Introduction

Motivated by the progress of artificial intelligence and internet of things (AIoT), an intense effort has been devoted to the fabrication and characterization of sensors and actuators to quickly detect and response the variations in the physical world. Among them, the Hall magnetic sensors fabricated from semiconductors are mostly used in contact-less sensors for linear and angular position, velocity and angular frequency, electrical current, etc., and play an important role in the era of AIoT1. In particular, the two-dimensional electron gas (2DEG) in Si-δ-doped III–V quantum-well (QW) channel possesses high electron mobility (μn), high thermal stability, and low noise, and hence is widely adopted as the active layer for high speed and high sensitivity electronic devices2,3,4,5,6. In contrast to the homogeneous bulk-doped structure7,8,9, the uniform modulation doping in barrier layers of III–V QW structure can supply charge carriers in the undoped active channel with high mobility due to less scattering from ionized dopants10. Furthermore, the single Si-δ-doping (modulation doping with single Si-doping plane)11,12,13,14,15,16 in the barrier layer with appropriate space layer thickness (tS) can offer effectively more 2DEG than the uniform modulation doping. As compared to the single Si-δ-doping, the dual and symmetric Si-δ-doping (modulation doping with two planes of Si evenly separating into two sides of the active channel with appropriate tS, see Fig. 1) can afford nearly a similar electron concentration in the active channel with low induced internal electric field due to the symmetric distribution of Si-δ-dopants and hence provide an even higher μn due to the further reduction of scattering effects17,18,19.

Figure 1
figure 1

(a) Cross-sectional view of the Si-δ-doped QW structure adopted in this work. (b) Schematic band diagram nearby QW of the structure.

Full size image

For Si-doped Alx′Ga1−x′As fabricated by metal organic chemical-vapor deposition (MOCVD) processes under As-rich environments, the Si dopants (with concentration NSi) usually occupy the group III sites20 and act either as shallow donors (NSD, with donor level ESD = EC – 5.8 meV) for normal substitution or as deep DX centers (NDD, with EDD = EC – 145 meV) for broken-bond configurations12,21,22,23,24, where EC is the conduction band edge. Assume that NSD + NDD = NSi, the ratio of NDD/NSi changes with Al mole fraction x′: for x′ < 0.20, NDD/NSi = 0; and for 0.20 < x′ < 0.40, NDD/NSi increases continuously with x′25. Although this relation is deduced from a homogeneous Si-bulk-doped structure, it still holds in the Si-δ-doping layer due to the similar occupation behaviors of Si atoms in the group III sites as long as the concentration of Si-dopants is below degenerate doping. To achieve a good confinement of 2DEG in the active channel, x′ = 0.3 is commonly chosen. Accordingly, with high enough temperature, the thermal activation of charge carriers released from DX centers and the transport of carriers across space layer between Si-δ-doped layer and active QW layer should be carefully characterized because both of them play a crucial role in the performance of the devices fabricated.

As depicted in Fig. 1, an undoped In0.15Ga0.85As was chosen as the active layer because it possesses a higher μn than GaAs. In InGaAs system, the increase of indium percentage will enhance μn, but it also enhances the lattice constant and reduces the bandgap. The enhancement of lattice constant in InGaAs increase the lattice mismatch between InGaAs active layer and AlGaAs barrier layer which will deteriorate the quality of the sandwiched active layer. The reduction of bandgap can induce the noise from thermally generated electron–hole pairs as the device is operating at high temperature. Based on the published data26, a compromise of 15% was chosen.

Here, a dual and symmetric Si-δ-doped Al0.3Ga0.7As/In0.15Ga0.85As/Al0.3Ga0.7As QW structure as depicted in Fig. 1 was fabricated to be used as the core-element in the micro-Hall magnetic sensors as shown in Fig. 2. The values of tS = 5, 10, and 15 nm were chosen. The temperature (T = 40 ~ 300 K) dependent Hall-effect analyses were conducted, and the electron sheet concentration n2D(T) and electron mobility μn(T) in the active channel were deduced. Then, the T-dependent charge carriers transport across the space layer together with the effects of tS on n2D(T) and μn(T) are discussed.

Figure 2
figure 2

(a) Configuration of the cross-like micro-Hall element. (b) I–V characteristics for a micro-Hall sample with tS = 5 nm.

Full size image

Results and discussion

Dynamic SIMS measurement

The Si-δ-doping in both sides of active layer provides the 2DEG in active channel, so at first the characteristics of the doping profile NSi(y) in-depth distribution for sample with tS = 5 nm was checked by dynamic secondary ion mass spectrometer (SIMS). As shown in Fig. 3, the asymmetric distribution of NSi(y) with respect to the center of each doping profile (at yδ =  ± 11 nm) can be understood as that the thermal energy caused by the primary ions bombardment could drive some in situ Si dopants to diffuse back. Therefore, an extra tiny residues were added to the falling edge of the δ-doping profile as compared to the leading edge. Therefore, by taking the leading edge into account only, one of the Si-δ-doping profiles as depicted in Fig. 3 can be modeled as a Gaussian distribution27 with short enough standard deviation s,

$$N_{Si} \left( y \right) \, = \, (N_{2D} /({\text2{\pi}}s^{{2}} )^{{{1}/{2}}} ) \times {\exp}[ - (y - y_{\delta} )/s)^{{2}} /{2}] \, + N_{0} ,$$
(1)
Figure 3
figure 3

NSi(y) measured from dynamic SIMS for a dual Si-δ-doping profile with tS = 5 nm.

Full size image

where N2D is the Si sheet concentration and N0 is the background. From a least squared fit to these dual Si-δ-doping profiles, the averaged values of N2D = (1.40 ± 0.07) × 1012 cm−2 and s = 2.00 ± 0.13 nm were obtained, and the two fitted Gaussian profiles were rather symmetric. The total sheet concentration of Si atoms from these dual δ-doping layers was equal to 2 × N2D = (2.80 ± 0.14) × 1012 cm−2 and this parameter was held as a constant for the samples fabricated with various tS in this work. Although the magnitude of s = 2 nm was small enough for tS = 10 and 15 nm (i.e., tS ≥ 5 s) such that NSi(y) can be reasonably regarded as a δ-doping profile, but for tS = 5 nm (≤ 2.5 s) the actual distribution of NSi(y) as depicted in Fig. 3 in the narrow regions close to active channel (for y = − 11 to − 6 nm and 6 to 11 nm) as well as the tiny amount of Si atoms penetrated into the active region (from y = − 6 to + 6 nm) cannot be neglected and their effects on the magnitude of n2D and on the transport properties will be addressed later.

Electron sheet concentrations n 2D(T) from Hall-effect analysis

Three types of samples are focused in this study with similar structure as Fig. 1a and identical Si-δ-doping profile as Fig. 3 except tS varied from 5, 10, to 15 nm. The electron sheet concentrations n2D (= IxBy/qVH) of these samples measured from Hall-effect analysis with decreasing- and then increasing-T modes (for T between 40 and 300 K) are depicted in Fig. 4. For tS = 5 nm, n2D(T) varied slightly with T, but for tS = 10 and 15 nm, n2D(T) varied drastically and exhibited an interesting hysteresis behavior. Besides, at 40 K, n2D for tS = 5 nm was 46% higher than n2D for tS = 10 and 15 nm. To explain the hysteresis behavior of n2D(T) and the effects of tS on n2D(T) and μn(T), the following arguments are proposed.

Figure 4
figure 4

n2D(T) measured from Hall-effect analysis for samples with tS = 5, 10, and 15 nm under decreasing- (▽) and then increasing-T (▲) measurements. The magnitude of the error-bar estimated from the measurement is less than the size of the symbol.

Full size image

Shallow donors versus deep DX centers

In our samples, the AlGaAs barrier layers (except at the δ-doping layers) and InGaAs active channel were undoped, the surface states were passivated by InGaP, and the intrinsic carrier concentrations from these III–V compounds were small enough to be neglected, so the measured n2D(T) was assumed dominantly from the dual Si-δ-doping layers. The Si dopants (NSi) in AlGaAs barrier layer can act either as shallow donors (NSD with ESD = EC – 5.8 meV) or as deep DX centers (NDD with EDD = EC – 145 meV)22. Based on the model mentioned above25, the ratios of NSD/NSi = 0.3 and NDD/NSi = 0.7 were calculated for Al0.3Ga0.7As. As shown in Fig. 4 and from the Gaussian fit, the total sheet concentration of Si atoms from dual δ-doping layers was 2 × N2D. Thus, 30% of the Si dopants (acted as shallow dopants) became nearly completely ionized at T = 300 K, and contributed a sheet electron density nSD ~ 0.3 × 2 × N2D = (0.84 ± 0.04) × 1012 cm−2. As depicted in Fig. 4, this value coincided with the lowest value of n2D observed for samples with tS = 10 and 15 nm at T = 40 K, i.e., in such situations n2D was composed of the electrons completely ionized from shallow donors (nSD). Therefore, for tS = 10 and 15 nm, the measured values of n2D > nSD at high T suggests that electrons partially ionized from the DX centers (nDD) must be taken into account, which gives nDD = n2D − nSD. From the ratio of nDD/(0.7 × 2 × N2D), which equals the unoccupied probability [1 − f(T, EDD)] of electrons at EDD, then the occupied probability for electrons at DX centers, f(T, EDD), can be evaluated from Fermi–Dirac statistics28 with

$$f\left( {T,E_{DD} } \right) \, = { 1}/\left[ {{1 } + {\exp}\left( {\left( {E_{DD} - E_{F} } \right)/kT} \right)} \right],$$
(2)

where EF is the Fermi energy level and kT is the thermal energy. Accordingly, the energy difference of EDD – EF can also be estimated. For samples with various tS, the associated parameters (n2D, nSD, nDD, f(EDD), and EDD – EF) are listed in Table 1 for T = 300 K.

Table 1 Associated parameters for evaluating EDD – EF at each Si-δ-doping layer and ΔV across tS for samples with various tS at 300 K.
Full size table

Simplified band diagrams modelling by two back-to-back capacitors

Unlike homogeneous bulk-doped materials, where the electrons and ionized donors appear in the same spatial location, for Si-δ-doped heterojunction QW the electrons are transferred to the QW region while the ionized donors remain in the δ-doped layers11,12,13,14,15,16,17,18,19. Because of the separation of charges, a model consisted of two back-to-back capacitors was chosen to calculate the internal transverse electric field (F) and electric potential difference ΔV across the space layer. Furthermore, a symmetric band diagram was assumed for simplicity for a dual and symmetric δ-doped layers17,18,19 as depicted in Fig. 5a. In such a case, the capacitor has a positive surface charge density + σ (= q × (nSD + nDD)/2) from ionized Si donors located at one of the δ-doping layers and a negative surface charge density − σ (= − q × n2D/2) accumulated by electrons at one edge of the active QW channel, and n2D = nSD + nDD is assumed as mentioned above. Accordingly, the values of F within the capacitor (= σ/ε0εr, where the dielectric constant εr = 12.24 is taken for Al0.3Ga0.7As29) and ΔV across tS were calculated for tS = 5, 10, and 15 nm, separately. These data at T = 300 K are also listed in Table 1. By taking the conduction band offset of 0.415 eV for the Al0.3Ga0.7As/In0.15Ga0.85As QW heterojunction30, the simplified conduction band diagrams for various tS are depicted as Fig. 5b. Based on these band diagrams derived from this simplified scheme, EF locates reasonably far away from EC at the Si-δ-doped layer, which suggests that the conductivity was dominantly by the 2DEG in the active channel for our samples with moderate Si-dopants (N2D = 1.40 × 1012 cm−2). Only for very high Si-doping such that EF is very close to EC, the parallel conductivity over the doped layers should be taken into account.

Figure 5
figure 5

(a) Simplified conduction band diagram deduced from the dual Si-δ-doping layers based on two back-to-back capacitors at 300 K. (b) Comparison of the corresponding band diagrams for samples with tS = 5, 10, and 15 nm.

Full size image

Origin of hysteresis curves of n 2D(T) for t S = 10 and 15 nm

As demonstrated in Fig. 5, the hysteresis curves of n2D(T) observed for tS = 10 and 15 nm in Fig. 4 could be understood as followings. In general, if electrons generated and recaptured from their donor states involve only one activation energy (as in the uniform homogeneous bulk-doped sample) and if the relaxation time is short enough, then as long as the thermal equilibrium is nearly reached, no hysteresis curve will be observed for decreasing- and increasing-T measurements. In this work, the time for the system stayed at each T before Hall-effect measurement was much larger than the relaxation time expected. Therefore, more than one activation energy associated with the charge carriers transfer were required to account for the hysteresis curves observed in Fig. 4. During a decreasing-T measurement from 300 to 280 K, the electrons in active QW channel (with energy around EF) even though encounter a large activation barrier (EA1 = 216 meV for tS = 15 nm as depicted in Fig. 5b) to return the donor states in Si-δ-doped layer, with a thermally assisted tunneling the transfer of electrons from active QW channel to the δ-doping layer can be rather efficient and then n2D(T) decreased with decreasing T (i.e., more DX levels become occupied, the probability of f(T, EDD) increased and ΔV(T) decreased). However, as T decreased from 280 to 200 K, with less thermally assisted tunneling the excess electrons are then persistently confined in the active 2D channel, n2D(T) and the corresponding f(T, EDD) and ΔV(T) all remained constant as shown in Fig. 6. As T further decreased from 200 to 40 K, the value of F induced by the non-equilibrium extra space-charges would now assist the electrons transfer from active channel to δ-doped layers, thus n2D(T) dropped again and the occupied probability of DX levels, i.e., f(T, EDD), started to increase and ΔV(T) across the capacitor reduced. At T = 40 K, f(T, EDD) reached nearly 100%, and in this situation n2D was totally dominated by the completely ionized shallow donors (nSD). On the other hand, during an increasing-T measurement from 40 to 150 K, the electrons thermally released from DX centers in Si-δ-doping layer were rather small as compared with nSD and thus n2D was very close to nSD (see Fig. 4), hence f(T, EDD) remained nearly 100% and ΔV(T) kept as a constant (see Fig. 6). For T increased from 150 to 280 K, the thermally released electrons from DX centers to conduction band in AlGaAs become noticeably and by a longitudinal applied bias these electrons then transferred to the active channel efficiently due to a small activation barrier (EA2 = qΔV = 95 ~ 117 meV for tS = 15 nm at T = 150 ~ 280 K, see Fig. 6b). Therefore, n2D(T) and ΔV(T) increased while f(T, EDD) decreased with increasing T. Because of these different activation barriers encountered with respect to decreasing- and increasing-T modes (EA1 versus EA2), a T-dependent hysteresis on n2D(T) was resulted as shown in Fig. 4 for tS = 10 and 15 nm. Besides, based on the model of two back-to-back capacitors, the values of F for these two samples were nearly alike due to the good match of n2D(T) as shown in Table 1 and Fig. 4. Hence, as depicted in Fig. 6, f(T, EDD) followed the same trend while ΔV(T) (= F•tS) acted differently for these two samples. Furthermore, as displayed in Fig. 4, the hysteresis curves ended at T ~ 280 K for these two samples. Note that the occupied probability for electron at DX center is related to an energy difference of EDD − EF. As listed in Table 1, the magnitudes of EDD − EF were nearly comparable to kT for T = 280 ~ 300 K. With the support of thermally assisted tunneling and a longitudinal applied bias, the transfer of electrons through the space layers became reversible. Because of the upper limit of our cryostat, no further experimental data are provided for T > 300 K.

Figure 6
figure 6

(a) f(T, EDD) and (b) ΔV(T) evaluated for Hall samples with tS = 10 and 15 nm.

Full size image

Effects from non-ideal Si-δ-doped profiles for t S = 5 nm

As also illustrated in Fig. 4, the fact of no hysteresis curve of n2D(T) observed for small tS = 5 nm suggests that the non-ideal Si-δ-doped profiles as shown in Fig. 3 was the culprit. For tS = 5 nm = 2.5 s, the actual distribution of NSi(y) near active QW (for y = − 11 to − 6 nm and 6 to 11 nm in Fig. 3) acted as a specific form of “modulation doping” with strong gradient toward QW channel and could not be regarded as an ideal δ-doping. Meanwhile, the band bending up as illustrated in Fig. 5a can shift EDD (corresponding to the Si DX levels in Al0.3Ga0.7As barrier layer with energy of EC – EDD = 145 meV) above EF. Therefore, those electrons from DX centers located in the vicinity close to QW with energy around EF can hop through these DX centers and fall into the QW channel. Based on the data at low enough T = 40 K as depicted in Fig. 4, n2D (≅ 1.23 × 1012 cm−2) for tS = 5 nm was higher than n2D (≅ 0.84 × 1012 cm−2, dominated by nSD) for tS = 10 and 15 nm by 46% [= (1.23 − 0.84)/0.84)]. The percentage of the extra electrons released from DX centers in this specific “modulation doping” was estimated around 20% [= (1.23 − 0.84) × 1012/(2 × N2D × 0.7)]. These electrons persistently confined within the active QW channel not only enhanced the total values of n2D but also compensated the T-dependence of n2D from the typical DX centers as mentioned above. Thus n2D(T) for tS = 5 nm turned out to be relatively weak T-dependence with respect to decreasing- and increasing-T modes and hence no hysteresis curve was observed.

Effects of t S on the T-dependence of electron mobility

The T-dependence of electron mobility μn(T) deduced from Hall-effect analysis on samples with various tS (= 5, 10 and 15 nm) is demonstrated in Fig. 7. For T = 40 K with less phonon scattering, μn is very sensitively dependent on the Coulombic scattering from the ionized Si-donors located at δ-doped layers separated by tS from the active channel. In addition, the very small amount of Si dopants entered into the In0.15Ga0.85As active channel (from y = − 6 to + 6 nm as shown in Fig. 5a) could further reduce μn. In viewing of these two factors, the sample with tS = 15 nm possessed a highest μn(40 K) = (5.41 ± 0.02) × 104 cm2 V−1 s−1 among these three types of samples. With increasing T, μn decreases dramatically due to phonon scattering in the active channel. Nevertheless, μn(300 K) for all dual and symmetric Si-δ-doped AlGaAs/InGaAs/AlGaAs QW samples fabricated by this work still possessed relatively high values of (6.43 ± 0.06) × 103, (7.17 ± 0.16) × 103 and (7.76 ± 0.03) × 103 cm2 V−1 s−1 for tS = 5, 10 and 15 nm, respectively. A high value of μn is one of the important figures of merit for high sensitivity magnetic sensor.

Figure 7
figure 7

μn(T) measured from Hall-effect analysis for samples with various tS under decreasing T (▽) and then increasing T (▲) measurements. The inset enlarges the corresponding values near room temperature. The magnitude of the error-bar estimated from the measurement is less than the size of the symbol.

Full size image

Conclusion

In this work, dual and symmetric Si-δ-doped AlGaAs/InGaAs/AlGaAs QW structures of various tS (= 5, 10 and 15 nm) were fabricated and characterized. At first, from a Gaussian fit on the dynamic SIMS data for the dual Si-doping profile, the two Si dopant-profiles were rather symmetric and N2D = 1.40 × 1012 cm−2 and s = 2.0 nm were obtained for each profile. Next, n2D(T) and μn(T) in the active QW channel of these samples were measured from Hall-effect analysis with decreasing- and then increasing-T modes. Interesting hysteresis curves of n2D(T) were observed for tS = 10 and 15 nm but not for tS = 5 nm. Because of the charge separation for the Si-δ-doped AlGaAs/InGaAs/AlGaAs QW structure, a simplified energy-band diagram based on two back-to-back charged capacitors was proposed to explain these phenomena. Due to the different activation barriers encountered respectively by electrons in the active QW and by electrons in the δ-doped layers during decreasing- and then increasing-T modes, a hysteresis on n2D(T) was obtained for tS = 10 and 15 nm. Besides, at T = 40 K the lowest value of n2D = 0.84 × 1012 cm−2 observed for samples with tS = 10 and 15 nm indicates that in such situations n2D was composed of the electrons completely ionized from shallow donors (nSD) which agrees well with the results from the proposed model based on experimental SIMS data. However, for small enough tS = 5 nm (i.e., tS ≤ 2.5 s), the actual distribution of NSi(y) near QW could not be regarded as an ideal δ-doping. The amount of Si atoms nearby the active QW channel acted as a specific form of “modulation doping” and effectively increased the level of n2D, and hence no hysteresis curve was observed. Finally, the effects of tS on μn(T) for these three structures were also addressed.

Methods

Fabrication of the epi-structure

The epi-layers of the Hall sample on a semi-insulating GaAs substrate depicted in Fig. 1 were fabricated by MOCVD under As-rich environments20 in the following steps. (1) At first, 5 nm InGaP and 10 nm GaAs buffer layers were deposited sequentially, followed by 10 pairs of 6 nm AlAs and 6 nm GaAs superlattice layers, and another 10 nm GaAs buffer layer to release the strain resulted from lattice mismatch. (2) After deposition of an undoped 55 nm Al0.3Ga0.7As barrier layer, the first Si-δ-doping layer with sheet concentration of 1.4 × 1012 cm−2 was achieved by injection a high SiH4 doping flow and followed by an undoped Al0.3Ga0.7As space layer with thickness tS. (3) The active layer was fabricated by In0.15Ga0.85As with thickness tA = 12 nm. (4) After an undoped Al0.3Ga0.7As space layer with thickness tS, the second symmetric Si-δ-doping layer was repeated, then followed by a 60 nm undoped Al0.3Ga0.7As barrier layer. (5) Finally, a passivation layer of 5 nm InGaP was deposited on top of the Hall element to reduce the effects from surface states and acted as an etching stopping layer (ESL). For ohmic contact with the active channel, after capped with 15 nm Si-doped GaAs cap layer, the standard ohmic contact were applied by adding a metal layer series of AuGe/Ni/Au31 of total thickness 350 nm, thermally driven to make a contact with the QW active channel, and finalized with a Ti/Au bonding pad. The dimensions of the cross-like micro-Hall element as illustrated in Fig. 2 were 420 × 420 μm2, with channel width W = 115 μm and length L = 350 μm. According to the I–V characteristics for a micro-Hall sample with tS = 5 nm as depicted in Fig. 2b, the charge transport properties along the active channels exhibited a good ohmic contact behavior and a nice symmetry.

Characterization

The characteristics of the Si-δ-doping in-depth distribution were first verified by dynamic mode of secondary ion mass spectroscopy (SIMS, outsourced by EAG Laboratories). Then the Hall-effect analysis was conducted by a Keithley 7065 system32 on the cross-like sample with 4 numbered terminals as depicted in Fig. 2, and the linear Ix–Vx characteristics in each active channel between terminals 1 ↔ 3 and 2 ↔ 4 were confirmed for Ix < 8 mA, respectively. With a steady current Ix = 1.0 mA applied between terminals 1 ↔ 3 under a perpendicular By = 5.0 kG, the Hall voltage across terminals 2 ↔ 4 were consecutively measured for ten times and averaged to give the value of VH1. By reversing the direction of Ix, VH2 was obtained similarly. Next, Ix was switched from terminals 1 ↔ 3 to 2 ↔ 4, and by repeating the above procedures VH3 and VH4 across terminals 1 ↔ 3 were obtained again. Finally, by reversing the direction of the B-field and following the same procedures, VH5 ~ VH8 were measured correspondingly. From the combination of these eight measurements to compensate the offset voltage due to any asymmetry of the cross-like Hall sample, the Hall coefficient RH was obtained24 and the carrier concentration n0 = 1/qRH was calculated. Furthermore, for resistivity measurement, the van der Pauw model was adopted with a current applied between terminals 1 and 2 and the voltage measured across terminals 3 and 4. By switching the direction of the current and then rotating the sequence of the contact terminals, again a total of eight measurements were taken separately, and the averaged resistivity ρ of the sample was evaluated. From n0 and ρ, the averaged value of μn in the active 2DEG channel was estimated. The temperature (T) of the sample during the Hall-effect analysis was controlled by a cryostat system. The cooling or heating rate was set at 2.0 K min−1 for each ΔT = 20 K, and then the system stayed at each T for 10 min to allow the sample to reach near equilibrium before Hall-effect measurement. The period for one set of Hall-data acquisition at each T took another 15 min.