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Yuto Uematsu, Takafumi Ishibe, Seiya Kozuki, [Takaaki Mano](https://orcid.org/0000-0002-6955-260X), [Akihiro Ohtake](https://orcid.org/0000-0002-3519-4613), [Hideki T. Miyazaki](https://orcid.org/0000-0003-4152-1171), [Takeshi Kasaya](https://orcid.org/0000-0002-1976-8760), Yuichiro Yamashita, Mutsunori Uenuma, Yoshiaki Nakamura

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[Film Thermoelectric Generator of Multiple 2-D Electron Gas](https://mdr.nims.go.jp/datasets/3b241ef9-f407-41b4-a894-47f882d8ea7b)

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1                                                                                                                    IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. xx, NO. x, xxx xxxx  Abstract—Two-dimensional electron gas (2DEG) is one of the promising approaches for high thermoelectric performance. However, high electrical resistance of the film originating from thin 2DEG conduction channel width and thick insulator layer is a bottleneck for obtaining high output power. In this study, we propose stacked GaAs 2DEG thermoelectric generator (TEG), which has a number of stacked channel structures for low electrical resistance. Our GaAs 2DEG TEGs with channels formed in triangular well exhibit ultrahigh thermoelectric power factor. In addition, the interfaces of the stacked 2DEG intensify phonon scatterings, resulting in the reduction of thermal conductivity. The stacked 2DEG TEGs exhibit 9 times higher sheet electrical conductivity than the unstacked 2DEG one, resulting in ~7.5 times higher output power of stacked 2DEG TEGs (5.1 nW) than that of unstacked 2DEG ones. Cross-sectional thermoelectric efficiency factor of stacked 2DEG TEGs reaches 3.7 W cm-2K-2. This value is the highest among various simple planar type film TEGs without cavities under the film. This TEG demonstration will open an avenue for the social application of 2DEG TEG.  Index Terms—Energy harvesting, thermoelectric materials, thermoelectric devices, two-dimensional electron gas.  I. Introduction ILM thermoelectric generator (TEG), which directly converts heat to electricity, has drawn much attention as one of the sustainable energy power sources for running internet of things sensors [1], [2]. Toward various application places, there have been a lot of studies about film TEGs on various substrates, such as processable Si, flexible polyimide, and transparent r-Al2O3 [3]-[7].  To enhance the performance of film TEG, it is essential to optimize the TEG structure and enhance the performance of the thermoelectric (TE) materials [8], [9].  The TE performance is strongly related to a dimensionless figure of merit zT; zT=S2T-1, where S is Seebeck coefficient,  is electrical conductivity,  is thermal conductivity, and T is absolute temperature. The central issue for high zT is the independent control of the intercorrelated TE three parameters. Nanostructuring approach has reduced  drastically, leading to high zT [10]-[16]. On the other hand, as for S2 enhancement [17]-[22], quantum confinement effect is well-known as one of the promising approaches [23]-[25]. For example, some groups have reported remarkable S2 enhancements using two-dimensional electron gas (2DEG) [26], [27]. In 2024, we proposed the multiplied 2DEG effect that S of multiple 2DEG (M-2DEG), where multiple subbands in triangular quantum well contribute to electrical conduction, is higher than that of single 2DEG (S-2DEG) with a single subband (Fig. 1(a)) [28]. This S enhancement is also confirmed even when we plot S as a function of sheet carrier concentration ns (Fig. 1(b)). This result implies that S enhancement by multiplied 2DEG effect [28] does not depend on the definition of the channel width of 2DEG. This effect brought the drastic enhancement of S2, which is beyond the theoretical prediction of S2 in the conventional 2DEG (Hicks and Dresselhaus theory [23]). This makes us expect that the TEG using M-2DEG can exhibit high output power Pout. However, high electrical resistance R of the film originating from thin channel width of 2DEG and thick insulator layer is a bottleneck for high Pout.  In this study, we propose the planar unileg-type stacked GaAs 2DEG TEG composed of M-2DEG film, which has stacked channel structures for low R, namely large cross-sectional area of electrical conduction layer (Fig. 1(c)). Our GaAs 2DEG TEGs with channels formed in triangular well exhibited ultrahigh S2 because of multiplied 2DEG effect. In addition, the out-of-plane  of the stacked GaAs 2DEG TEGs was ~3 times lower than that of GaAs bulk because of the intensified interface phonon scattering. This leads to the increase of T difference T between hot and cold side. The stacked 2DEG TEGs exhibited 9 times higher sheet electrical conductivity than the unstacked 2DEG TEGs with only one 2DEG channel, resulting in higher Pout of stacked 2DEG TEGs than that of unstacked 2DEG TEGs. We evaluated the TE efficiency factor ; =Pmax/(A×T2), where Pmax is the maximum value of Pout, A is the area of the TEG. The  calculated using the total cross-sectional area of the TE legs of the planar unileg-type stacked GaAs 2DEG film TEGs reached 3.7 W cm-2K-2, which was the highest among various simple planar type film TEGs without cavities under the film. This Film Thermoelectric Generator of Multiple Two-Dimensional Electron Gas Yuto Uematsu, Takafumi Ishibe, Seiya Kozuki, Takaaki Mano, Akihiro Ohtake, Hideki T. Miyazaki, Takeshi Kasaya, Yuichiro Yamashita, Mutsunori Uenuma, and Yoshiaki Nakamura F Manuscript received xxxxx, x, xxxx. This work was supported by Grant-in-Aid for Scientific Research A (Grant No. 23H00258), JSPS Fellows (T22KJ2052), and "Advanced Research Infrastructure for Materials and Nanotechnology in Japan (ARIM)" of the Ministry of Education, Culture, Sports, Science and Technology (MEXT) (JPMXP1223NM5062). (Corresponding author: Yoshiaki Nakamura.) Y. Uematsu, T. Ishibe, S. Kozuki, and Y. Nakamura are with the Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan (e-mail: nakamura.yoshiaki.es@osaka-u.ac.jp).  T. Mano, A. Ohtake, H. T. Miyazaki, and T. Kasaya are with the National Institute for Materials Science, Ibaraki 305-0047, Japan. Y. Yamashita is with the National Institute of Advanced Industrial Science and Technology, Ibaraki 305-8565, Japan. M. Uenuma is with the National Institute of Advanced Industrial Science and Technology, Saga 841-0052, Japan. UEMATSU et al.: FILM THERMOELECTRIC GENERATOR OF MULTIPLE TWO-DIMENSIONAL ELECTRON GAS 2 study highlights that stacked GaAs 2DEG TEGs become a promising TEG. II. EXPERIMENTAL  A. Formation of GaAs 2DEG Films Figure 2(a) shows the structures of stacked M-2DEG films (9 times stacked 2DEG channels) and unstacked M-2DEG films. These samples were epitaxially grown on GaAs(001) substrates using molecular beam epitaxy in the following processes. To obtain clean surfaces of undoped GaAs(001) substrates, undoped GaAs (300 nm) initial layers were formed on the GaAs substrates. Subsequently, as the buffer layers, GaAs/Al0.3Ga0.7As superlattice layers were formed on the undoped GaAs/GaAs substrates by alternately depositing GaAs (10 nm) and Al0.3Ga0.7As (10 nm) 20 times. In the case of stacked 2DEG films, the unit structure of undoped Al0.3Ga0.7As (2 nm)/ undoped GaAs (150 nm)/ undoped Al0.3Ga0.7As (2 nm)/ Si-doped Al0.3Ga0.7As (40 nm, dopant concentration: 7×1017 cm-3) layers was formed 4 times repeatedly on the buffer layers. Finally, stacked layers of undoped Al0.3Ga0.7As (2 nm)/ undoped GaAs (150 nm)/ undoped Al0.3Ga0.7As (2 nm)/ Si-doped Al0.3Ga0.7As (60 nm, dopant concentration: 7×1017 cm-3) and GaAs (10 nm) cap layers were formed. 2DEG channels were formed at the AlGaAs/GaAs interfaces of the top and bottom of 150 nm undoped GaAs layers except for the first undoped GaAs layer just above the buffer layer. In the case of unstacked 2DEG films, the structure of undoped Al0.3Ga0.7As (2 nm)/ undoped GaAs (100 nm)/ undoped Al0.3Ga0.7As (2 nm)/ Si-doped Al0.3Ga0.7As (80 nm, dopant concentration: 7×1017 cm-3) layers was formed once on buffer layers. Finally, GaAs (10 nm) cap layer was formed. In the epitaxial growth, two kinds of growth temperatures were used: 823 and 853 K. To suppress the diffusion of Si (ionized impurity) into the 2DEG channel, we formed the following two layers at low T of 823 K: (1) the Si-doped Al0.3Ga0.7As layers and (2) the 2 nm undoped Al0.3Ga0.7As layers on the Si-doped Al0.3Ga0.7As ones. On the other hand, the other layers were formed at high T of 853 K to enhance the crystallinity.  The energy band diagrams and the carrier concentrations (n) along the depth (z) direction were calculated using Schrödinger-Poisson equation self-consistently computed by 1D Poisson [29], as shown in Figs. 2(b) and (c). It was clearly found that 2DEG channels were formed at the interfaces of AlGaAs/GaAs by modulation doping. A width of the 2DEG channel was defined as 8 nm by the full width at half maximum of z-n profiles, the definition of which is widely used in the previous studies [24], [28]. Therefore, the total thickness of 2DEG layers (t2DEG) in stacked 2DEG and unstacked 2DEG films were simply estimated to be 72 nm (8 nm×9 channels) and 8 nm, respectively.   Fig. 1.  (a) The schematic illustrations of M-2DEG in triangular quantum well and S-2DEG in rectangular quantum well. (b) S-ns plot of M-2DEG and S-2DEG. The numbers around the experimental data points represent the channel width of 2DEG. (c) The schematic illustration of stacked 2DEG with triangular quantum wells and its energy band diagram.  Fig. 2.  (a) Structures of stacked 2DEG film and unstacked 2DEG film on the buffer layers of GaAs (10nm) /Al0.3Ga0.7As (10 nm) × 20 times on GaAs (300 nm) on GaAs(100) substrate. (b) Energy band structures of stacked 2DEG film and (c) unstacked 2DEG film.  3                                                                                                                                             IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. xx, NO. x, xxx xxxx B. Fabrication of GaAs 2DEG TEGs The fabrication processes of the planar unileg-type stacked GaAs 2DEG film TEG are shown in Figs. 3(a)-(d). TE legs of stacked 2DEG film were formed by wet etching with the mixed solution of H3PO4 (85 wt%, 2 mL), H2O2 (35 wt%, 2 mL), and H2O (16 mL). Then, the inclined facets at the sides of stacked 2DEG TE legs appeared (Fig. 3(e)), and the AlGaAs/GaAs interfaces with 2DEG channels were bared. These bared structures enable us to easily make ohmic contact with all the 2DEG channels (Fig. 3(f)). To make ohmic contact with 2DEG channels, the stacked metal electrodes of Ni (5 nm)/AuGe (50 nm)/Ni (30 nm)/Au (100 nm) were deposited on the both ends of TE legs, and were annealed for 90 seconds at 450ºC. These isolated TE legs with ohmic electrodes were serially connected with wire electrodes of Ti (30 nm)/Au (400 nm).  The plan-view photograph of the fabricated stacked 2DEG TEG and its schematic illustration are shown in Figs. 3(g) and (h), respectively. The size of the TE legs: length Ltl × width Wtl × thickness ttl in unstacked (stacked 2DEG TEG) is 4 mm ×500 m ×0.19 m (4 mm × 500 or 200 m × 1 m). The numbers of TE legs ntl of the TEGs with Wtl = 500 m and with Wtl = 200 m are 12 and 21, respectively. The size of the Ti/Au wire electrodes (length Lw × width Ww × thickness tw) is 4 mm × 100 m × 0.43 m. The size of Ti/Au wire (length Lw2 × width Ww2 × thickness tw2) between the TE leg and the wire electrode is 50 m × 300 m × 0.43 m. C. Measurement Method of TE Properties Sheet electrical conductivity s, ns, and carrier mobility  of the unstacked 2DEG and the stacked 2DEG films were measured using van der Pauw method and Hall effect measurement along in-plane direction.  and n were obtained by dividing the measured s and ns by  t2DEG. S was measured using ZEM-3 (ADVANCE RIKO Inc.).  along the out-of-plane direction was measured using time domain thermoreflectance (TDTR) method. During the measurement of the output voltages Vout, the output current Iout, and the Pout of GaAs 2DEG TEGs at room temperature, T was applied by Peltier heater. T distribution in the GaAs 2DEG TEG was measured using InfraScope MWIR temperature mapping microscope (Quantum Focus Instruments Co.). III. RESULTS AND DISCUSSION A. TE Properties of GaAs 2DEG Films T dependences of TE properties of unstacked and stacked 2DEG films were measured (Fig. 4). When estimating n and  of 2DEG films, we used the following t2DEG: 72 nm (8 nm×9 channels) for stacked 2DEG films; 8 nm for unstacked 2DEG films. Comparing the TE properties of unstacked and stacked 2DEG films in the measurement T range, there were almost no differences in n, , , and S (Figs. 4(a)-(d)). Therefore, both unstacked and stacked 2DEG films exhibited almost the same S2 in the measurement T range (Fig. 4(e)). We discuss the tendencies of the TE properties against T. As shown in Fig. 4(a), both 2DEG films showed almost no T dependences of n. Figures 4(b) and (c) are T dependences of  and , respectively.  values of unstacked and stacked 2DEG films drastically decreased with increasing T. The strong T dependences of  and the extremely high  values are consistent with the results reported by previous studies about modulation doped 2DEG [28], [30]. This indicates that in both 2DEG films, carriers travel in 2DEG channels without ionized impurity. Because of strong T dependences of , unstacked and stacked 2DEG films also showed strong T dependences of  (Fig. 4(c)). On the other hand, S values of unstacked 2DEG and stacked 2DEG films increased monotonically with increasing T. This is conventional tendency, which is because the average energy per unit conducting electron increased with T increase (Fig. 4(d)).  The correspondence between the TE properties of unstacked and stacked 2DEG films demonstrates that TE properties of all the 2DEG channels in stacked 2DEG film are the same as that in unstacked 2DEG film. This indicates that stacked 2DEG films have almost nine times larger s than unstacked 2DEG films, as shown in Fig. 4(f). s is almost proportional to the number of channels. Therefore, it is expected that stacked 2DEG TEG exhibits higher Pout than unstacked 2DEG films because of the lower R.  The out-of-plane thermal conductivity, o of the stacked 2DEG film was measured to be 15.9 ± 0.4 Wm-1K-1, which is ~3 times smaller than  of GaAs bulk (~54 Wm-1K-1) [31], as   Fig. 3.  (a-d) Fabrication processes of stacked 2DEG TEG: (a) Growth of stacked 2DEG film, (b) TE legs by wet etching, (c) Deposition of ohmic electrode, and (d) Deposition of wire electrode. (e) SEM image and (f) schematic illustration of ohmic electrode deposited at the ends of TE legs. (f)  Schematic illustration of electron conduction in 2DEG channels. (g) Plan-view photograph and (h) schematic illustration of the fabricated stacked 2DEG TEG with Wtl = 500 m. UEMATSU et al.: FILM THERMOELECTRIC GENERATOR OF MULTIPLE TWO-DIMENSIONAL ELECTRON GAS 4 shown in Fig. 4(g). This  reduction is likely attributed to the intensified interface phonon scattering in the stacked layer structures of stacked 2DEG films. To estimate the TE performance, it is required to discuss  and S2 along the same measurement direction. For this reason, we estimate the in-plane thermal conductivity, i with reference to the previous results by another group. According to the previous study, the anisotropy rate of  (i /o) in AlAs/GaAs superlattice was reported to be ~1.2-1.3 [32]. By using this reported rate, the i of stacked 2DEG films was estimated to be ~20 Wm-1K-1. Although the i of stacked 2DEG films was higher than the o, the i of stacked 2DEG films was sufficiently lower than  of GaAs bulk. This low i, which is an estimated value, play an important role in applying large T in terms of the thermoelectric devices.   B. Performance of GaAs 2DEG TEGs Figure 5(a) shows Vout and Pout of the planar unileg-type stacked GaAs 2DEG film TEG composed of stacked 2DEG films with Wtl = 500 m as a function of Iout. During the measurement, the T at the cold side of the TE leg was kept to be 320 K. When increasing T, open-circuit Vout Voc and short-circuit Iout increased, resulting in the enhancement of the maximum Pout, Pmax. At T=4.9 K, Voc was obtained to be ~14 mV, which corresponded to the Voc calculated from the |S| and ntl of the stacked 2DEG film (|S|= 230 VK-1, ntl=12). This correspondence proves that the applied T was accurately measured and there was no influence of thermal interfacial resistance.  Figure 5(b) summarizes Pmax values of the three TEGs (stacked 2DEG TEGs with Wtl=500 m and Wtl=200 m, and unstacked 2DEG TEG with Wtl=500 m) as a function of T. We simultaneously plotted the theoretical T-Pmax curves calculated using the following equations:  𝑃max =(𝑉oc)24𝑅(1),  𝑉oc = 𝑛tl𝑆Δ𝑇 (2),  Fig. 4.  Temperature dependences of (a) n, (b) , (c) , (d) S, (e) S2, (f) s of stacked and unstacked 2DEG films. (g) o and i of stacked 2DEG films and  of GaAs bulk [31].  Fig. 5.  (a) Iout dependences of Vout (the open circles) and Pout (the solid circles) of the stacked 2DEG TEG with Wtl = 500 m at different temperatures. (b) T dependences of Pmax of the stacked 2DEG TEGs and the unstacked 2DEG TEG. The broken lines are calculated Pmax and the solid circles are experimental values of Pmax. 5                                                                                                                                             IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. xx, NO. x, xxx xxxx  𝑅 = 𝑛tl (1𝜎tl𝐿tl −𝑊w2𝑡tl𝑊tl+1𝜎Au(𝐿w −𝑊w2𝑡w𝑊w+2𝐿w2 +𝑊w +𝑊tl𝑡w2𝑊w2)) (3),  where tl and Au are the electrical conductivity of TE leg and Au electrode. Au was experimentally obtained to be ~7×103 -1 cm-1 by measuring R of the individual Au wire electrode. Using equation (3) with the designed and the measured parameters, the R values of the TEGs were estimated as follows: 45 k for the stacked 2DEG TEGs with Wtl=200 m; 11 k for the stacked 2DEG TEGs with Wtl=500 m; 95 k for the unstacked 2DEG TEG.  When increasing T, the experimental Pmax values of all the TEGs increased. In all the TEGs, the experimental data agreed well with the theoretical T-Pmax curves in the T region of less than 4 K (Fig. 5(b)). On the other hand, there was the slight Pmax difference between the experiment and the calculation at the T of ~5 K. This could come from the measurement error of the contact resistance or T.  The stacked 2DEG TEGs exhibited ~7.5 times higher Pmax values than the unstacked 2DEG TEG at almost the same T. This higher Pmax is attributed to the lower R of the stacked 2DEG TEGs than that of the unstacked 2DEG TEG. The Pmax of the stacked 2DEG TEGs with Wtl=500 m was higher than that of the stacked 2DEG TEGs with Wtl=200 m. From the equations (1)-(3), it is considered that low R and high ntl bring high Pmax in the case of the material with the same S. In our experiment, the stacked 2DEG TEG with Wtl=500 m had smaller R of 11 k and ntl of 12 than the stacked 2DEG TEG with Wtl=200 m (R of 45 k and ntl of 21). Namely, in our TEG design, lower R is found to be more effective for enhancing Pmax than larger ntl. In all the TEGs, the experimental Pmax values at the T of ~5 K were obtained as follows: 3.9 nW for the stacked 2DEG TEGs with Wtl=200 m; 5.1 nW for the stacked 2DEG TEGs with Wtl=500 m; 0.7 nW for the unstacked 2DEG TEG. We evaluate the TEG performance by comparing the TE efficiency factor  of the planar TEG with those of previous studies without cavities. In the previous studies,  was calculated using two kinds of definition of A: (1) the bottom area of TEG on the substrate and (2) the total cross-sectional area of the TE legs, as shown in Fig. 6. The structures of the planar TEG are shown in Table I. The  values calculated using the bottom area of TEG and the total cross-sectional area were defined as the bottom TE efficiency factor B and the cross-sectional TE efficiency factor C, respectively. The stacked 2DEG TEGs exhibited higher B and C than the unstacked 2DEG TEG (Fig. 6). The stacked  Fig. 6.  Comparison of B and C of our GaAs 2DEG TEGs with other planar type film TEGs without cavities [6], [33]-[42].  TABLE I STRUCTURES OF THE PLANAR THIN FILM THERMOELECTRIC DEVICES WITHOUT CAVITIES Reference p-type n-type Thickness of legs Length of legs Width of legs number of legs This Work - Unstacked GaAs 2DEG 0.19 m 4 mm 0.5 mm n: 12 This Work - Stacked GaAs 2DEG 1 m 4 mm 0.5 mm n: 12 This Work - Stacked GaAs 2DEG 1 m 4 mm 0.2 mm n: 21 [33] Si0.8Ge0.2 Si0.8Ge0.2 10 m 21.5 mm 3 mm p: 3, n: 3 [34] Si0.85Ge0.15 Si0.85Ge0.15 0.05 m 3 mm 2.5 mm p: 1, n: 2 [35] Mg2Sn0.8Ge0.2 Bi 0.27 m 12 mm 0.15 mm p: 36, n: 36 [36] Pb0.925Yb0.075Te Pb0.925Yb0.075Se0.2Te0.8 0.95 m 10 mm 0.4 mm p: 10, n: 10 [37] Bi0.4Sb1.6Te3 Bi2Te2.7Se0.3 1 m 15 mm 1 mm p: 7, n: 7 [38] Sb2Te3 Bi2Te3 30 m 10 mm 0.3 mm p: 10, n: 10 [39] Bi0.5Sb1.5Te3 Bi2Te3 0.3 m 7.8 mm 1 mm p: 4, n: 4 [40] Bi0.5Sb1.5Te3 Bi2Te3 0.6 m 15 mm 5 mm p: 3, n: 3 [6] Ag0.005Bi0.5Sb1.5Te3 - 0.75 m 15 mm 5 mm p: 4 [41] Bi0.5Sb1.5Te3 Bi2Te2.4Se0.6 4 m 20 mm 0.67 mm p: 15, n: 15 [42] - PVP/Ag2Se composite 4.8 m 20 mm 5 mm n: 6  UEMATSU et al.: FILM THERMOELECTRIC GENERATOR OF MULTIPLE TWO-DIMENSIONAL ELECTRON GAS 6 2DEG TEGs with Wtl = 500 m and with Wtl = 200 m had the highest B of 6.3×10-4 Wcm-2K-2 and the highest C of 3.7 Wcm-2K-2, respectively. Furthermore, the B and C values of these stacked 2DEG TEGs were the highest among various simple planar type film TEGs without cavities under the film [6], [33]-[42]. These larger  values and the highly sophisticated fabrication process of GaAs highlight that 2DEG TEG composed of GaAs becomes a promising power source toward the social application.  IV. CONCLUSION We demonstrated the operation of stacked GaAs 2DEG TEGs composed of M-2DEG films. The stacked GaAs 2DEG exhibited almost the same S2 as the unstacked GaAs 2DEG. This indicates that 2DEG channels with the same electrical performance are formed even if the triangular well is repeatedly stacked. Thanks to the stacked structure, which intensifies interface phonon scattering, both o and i of the stacked GaAs 2DEG were ~3 times lower than  of GaAs bulk. The stacked 2DEG TEGs exhibited 9 times higher s than the unstacked 2DEG TEGs, resulting in ~7.5 times higher Pmax of stacked 2DEG TEGs than that of unstacked 2DEG TEGs. 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