctions and were attributed to a spatially inhomogeneous junction potential, this explanation has not been considered for SiGe HBTs. Here, we experimentally investigate this hypothesis by characterizing the collector current ideality factor and built-in potential of a SiGe HBT vs temperature using a cryogenic probe station. The temperature dependence of the ideality factor and the relation between the built-in potential as measured by capacitance–voltage and current–voltage characteristics are in good qualitative agreement with the predictions of a theory of electrical transport across a spatially inhomogeneous junction. These observations suggest that inhomogeneities in the base–emitter junction potential may contribute to the cryogenic non-idealities. This work helps to identify the physical mechanisms limiting the cryogenic microwave noise performance of SiGe HBTs.

Silicon–germanium heterojunction bipolar transistors (HBTs) are widely used in microwave applications such as high-speed communications and radar systems owing to their competitive microwave performance, low cost, and ease of integration compared with III–V compound semiconductor devices.1 Technological advances such as reduced emitter widths, decreased base resistances and extrinsic capacitances, and advanced epitaxial techniques have enabled microwave noise performance approaching that of III-V high electron mobility transistors.2,3 As a result, cryogenic SiGe HBTs have recently been considered for applications in quantum computing and radio astronomy.4–6 

The cryogenic microwave performance of SiGe HBTs has been investigated following their initial development in the 1980s7 as various performance metrics such as transconductance and noise figure improve with cooling. However, below 77 K, these improvements are observed to plateau with decreasing temperature,8–10 corresponding to a temperature-dependent collector current ideality factor n ( T ) that greatly exceeds unity at cryogenic temperatures.11,12 This behavior differs markedly from the predictions of drift-diffusion theory for fabricated p–n junctions for which n is close to unity and independent of temperature, and the transconductance increases inversely with temperature.13,14 Because the minimum noise temperature is directly proportional to n ( T ) in the limit of f / f t 1 and low base resistance (see Eq. 5.1 in Ref. 10), identifying the physical origin of the discrepancies is necessary to improve the cryogenic microwave noise performance of SiGe HBTs.

The cryogenic non-ideal behavior has been attributed to various mechanisms including quasiballistic transport,9,10 direct tunneling,11 or trap-assisted tunneling.15 However, a theoretical study has reported that quasiballistic electron transport cannot explain the observed collector cryogenic non-idealities.16 Recent works indicate that direct tunneling across the base can account for the non-idealities in highly scaled devices.5,11,12,17 At the same time, non-idealities have been observed in devices with base widths of 100 nm5,9 for which direct tunneling is negligible. This observation suggests that, in addition to tunneling, another mechanism may contribute to cryogenic non-ideal current–voltage characteristics in SiGe HBTs.

In a different context, similar anomalies have been observed and extensively investigated in Schottky diodes,18–21 and they were ultimately attributed to spatial inhomogeneities in the built-in potential Φ B I.22–25 Although semiconductor junctions are often modeled as uniform across their lateral area, in fact various imperfections exist, which affect the local electronic structure of the junction, a point that was recognized as early as 1950.26 Even at epitaxial interfaces, it was found that different crystallographic orientations27–29 or the presence of dislocations30,31 can lead to potential barrier height variations on the order of hundreds of mV. In Schottky junctions, these inhomogeneities have been directly observed using ballistic electron emission microscopy.32,33 Various theories and numerical analyses of the electrical characteristics of inhomogeneous junctions have been reported and lead to compatible conclusions.22–24 In particular, the theory of Werner and Güttler makes several predictions regarding the temperature dependence of the ideality factor and the relation between the built-in potential as measured by different methods.23,34,35 However, an experimental test of these predictions for SiGe HBTs has not yet been reported.

Here, we perform this experimental investigation by characterizing the collector current ideality factor and built-in potential of a SiGe HBT from room to cryogenic temperatures. We find that the measured temperature dependence of the ideality factor and the relation between built-in potential as determined by capacitance–voltage and current–voltage characteristics are compatible with the theoretical predictions. This observation suggests that inhomogeneities in the base–emitter junction potential could be a mechanism affecting the cryogenic current–voltage characteristics in SiGe HBTs. We discuss how the existence of barrier inhomogeneities could be further confirmed. Our work advances efforts to improve the cryogenic electrical characteristics and, hence, microwave noise performance of SiGe HBTs.

The theory of Werner and Güttler describes electrical transport over the lateral area of a Schottky junction with a spatially inhomogeneous barrier characterized by a mean barrier height and a variance.23 The distribution of the barrier heights is taken to be Gaussian, an assumption that is supported by local measurements of the barrier heights (Fig. 19.12 in Ref. 13). The variance in the distribution is assumed to decrease with increasing junction bias due to the pinch-off of low-barrier patches of dimension less than the depletion length, a concept that was originally introduced in Ref. 24 and later developed in Ref. 22.

The theory makes several predictions regarding the trends of electrical characteristics with temperature and other parameters in junctions exhibiting voltage-independent ideality factors n ( T ). In particular, n ( T ), as determined from the slope of I V characteristics, is predicted to vary with temperature according to
(1)
where k is Boltzmann’s constant, q is the electric charge, and ρ 2 and ρ 3 are constants describing a linear variation in the mean barrier height, ϕ B ¯, and variance, σ s 2, with junction voltage U [c.f. Eq. (23) in Ref. 23]
(2a)
(2b)

A plot of n ( T ) 1 1 vs T 1 should, therefore, yield a line over some range of temperatures if any temperature-dependence of ρ 2 and ρ 3 is negligible.

In addition, the effective built-in potential Φ B I can be measured in two ways. From C B E V B E characteristics, Φ B I ( C V ) can be obtained by fitting the variation in depletion capacitance with junction voltage using C B E ( V B E ) = C B E , 0 ( 1 V B E / Φ B I ) m, where C B E , 0 is the zero-bias junction capacitance and m is an exponent that depends on the doping profile at the junction.14 

On the other hand, from I C V B E characteristics, the potential barrier for transport, Φ B I ( I V ), relative to its value at some temperature, can be obtained by extrapolating the measured collector current to zero bias using the diode expression,
(3)
where I S = A exp ( q Φ B I ( I V ) / k T ) is the transport saturation current and A is a constant prefactor that depends on device-specific characteristics (Sec. 4.2.1 in Ref. 1). From this expression, Φ B I ( I V ) can be extracted relative to Φ B I , T R T ( C V ), the value determined by C–V measurements at T R T = 300 K, using
(4)

We have neglected the polynomial temperature-dependence of the prefactor as it only makes a log-scale correction to the built-in potential that does not alter our conclusions.

The inhomogeneous junction theory predicts that the barrier measured in these two ways should differ in magnitude and temperature dependence owing to the fact that current depends on V B E exponentially, while the capacitance varies with V B E with a weaker polynomial dependence. The barrier as determined through C V characteristics is, therefore, typically interpreted as the mean barrier height, while that determined from I V characteristics is often less than the mean value due to the larger contribution from low-barrier regions.23,36 The theory gives a relation between these barrier heights as [c.f. Eq. (14) in Ref. 23]
(5)

Considering the form of n ( T ) in Eq. (1), this relation is compatible with the empirical observation that Φ B I ( C V ) n ( T ) Φ B I ( I V )37 (see also Sec. V in Ref. 23).

These predictions can be tested on SiGe HBTs of interest in this work using a cryogenic probe station to measure the I C V B E and C B E V B E characteristics. Although the inhomogenous junction theory and measurement approach were developed for Schottky junctions, it is applicable to the base–emitter junction of an HBT due to the following considerations. First, the theory does not make any specific assumption that the junction is a Schottky junction, but rather only that an inhomogeneous barrier to current transport exists that is characterized by a Gaussian height distribution. Second, the temperature dependence of the junction saturation current is exponential in both cases as the key transport mechanism is thermal charge injection, with the exponent being proportional to the effective barrier height for charge injection, Φ B I ( I V ). Therefore, analysis of the current–voltage characteristics as described above will yield Φ B I ( I V ) equally well for SiGe HBTs as for Schottky junctions. We note that this charge injection approach to model the current–voltage characteristics is relatively simple compared to generalized-integral charge-control relation (GICCR) models employed for HBTs.38 However, GICCR models also exhibit discrepancies with experiment below 100 K (Fig. 6.5(a) in Ref. 39), suggesting that the physical origin of current–voltage discrepancies is not due to the simplicity of the charge injection model.

Finally, for the C B E V B E characteristics, although the mechanisms of current transport in the forward active regime differ between Schottky junctions and HBTs, the electrostatics of the space charge region and the associated depletion capacitance are identical between the devices (Chaps. 6 and 7 in Ref. 40). Therefore, the built-in potential can be determined by the dependence of C B E on V B E at reverse or low-forward biases, in which case C B E is dominated by the depletion capacitance. This procedure was recently applied to determine the built-in potential of SiGe HBTs.12 

We extracted Φ B I from C B E V B E and I C V B E characteristics from 20 to 300 K on a SiGe HBT (SG13G2, IHP). The discrete transistors were probed in a custom-built cryogenic probe station.41,42 We employed nickel/tungsten probes (40A-GSG-100-DP, GGB Industries) that are suitable for probing Al pads. I C V B E characteristics were performed at a constant collector voltage V C E = 1 V to provide a collector current above the minimum resolution of our measurement setup (10 nA). The current range used for this fitting is limited to 0.2 mA, below the high-injection regime, to exclude effects of series resistance, self-heating, and the Early and reverse Early effects. Inclusion of the reverse Early effect in the extraction of the built-in potential was found to alter the extracted Φ B I ( I V ) by only a few percent as it makes only a log-scale correction. Due to the difficulty in distinguishing periphery from area currents, we assumed that the area current is dominant following other studies of cryogenic SiGe HBTs.5,10–12,17

Following standard procedure,12,13 C B E V B E characteristics were obtained using a vector network analyzer (VNA, Keysight E5061B). In reverse-bias and low-forward bias regimes, the Y-parameters are given by Y 11 = g B E + j ω ( C B E + C B C ) and Y 12 = j ω C B C, where C B C is the base–collector depletion capacitance. The base–emitter capacitance can, therefore, be expressed as C B E = ( ( Y 11 + Y 12 ) ) / 2 π f. V B E was restricted to [ 0.5 V, + 0.5 V] to ensure that the measured capacitance was dominated by the depletion capacitance. V B C = 0 V was held constant to ensure C B C was constant while V B E was swept. The Y parameters were measured in 1–3 GHz, and the extractions were performed at 2.4 GHz.

At these frequencies, it was observed that the imaginary part of Y 11 was linear in frequency, indicating purely capacitive behavior. Short-Open-Load-Through calibration was performed on a CS-5 calibration standard at each temperature, and the shunt parasitic capacitance at the input of the device was de-embedded using an OPEN structure. The intermediate-frequency bandwidth (1 kHz) and frequency points (every 0.2 GHz) were selected to limit the total sweep time to less than 15 s to avoid drift. At each bias, Y-parameters were swept across frequency and ensemble-averaged 10 times. Φ B I was extracted from a sweep of C B E vs V B E by fitting the parameters Φ B I, C B E , 0, and m using a trust region reflective algorithm from the SciPy library.43  Φ B I was constrained to [0.5 V, 1.2 V], C B E , 0 was constrained between the minimum and maximum values of the sweep, and m was constrained to [0, 1].

Figure 1(a) shows the collector current I C vs V B E at various temperatures between 20 and 300 K. Consistent with prior findings,5,10,11 the measurements exhibit deviations from drift-diffusion theory at cryogenic temperatures, with the current–voltage characteristic plateauing to a temperature-independent curve below 60 K. We plot the extracted n ( T ) as T e f f = n ( T ) T p h y s vs T p h y s in Fig. 1(b). T e f f is observed to plateau to 100 K due to n ( T ) > 1 at cryogenic temperatures, as has been reported previously.10 

FIG. 1.

(a) Measured I C vs V B E for various temperatures. The characteristics become independent of temperature at cryogenic temperatures. (b) T e f f = n ( T ) T p h y s vs T p h y s from measurements (symbols) and diode theory (line), indicating the non-ideality of the base–emitter junction at cryogenic temperatures.

FIG. 1.

(a) Measured I C vs V B E for various temperatures. The characteristics become independent of temperature at cryogenic temperatures. (b) T e f f = n ( T ) T p h y s vs T p h y s from measurements (symbols) and diode theory (line), indicating the non-ideality of the base–emitter junction at cryogenic temperatures.

Close modal

We next examine the C B E V B E characteristics. Figure 2(a) plots ( Y 11 + Y 12 ) / ω vs f for various V B E at 300 K, where ω = 2 π f. A narrowed frequency range from the 1–3 GHz measurements is plotted to aid in distinguishing the curves. The de-embedded base–emitter capacitance C B E is directly obtained from this plot by averaging across the frequency range. Figure 2(b) plots the resulting C B E vs V B E at seven representative temperatures vs voltage along with the fitted curves. The error bars, representing the 2 σ error in C B E, are obtained from the 10 C B E V B E sweeps performed at each temperature.

FIG. 2.

(a) Measured ( Y 11 + Y 12 ) / ω (symbols) vs f at 300 K for various V B E in steps of 0.1 V. The lines are guides to the eye. (b) Measured (symbols) and fitted (solid lines) intrinsic C B E capacitance vs V B E from 20 to 300 K. Parasitic capacitances are de-embedded from the Y-parameter measurements as detailed in Sec. II B. As a representative example, the fit for 300 K yields Φ B I = 0.83 V and m = 0.10.

FIG. 2.

(a) Measured ( Y 11 + Y 12 ) / ω (symbols) vs f at 300 K for various V B E in steps of 0.1 V. The lines are guides to the eye. (b) Measured (symbols) and fitted (solid lines) intrinsic C B E capacitance vs V B E from 20 to 300 K. Parasitic capacitances are de-embedded from the Y-parameter measurements as detailed in Sec. II B. As a representative example, the fit for 300 K yields Φ B I = 0.83 V and m = 0.10.

Close modal

The magnitudes of the measured capacitances are on the same order as other reports on the same device type (SG13G2) used here.12 We note a non-monotonic trend in the capacitance data of Fig. 2(b) with temperature. This trend was also observed in Fig. 4(a) in Ref. 12, and it can be attributed to variations in parasitic pad capacitances on landing, which overwhelm any intrinsic temperature dependence of the capacitances. However, the magnitude of the capacitance is not used to determine Φ B I ( C V ), and hence the parasitics have no effect on our conclusions.

These data are next analyzed to obtain Φ B I from the I C V B E and C B E V B E characteristics according to the methods in Sec. II A. At 300 K, Φ B I ( C V ) is found to be 0.83 V, in good agreement with Ref. 12. This value is specified as the room temperature value for Φ B I ( I V ). Figure 3(a) plots the Φ B I from both measurements vs T p h y s. For Φ B I ( C V ), the error bars represent the 2- σ error, obtained by performing fits to 100 C B E V B E sweeps with errors randomly determined based on a normal distribution defined by the uncertainty in the measured C B E. The extracted Φ B I ( C V ) is observed to weakly increase with decreasing temperature, consistent with observations for similar HBT devices12 and Schottky diodes.37 In contrast, Φ B I ( I V ) demonstrates a qualitatively stronger dependence on temperature than Φ B I ( C V ), exhibiting a lower magnitude at cryogenic temperatures as previously observed in Schottky diodes13,37 (also compare to Fig. 3 in Ref. 34). The variation in Φ B I ( I V ) with temperature is significantly stronger than that of the emitter and base bandgaps,14 suggesting that another mechanism is responsible for the observed temperature trend.

FIG. 3.

(a) Built-in potential Φ B I vs physical temperature from C B E V B E (black circles) and I C V B E (orange triangles) measurements. Also plotted is n ( T ) Φ B I ( I V ) (green squares), which is predicted to agree with Φ B I ( C V ).23 Good agreement is observed. (b) n ( T ) 1 1 vs inverse physical temperature T 1 for measured data on the SG13G2 (base width < 20 nm) with a linear fit to the data in 40–100 K following the prediction in Ref. 23. Data obtained from older-generation devices (Fig. 5.8 in Ref. 10) are also shown.

FIG. 3.

(a) Built-in potential Φ B I vs physical temperature from C B E V B E (black circles) and I C V B E (orange triangles) measurements. Also plotted is n ( T ) Φ B I ( I V ) (green squares), which is predicted to agree with Φ B I ( C V ).23 Good agreement is observed. (b) n ( T ) 1 1 vs inverse physical temperature T 1 for measured data on the SG13G2 (base width < 20 nm) with a linear fit to the data in 40–100 K following the prediction in Ref. 23. Data obtained from older-generation devices (Fig. 5.8 in Ref. 10) are also shown.

Close modal

We now examine the agreement between the data and the predictions from the inhomogeneous junction theory of Ref. 23, as summarized in Sec. II A. First, qualitatively, Φ B I ( C V ) and Φ B I ( I V ) are predicted to differ, with Φ B I ( I V ) expected to exhibit a stronger temperature dependence and be smaller in magnitude than Φ B I ( C V ). This behavior is observed in Fig. 3(a). More quantitatively, for the predicted temperature-dependence of n ( T ) from the theory, it is expected that Φ B I ( C V ) n ( T ) Φ B I ( I V ) (Sec. V in Ref. 23). The product n ( T ) Φ B I ( I V ) is also plotted in Fig. 3(a), demonstrating good agreement with Φ B I ( C V ).

Second, Eq. (1) predicts that n ( T ) 1 1 vs T 1 should be a straight line over some range of temperatures, assuming ρ 2 and ρ 3 to be independent of temperature. Figure 3(b) plots this quantity for the present device and other devices with data obtained from Fig. 5.8 in Ref. 10. For all the devices, the expected trend is observed over a temperature range that is comparable in relative width to that in Fig. 9 in Ref. 23, confirming the prediction.

Two regimes of deviation from the linear trend are observed at high and low temperatures. At high temperatures 300 K, n ( T ) 1 1 plateaus to zero for all the devices, corresponding to an ideal junction with n = 1. This deviation was also observed in Fig. 8 in Ref. 23 and is expected since n 1 for fabricated junctions, meaning n ( T ) 1 1 0. It could be explained by a temperature dependence of ρ 2 and ρ 3, which was neglected in this work and in Ref. 23.

The SG13G2 also exhibits a deviation from the linear trend at low temperature ( 20 K). This discrepancy could be attributed to the presence of a direct tunneling current that has been previously reported to exist in highly scaled devices.11,12 The present device has a base width less than 20 nm,44 and so direct tunneling could occur. The older-generation devices exhibit the linear trend down to 20 K, which is compatible with the absence of tunneling current in these devices with larger base widths. Despite these deviations, overall the experimental trends are in good qualitative agreement with the theoretical predictions of the inhomogeneous barrier theory.

Semi-quantitative information regarding the variation of the barrier height variance with bias for the present device can be obtained from the linear fit in Fig. 3(b). If ρ 2 + ρ 3 / ( 2 k T / q ) 1, the theory of Ref. 23 described in Sec. II A reduces to the T 0 model for non-ideal junctions, which has been extensively studied in the Schottky junction literature.19,22,23 In this case, the slope of the linear fit in Fig. 3(b) is simply T 0. Performing this fit for the present data yields T 0 30 K, a value that is generally compatible with values for Schottky diodes compiled in Ref. 23. T 0, in turn, can be linked to ρ 3 as (Eq. 33 in Ref. 23)
(6)

We obtain a value ρ 3 = 5.2 mV. The magnitude of this value indicates that changes in standard deviation of the potential barrier height distribution with base–emitter bias of less than a percent of the mean barrier height are sufficient to account for the observed electrical anomalies.

The agreement of our data with the predictions of the inhomogeneous barrier theory suggests that lateral inhomogeneities in the base–emitter junction potential could contribute to the cryogenic electrical anomalies. Additional evidence for the barrier inhomogeneity hypothesis could be obtained using techniques such as ballistic emission electron microscopy (BEEM), which directly measures the spatial profile of the built-in potential.33 However, applying this method to SiGe HBTs would require specialized samples to be prepared, which are compatible with the measurement technique. Other methods that would be applied without requiring specialized samples might include transmission electron microscopy or a combination of device physics simulations and electrical measurements. However, none of these methods give as direct evidence as BEEM.

The materials-scale origin of the inhomogeneities in HBTs could be crystal defects such as dislocations, Ge clusters,45 or electrically active carbon defects.46 Non-uniform Ge content over a few nanometers in SiGe p-wells with Ge concentration 30% has been reported to lead to the degradation of electrical properties such as hole mobility.45 Trap states associated with C impurities have also been detected in modern HBTs.46 In the SG13G2, the peak Ge concentration in the base is 28%,47 and the value in the vicinity of the base–emitter junction is, therefore, expected to be in the tens of percent based on the SIMS profile of similar device reported in Fig. 1.17 in Ref. 44. These concentrations are sufficiently high for Ge clusters to potentially form. Additionally, the C doping is on the order of 10 20 cm-3,1 and electronic trap states associated with the doping have been previously identified.46 These defects could be the origin of an inhomogeneous base–emitter junction potential.

If the presence of spatial inhomogeneities across the emitter area is verified, a less aggressive Ge doping concentration and profile, especially in narrow-base SiGe HBTs, could decrease the concentration of these imperfections and thereby lead to a more uniform base–emitter junction potential. However, care would need to be taken to avoid negatively impacting the high-frequency properties of the device. Additionally, direct tunneling may pose a fundamental obstacle to improve the cryogenic electrical ideality of highly scaled devices. Further study is needed to distinguish the various mechanisms and identify the limits to cryogenic electrical ideality and, hence, microwave noise performance.

We have reported a characterization of the collector-current ideality factor and built-in potential vs temperature of a SiGe HBT. The observed trends with temperature and between measurement techniques agree with a theory of electrical transport across a spatially inhomogeneous junction, suggesting that barrier inhomogeneities may contribute to cryogenic electrical non-idealities in these devices. This work advances efforts to improve the cryogenic microwave noise performance of SiGe HBTs.

The authors thank Akim Babenko, John Cressler, Nicolas Derrier, Xiaodi Jin, Pekka Kangaslahti, Holger Rücker, Michael Schröter, and Sander Weinreb for useful discussions. This work was supported by NSF Award No. 1911926 and by JPL PDRDF Project No. 107978.

The authors have no conflicts to disclose.

Nachiket R. Naik: Conceptualization (equal); Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal). Bekari Gabritchidze: Data curation (equal); Methodology (equal). Justin H. Chen: Data curation (supporting); Writing – review & editing (equal). Kieran A. Cleary: Project administration (equal); Writing – review & editing (equal). Jacob Kooi: Funding acquisition (equal); Methodology (equal); Writing – review & editing (equal). Austin J. Minnich: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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