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[Yuto Uematsu](https://orcid.org/0009-0001-8161-6258), [Takafumi Ishibe](https://orcid.org/0000-0002-8662-875X), [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), [Yoshiaki Nakamura](https://orcid.org/0000-0002-5387-1630)

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[Anomalous enhancement of thermoelectric power factor in multiple two-dimensional electron gas system](https://mdr.nims.go.jp/datasets/84de7f6a-de15-496d-8113-a661555f0452)

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Anomalous enhancement of thermoelectric power factor in multiple two-dimensional electron gas systemArticle https://doi.org/10.1038/s41467-023-44165-3Anomalous enhancement of thermoelectricpower factor in multiple two-dimensionalelectron gas systemYuto Uematsu 1, Takafumi Ishibe 1, Takaaki Mano 2, Akihiro Ohtake 2,Hideki T. Miyazaki 2, Takeshi Kasaya 2 & Yoshiaki Nakamura 1Toward drastic enhancement of thermoelectric power factor, quantum con-finement effect proposed by Hicks and Dresselhaus has intrigued a lot ofresearchers. There has been much effort to increase power factor using step-like density-of-states in two-dimensional electron gas (2DEG) system. Here, wepay attention to another effect caused by confining electrons spatially alongone-dimensional direction: multiplied 2DEG effect, where multiple discretesubbands contribute to electrical conduction, resulting in high Seebeckcoefficient. The power factor of multiple 2DEG in GaAs reaches the ultrahighvalue of ~100 μWcm−1 K−2 at 300K. We evaluate the enhancement rate definedas power factor of 2DEG divided by that of three-dimensional bulk. Theexperimental enhancement rate relative to the theoretical one of conventional2DEG reaches anomalously high (~4) in multiple 2DEG compared with those invarious conventional 2DEG systems (~1). This proposed methodology forpower factor enhancement opens the next era of thermoelectric research.Human beings have been seeking a powerful solution to the energycrisis. Thermoelectric (TE) material, which enables the direct conver-sion between waste heat and electricity, is attracting worldwideinterests as one of the sustainable power sources1,2. The TE perfor-mance is quantified by a dimensionless figure-of-merit ZT; ZT = S2σT/κ,where S is Seebeck coefficient, σ is electrical conductivity, κ is thermalconductivity, T is absolute temperature, and S2σ is power factor (PF).The ZT increase has been done by two approaches: κ reduction or PFenhancement3–15. In 2000s, nanostructuring approach intensifiedinterface phonon scattering, decreasing κ drastically. Some studiesachieved 100–200 times smaller κ by introducing nanostructures,making a big impact on TE research3–8. On the other hand, in the ever-reportedmethodologies of PF enhancement, the enhancement rate ofPF is achieved to be several times (2–3 times for energy filteringeffect10–12, 1.5–2 times for resonant scattering effect13,14). Epoch-makingmethodologies for PF enhancement have been expected for furtherincrease in thermoelectric performance.In 1993, Hicks and Dresselhaus proposed the concept of PFenhancement by quantum confinement effect16; e.g. step-likedensity of states (DOS) in two-dimensional electron gas (2DEG)system increases S (step-like DOS effect) (Supplementary Note 1).Since then,much effort has beenmade to experimentally demonstratePF enhancement by quantum confinement effect17–22. In 2018,Zhang et al. experimentally observed an evident feature of 2DEGin SrTiO323: the phenomenon of S enhancement brought by decreasingt2DEG/λ, where t2DEG is 2DEG channel thickness and λ is thethermal de Broglie wavelength23–25. Furthermore, PF enhancementhas been tried by a combination of step-like DOS effect for highS and modulation doping effect for high carrier mobility μ (Fig. 1a)17,21.Toward further high enhancement rate R2D/3D defined asR2D/3D = PF2DEG/PF3D, where PF2DEG is PF of 2DEG and PF3D is PF ofthree-dimensional (3D) bulk, it is strongly demanded to obtain moredrastic increase of R2D/3D as a function of t2DEG/λ than theoreticalfunction R2D/3D ((R2D/3D)th) reported in the previous study24 (Fig. 1d).Although step-like DOS effect by quantum confinement has beenspotlighted so far, we pay attention to another effect caused byquantumconfinement effect:multiple discrete subbandswith step-likeDOS. Provided that multiple subbands with step-like DOS at higherReceived: 9 June 2023Accepted: 3 December 2023Check for updates1Osaka University, 1-3 Machikaneyama-cho, Toyonaka, Osaka 560-8531, Japan. 2National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan. e-mail: nakamura.yoshiaki.es@osaka-u.ac.jpNature Communications |          (2024) 15:322 11234567890():,;1234567890():,;http://orcid.org/0009-0001-8161-6258http://orcid.org/0009-0001-8161-6258http://orcid.org/0009-0001-8161-6258http://orcid.org/0009-0001-8161-6258http://orcid.org/0009-0001-8161-6258http://orcid.org/0000-0002-8662-875Xhttp://orcid.org/0000-0002-8662-875Xhttp://orcid.org/0000-0002-8662-875Xhttp://orcid.org/0000-0002-8662-875Xhttp://orcid.org/0000-0002-8662-875Xhttp://orcid.org/0000-0002-6955-260Xhttp://orcid.org/0000-0002-6955-260Xhttp://orcid.org/0000-0002-6955-260Xhttp://orcid.org/0000-0002-6955-260Xhttp://orcid.org/0000-0002-6955-260Xhttp://orcid.org/0000-0002-3519-4613http://orcid.org/0000-0002-3519-4613http://orcid.org/0000-0002-3519-4613http://orcid.org/0000-0002-3519-4613http://orcid.org/0000-0002-3519-4613http://orcid.org/0000-0003-4152-1171http://orcid.org/0000-0003-4152-1171http://orcid.org/0000-0003-4152-1171http://orcid.org/0000-0003-4152-1171http://orcid.org/0000-0003-4152-1171http://orcid.org/0000-0002-1976-8760http://orcid.org/0000-0002-1976-8760http://orcid.org/0000-0002-1976-8760http://orcid.org/0000-0002-1976-8760http://orcid.org/0000-0002-1976-8760http://orcid.org/0000-0002-5387-1630http://orcid.org/0000-0002-5387-1630http://orcid.org/0000-0002-5387-1630http://orcid.org/0000-0002-5387-1630http://orcid.org/0000-0002-5387-1630http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-44165-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-44165-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-44165-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-44165-3&domain=pdfmailto:nakamura.yoshiaki.es@osaka-u.ac.jpenergy, which are formed by quantum confinement in two-dimensional electron gas (2DEG) systems26,27, contributed to elec-trical conduction, S would be substantially enhanced because theparticipation rate of higher-energy carriers in the carrier conductionbecomes larger (Fig. 1b, c); we callmultiplied two-dimensional electrongas effect (M2DE). In this study, we choose GaAs as a material todemonstrate M2DE. Therein, quantum confinement effect easilyappears because t2DEG/λ of GaAs becomes sufficiently small for 2DEGeven in relatively large t2DEG due to its relatively long λ. In addition,GaAs, which is applied to photonic devices such as vertical cavitysurface emitting laser for smart phone, is an ideal material in terms ofsocial application.Here, we demonstrate thatM2DE brings drastic PF enhancementas follows. We form GaAs triangular quantum well (TQW) with M2DEin addition to modulation doping effect and step-like DOS effect(Fig. 1b, c)28. TQW samples exhibit higher S than rectangular quantumwell (RQW) samples without M2DE or with almost no M2DE whencomparing S values under the situation that the channel width tch ofTQW is equal to the well width twell of RQW. This indicates thatmultiple 2DEG (M-2DEG) in TQW with M2DE is more promising thanconventional single 2DEG (S-2DEG) in RQW without M2DE. The PF ofM-2DEG reaches the maximum value of ~100 μW cm−1 K−2 at n of~1×1018 cm−3 at 300K, which is in a class of ultrahigh PF. Thanks toM2DE, M-2DEG shows more drastic increase of R2D/3D with decreas-ing t2DEG/λ than S-2DEG (Fig. 1d). The experimental R2D/3D ((R2D/3D)ex)relative to the theoretical R2D/3D without M2DE ((R2D/3D)th) is anom-alously high in M-2DEG compared with those in various conventional2DEG systems (~1) (Fig. 1e)17,20–24,29–33. Therein, the layered materialsare excluded owing to the difficulty in discussing the contribution ofM2DE in the layered materials because the electronic band structurerelated to the layer number34,35 influences on the TE properties. Thisproposedmethodology for PF enhancement opens the next era of TEresearch.ResultsSample structures and calculated energy band diagramsThe TQW and RQW samples were formed for M-2DEG and S-2DEG,respectively, using molecular beam epitaxy (MBE). Illustrations ofsample structures and simple band diagrams are shown in Fig. 2a, b,where conduction band bottomof 3D GaAs (Ec), carrier energy (E) andthe bottom energy of i-th subband (Ei). The index i (i = 1, 2,…) is thesubband number, where the subband bottomwith the smaller numberof i positions at the lower energy level. In general RQW, the energydifference between discrete subband bottoms (Ei+1-Ei) is mono-tonically increasingwith increase in the i value. Therefore, unlike TQW,it is expected that one subband (or two subbands) can only exist in thepresent AlGaAs/GaAs/AlGaAs RQW with ~0.2 eV barrier height whenthe step-like DOS appears due to the sufficiently small twell, indicatingS-2DEG system (Supplementary Note 2). Modulation doping was per-formed for both samples by inserting Si-doped Al0.3Ga0.7As layers ascarrier suppliers. In TQWand RQW, 2DEG channels were formed at theinterfaces of undoped GaAs/Al0.3Ga0.7As spacer and in the quantumwell ofGaAs layers sandwiched by twoAl0.3Ga0.7As layers, respectively.In TQW, electron Hall concentration n values of channels were tunedby controlling the thicknesses of spacer layers tsp (0, 2, 30, 60, and90nm). The control of tsp also changed the energy band structure36,bringing the tch variation from 8 to 18 nm. In RQW, twell was controlledfrom 3 to 12 nm.�Fig. 1 | Power factor S2σ enhancement by multiplied 2DEG effect (M2DE).a Schematic illustration of single 2DEG (S-2DEG) in rectangular quantum well(RQW), where modulation doping effect increases carrier mobility μ and step-likedensity-of-states (DOS) effect originated in quantum confinement effect increasesSeebeck coefficient S. b Schematic illustration of multiple 2DEG (M-2DEG) in tri-angular quantum well (TQW), where M2DE bringing high S appears in addition tomodulationdoping effect for highμ and step-likeDOSeffect for high S. c Schematicillustration of S2σ enhancement by three effects: modulation doping effect, step-like DOS effect, and M2DE. d The enhancement rate of S2σ (R2D/3D) as a function ofthe 2DEG channel thickness/de Broglie wavelength. The solid triangles, the solidsquares, and the open marks are R2D/3D values of M-2DEG with M2DE (This study),S-2DEG without M2DE or with almost no M2DE (This study), and 2DEG withoutM2DEorwith almost noM2DE (Preceding studies by othergroups: PbTeRQW17 (theopen circles), PbTeRQW18 (the open triangles), Si RQW19 (the opendiamonds), SiGeRQW21 (the open squares)), respectively. The solid line represents the theoreticalR2D/3D without M2DE (R2D/3D)th24 which is consistent with the data of S-2DEG andpreceding data by the other groups. The broken line denotes R2D/3D = 1 corre-sponding to the performance of 3D materials. e Experimental R2D/3D (R2D/3D)exdivided by theoretical R2D/3D without M2DE (R2D/3D)th. In this work, GaAs TQW (thered star) and GaAs RQW (the red square). In the preceding results, GaN TQW30–32(the blue triangles), PbTe RQW17 (the purple square), SiGe RQW21 (the brownsquare), ZnO TQW33 (the light blue triangle), SrTiO3 TQW20 (the pink triangle), andSrTiO3 RQW20,22 (the pink squares).Article https://doi.org/10.1038/s41467-023-44165-3Nature Communications |          (2024) 15:322 2We reveal that multiple subbands can contribute to electricalconduction in TQW, not in RQW. As examples of calculationmodel, weconsider the samples of RQWwith twell = 4, 12 nm (Fig. 2c, d), and TQWwith tch = 8, 15 nm (Fig. 2e, f). The energy band diagrams and the cal-culated carrier distribution, ncal were obtained by self-consistentcomputation using one-dimensional Poisson-Schrödinger equation37.It was found that the TQW is formed at the interface of undopedGaAs/AlGaAs spacer. Therein, multiple subbands locate near Fermi energyEF. For example, some subbands locate in the range of E-EF < ~ 0.1 eV(Fig. 2e, f) in the TQW unlike only one or two subbands in the RQW(Fig. 2c, d, Supplementary Note 2).Theoretical demonstration of M2DETo clarify the contribution of carrier existing at each subband toelectrical conduction, we calculated the occupation ratioRO defined asRO=ni/nt, whereni is sheet carrier concentration at the i-th subband andnt is the sum of ni. The expressions for ni and nt are described asfollows:ni =Z 1E if 0 Eð ÞDi Eð ÞdE ð1Þnt =Xini ð2Þwhere f0(E) is the Fermi–Dirac distribution function.Di(E) is DOS at thei-th subband, which is described asm/πℏ2. Therein,m is effective massof carrier and ℏ is Dirac constant. It is found that RO is the function of Eifrom Eqs. (1) and (2). Figure 2g shows RO at the i-th subband. As Ei-EFincreased, RO decreased nearly exponentially, which is coming fromthe energy dependence of the Fermi–Dirac distribution function.However, some RO values at the i-th subband (i > 1) seem to berelatively high. This implies that multiple subbands can contribute toelectrical conduction38. Thus, it is expected that M2DE can appearin TQW.We theoretically demonstrate S enhancement by M2DE in TQW.As an example of calculation model, we consider the sample withtch = 15 nm. Theoretical S was calculated under parabolic band for2DEG and bulk, and relaxation time approximations on the basis ofBoltzmann transport theory (details available in Methods). In thesummation of i-th subband in the calculation, it is enough to considerup to themaximum i-th subband contributing to electrical conductionalthough it is ideal to consider up to infinity. Therefore, when thecontribution of the i-th subband is summated until im, we investigatedthe relationship between S and im (Fig. 2h), which was calculated usingphysical parameters39,40 displayed in Table 1. S was saturated in therange of im> 20 because of less contribution of subbandswith i > 20 toelectrical conduction. This saturation indicates that im of 20 is criticalvalue (iC) to calculate S accurately. Namely, it is enough to calculate Susing the i of less than iC (in this case, 20). In this study, calculations ofTE properties for TQW samples with various tch were also performedwith im ~ iC for sufficient calculation accuracy (Supplementary Note 3).�Fig. 2 | Sample structure illustrations, calculated energy band diagrams, andtheoretical demonstrationofmultiplied 2DEGeffect (M2DE). a,b Illustrations ofsample structures and energy band diagrams of rectangular quantum well (RQW)(a) and triangular quantum well (TQW) samples (b)48. c–f Calculated energy banddiagrams of RQW with the well width twell = 4 nm (c), 12 nm (d) and TQW sampleswith the channel width tch = 8 nm (e), 15 nm (f). The solid black lines: conductionband bottom of 3D GaAs (Ec), the broken black lines: Fermi energy EF, the solid redlines: the bottom energy of i-th subband (Ei) (for simplicity, Ei with i < 7 are dis-played), and the solid blue line: the calculated carrier distribution ncal as a functionof z. z is the distance from the interface of undoped GaAs/AlGaAs spacer along thedirection perpendicular to the sample surface. g The carrier occupation ratio RO asa function of Ei-EF in the TQW sample with tch = 15 nm. h Calculated Seebeck coef-ficient S as a function of im in the TQW sample with tch = 15 nm, when the con-tribution of the i-th subband is considered until im.Table 1 | Parameters used in the calculationParameter Symbol Value39,40Effective mass m 0.067m0 kgFree electron mass m0 9.11×10−31 kgRelative high-frequency dielectric constant κ∞ 10.89Relative static dielectric constant κ0 13.18Longitudinal optical phonon energy ℏωLO 36.5meVDeformation potential constant DA 13.5 eVLongitudinal elastic constant cL 1.4×1011 N m−2Article https://doi.org/10.1038/s41467-023-44165-3Nature Communications |          (2024) 15:322 3In the sample with tch = 15 nm (Fig. 2h), the saturated S value in thecalculation withmultiple subbands (im> 20) was ~1.7 times higher thanthat in the calculation with single subband (i = 1), namely the calcula-tion without M2DE. This theoretically proves that M2DE substantiallyenhances S.Thermoelectric propertiesExperimental and calculated TE properties of M-2DEG and S-2DEG areshown in Fig. 3. Therein, theoretical calculation of S and μ was per-formed under parabolic band and relaxation time approximations onthe basis of Boltzmann transport theory41. The details of carrier scat-tering models and used parameters are written in the section ofNumerical calculation and Table 1 respectively. Figure 3a, b show S andμ as a function of n at 300K, respectively. When estimating n ofM-2DEG in TQW, we defined the tch as FWHM of the carrier con-centration distribution along the perpendicular direction to substratesurface (Supplementary Note 4)20. In M-2DEG (TQW), n was tuned bycontrolling tsp. As shown in Fig. 3a, when decreasing tsp (tch), n wasincreased because of increase of carrier supply from Si-dopedAl0.3Ga0.7As layers. The S values of M-2DEG (the solid red triangles)were compared with that of 3D GaAs film (the solid black circle) thatdoes not have modulation doping effect, step-like DOS effect, andM2DE.Weplotted the calculation curveof 3DGaAs42which reproducesthe experimental value of 3D GaAs film. When comparing them at thesame n, M-2DEG exhibited higher S than the calculation curve of 3DGaAs. To demonstrate S enhancement by M2DE experimentally, wemeasured S values of conventional S-2DEG in RQW samples (the solidblue squares) without M2DE or with almost no M2DE for comparison.When varying twell from 12 to 3 nm, S values of S-2DEG were graduallyincreased because of step-like DOS effect. This tendency was wellreproduced by the S calculation for S-2DEG (the open blue squares).Thus, not only M2DE but also step-like DOS effect causes S enhance-ment, making it difficult to understand the physical mechanism of Senhancement. To discuss the differencebetween the twoeffects, let uscompare S values ofM-2DEGwith those of S-2DEG. At almost the samen, the M-2DEG with tch of ~8 nm exhibited higher S than S-2DEG withtwell of ~8 nm, while in the stronger confinement case of small width(twell ~ 3 nm) in RQW, high S was obtained to be comparable to that inthe case of 8 nmwidth in TQW. This is because S enhancement appearsin M-2DEG (TQW) over a wide range of confinement width, unlike��� �����Fig. 3 | Thermoelectric properties. a, b Carrier concentration n dependences ofSeebeck coefficient S (a) and carrier mobility μ (b) measured at 300K in multiple2DEG (M-2DEG) in triangular quantum well (TQW) with multiplied 2DEG effect(M2DE) (the solid red triangles), single 2DEG (S-2DEG) in rectangular quantumwell(RQW)withoutM2DEorwith almost noM2DE (the solid blue squares), respectively.The calculation data for M-2DEG (the open yellow triangles) and S-2DEG (the openblue squares) are also plotted simultaneously. For comparison with the data in 3DGaAs without 2DEG, the experimental value (the solid black circle) and calculationcurves (the broken lines) of 3DGaAs are simultaneously plotted. The channel widthtch of M-2DEG and the well width twell of S-2DEG are displayed around the experi-mental data points. The inset in (a) is an enlarged n-S plot: experimental and cal-culated n dependences of S in S-2DEG. c Temperature T dependences of μ inM-2DEG with tch = 8 (the solid diamonds) and 15 nm (the solid triangles), 3D GaAsfilm without 2DEG (the solid circles).We also simultaneously plotted the calculatedT-μ curves of M-2DEG with tch = 8 (the open diamonds) and 15 nm (the open tri-angles).d, e n dependences of electrical conductivity σ (d) and power factor S2σ (e)at 300K in M-2DEG with M2DE (the solid triangles), S-2DEG without M2DE or withalmost no M2DE (the solid squares), respectively. The experimental data (the solidcircles) and the calculation curves (the broken lines) of 3DGaAs are simultaneouslyplotted. The dotted curves in (e) are eye-guides for M-2DEG (red) and S-2DEG(blue). In (e), the tch of M-2DEG and twell of S-2DEG are displayed around theexperimental data points. The insets show the density of states (DOS) of M-2DEG(with M2DE) and S-2DEG (without M2DE).Article https://doi.org/10.1038/s41467-023-44165-3Nature Communications |          (2024) 15:322 4S-2DEG (RQW) with strong twell dependence, which is also confirmedby the calculation (Supplementary Note 5). Furthermore, the S calcu-lation (the open yellow triangles) including M2DE in M-2DEG agreedwith the experimental n-S data (Fig. 3a) and T–S data (SupplementaryNote 6), which is the theoretical evidence that M2DE appears. Thus, Senhancement by M2DE was demonstrated both experimentally andtheoretically.As well as S, μ values of M-2DEG were compared with the calcu-lation curve of 3DGaAs (Fig. 3b). When comparing them at the same n,M-2DEG with modulation doping effect exhibited higher μ than thecalculation curve of 3D GaAs without modulation doping effect whichreproduces the experimental value of 3D GaAs film. Furthermore, theexperimental μ data agreed with the μ calculation including M2DE inaddition to modulation doping effect and step-like DOS effect forM-2DEG.WealsoobtainedT–μdata in theT rangeof 80-300K (Fig. 3c).The experimental T-μ data of M-2DEG with tch of 8 nm were comparedwith those of 3D GaAs, where compared samples had almost the samen of ~1 × 1018 cm−3 at 300K. Then, μ values of the M-2DEG drasticallyincreased as T decreased, while μ of 3D GaAs film did not depend onthe T. The tendency of experimental data in theM-2DEGwas explainedby the theoretical T–μ curve of M-2DEG (the open marks in Fig. 3c),where the dominant scattering is polar optical phonon scattering dueto the almost no ionized impurity scattering unlike 3D GaAs withionized impurities. The orders of magnitude higher mobility at lowtemperature is reported as the result from modulation dopingeffect43,44. These results strongly support that the modulation dopingeffect, step-like DOS effect, and M2DE appear in M-2DEG.On the other hand, as shown in Fig. 3b, S-2DEG with modulationdoping effect also exhibited higher μ than the calculation curve of 3DGaAs without modulation doping effect at the same n. When varyingtwell from 12 to 3 nm, μ values of S-2DEGweremonotonically decreasedbecause of increase of interface carrier scattering rate. This tendencywaswell reproduced by the μ calculation for S-2DEG. Thus, S-2DEG hasa trade-off relationship between S and μwith respect to twell, making itdifficult to realize ultrahighPF. In contrast, highμ values ofM-2DEGdidnot depend on the tch within the range of 8–18 nm. Therefore, M2DE isexpected along with the high μ of ~6000 cm2V−1s−1. Namely, M-2DEGhas a high potentiality of exhibiting ultrahigh PF by simultaneousenhancement of S and μ.Figure 3d shows σ as a function of n at 300K. M- and S-2DEGexhibitedhigher σ values than 3DGaAs at almost the samenbecauseofhigher μ. As for the σ tendency against n, there was a significant dif-ferencebetweenM- and S-2DEG; σofM-2DEG increased asn increased,while σ of S-2DEG did not depend on the n. The increasing σ tendencyof M-2DEG is explained by constant μ tendency against n (Fig. 3b). Onthe other hand, constant σ tendency of S-2DEG is attributed to thedrastically-decreasing μ tendency against n. When σ values of M-2DEGwith tch of 8 nmwere comparedwith thoseof S-2DEGwith twell of 3 nm,where both samples exhibited the equivalent S values at almost thesame n, M-2DEG had approximately 3 times higher σ than S-2DEG. Thisindicates that M-2DEG is more promising than S-2DEG in terms ofsimultaneous realization of high S and high σ.Experimentally observed anomalouspower factor enhancementin M-2DEGFigure 3e shows PF as a function of n at 300K. Both M- and S-2DEGexhibited higher PF than 3D GaAs at almost the same n. Whendecreasing twell from 12 to 3 nm in RQW, PF values of S-2DEG increasedmonotonically because S was substantially increased by step-like DOSeffect. A remarkable fact is that M-2DEG always exhibitedmuch higherPF than S-2DEG because of M2DE. In Fig. 3a, b and e, at the ~(1-2) ×1018 cm−3, higher PF in M-2DEG comes from higher μ in TQW, where Svalues of TQWandRQWare comparable, while at ~4 × 1017 cm−3, higherPF in M-2DEG is due to higher S in TQW, where μ values of TQW andRQW are comparable. This is because there is a trade-off relationshipbetween S andμ in RQW.On theother hand, S andμ are simultaneouslyenhanced in TQWwith M2DE. As a result, the maximum PF of M-2DEGreached ~100 μW cm−1 K−2 at n of ~1 × 1018 cm−3 at 300K, which is in aclass of ultrahigh PF. Thanks to the S enhancement byM2DE alongwiththe high μ, M-2DEG showedmore drastic increase ofR2D/3Dwith t2DEG/λdecrease than S-2DEG (Fig. 1d). As a result, M-2DEG exhibited thehighest (R2D/3D)ex/(R2D/3D)th among various 2DEG systems (Fig. 1e),which was anomalously high like singularity compared with those invarious 2DEG systems (~1). This highlights that M2DE can bring ultra-high PF beyond conventional 2DEG.DiscussionIn summary, we demonstrated that M2DE caused by the quantumconfinement effect brings drastic PF enhancement. M-2DEG withM2DE in addition tomodulation doping effect and step-likeDOS effectexhibited higher S than conventional S-2DEG without M2DE or withalmost no M2DE over a wide range of confinement width. The PF ofM-2DEG reached the maximum value of ~100 μW cm−1 K−2 at n of ~1 ×1018 cm−3 at 300K, which is in a class of ultrahigh S2σ. Thanks to M2DE,M-2DEG exhibited the highest (R2D/3D)ex/(R2D/3D)th among variousconventional 2DEG systems except for the layered materials with theelectronic band structure depending on the layer number (Fig. 1e).This value was anomalously high like singularity compared with thosein various conventional 2DEG systems. This study presented themethodology enabling the drastic PF enhancement based on quantumconfinement effect, which opens the next era of TE research.MethodsSample preparationTQW samples were formed using MBE in the following process. Toobtain clean surfaces of undoped GaAs(001) substrates, undopedGaAs (300nm) initial layers were grown on the GaAs substrates. Sub-sequently, as the buffer layers, GaAs/Al0.3Ga0.7As superlattice layerswere grown on the undoped GaAs (300 nm)/GaAs substrates byalternately depositing GaAs (10 nm) and Al0.3Ga0.7As (10 nm) 20 times.On the buffer layers, undoped GaAs (1000 nm) layers with high crys-tallinity were grown. These layers were grown at 893K. After thegrowth of Al0.3Ga0.7As spacer layers on the undoped GaAs (1000 nm)layers at 893 K, Si-doped Al0.3Ga0.7As (dopant concentration: 5 ×1017 cm−3, thickness: 80 nm) layers were grown at 823 K to supply car-rier to the interface of undoped GaAs (1000 nm)/Al0.3Ga0.7As spacer.The nwas tuned by controlling tsp (0, 2, 30, 60, and 90 nm). Finally, toprevent the oxidation of samples, the sample surfaces were capped bydepositing GaAs layers (10 nm) at 823 K.For reference, RQW samples without M2DE or with almost noM2DEwere formed. UndopedGaAs (500nm) layersweregrownon theGaAs(001) substrates. Subsequently, as the buffer layers, GaAs/AlAssuperlattice layers were grown on the undoped GaAs (500nm)/GaAssubstrates by alternately depositing GaAs (2 nm) and AlAs (2 nm) 100times. On the buffer layers, undoped Al0.3Ga0.7As (20 nm) barrier lay-ers, GaAs (3, 4, 5, 6, 8, 10, and 12 nm) layers, and Al0.3Ga0.7As (2, 10 nm)spacer layers were grown in a sequential order. The growths of theselayers were carried out at 873 K. After that, as the carrier suppliers toGaAs wells, Si-doped Al0.3Ga0.7As (dopant concentration: 7 × 1017 cm−3,thickness: 80 nm) layers were grown at 813 K on the Al0.3Ga0.7As (2 or10 nm) spacer layers. Finally, to prevent the oxidation of samples,capping GaAs layers (10 nm) were formed at 813 K.Thermoelectric property measurementsThe stacked structures of AuGe/Ni/Au were formed on the samples aselectrodes. To make ohmic contact, the samples were annealed at723K for 90 s. Sheet electrical conductivity and sheet carrier con-centration were measured using the van der Pauw method and Halleffect measurement, respectively. σ and n are obtained by dividingmeasured sheet electrical conductivity and sheet carrier concentrationArticle https://doi.org/10.1038/s41467-023-44165-3Nature Communications |          (2024) 15:322 5by twell or tch20. In our Hall effect measurement, we used 2401 source-meter (Keithley) as source measure unit, and the range of magneticfield is from −0.5 T to 0.5 T. Therein, the errors of n and μ are about13%. Swas measured using ZEM-3 (ADVANCE RIKO Inc.)45,46, where thetemperature difference is applied along the in-plane direction, and thedifferences of temperatures and the electric voltages between twopoints on the films were obtained by thermocouple probes. The con-tribution of Si-doped Al0.3Ga0.7As layer to electrical conduction wasremoved using the parallel conductionmodel (SupplementaryNote 7).Numerical calculationTheoretical T–μ, n–μ, and n–S curves were calculated under effectivemass and relaxation time approximations on the basis of Boltzmanntransport theory as follows:S= � 1eTPiR1�1 E � E i� �E � EF� � ∂f 0∂E τi E � E i� �Di E � E i� �dEPiR1�1 E � E i� � ∂f 0∂E τi E � E i� �Di E � E i� �dEð3Þμ=emPiR1�1 E � E i� � ∂f 0∂E τi E � E i� �Di E � E i� �dEPiR1�1 E � E i� � ∂f 0∂E Di E � E i� �dEð4Þwhere e is the elementary charge and τi is the total carrier relaxationtimeat the i-th subband.Di(E) was simply assumed as a step function. 1/τi is described as the sum of each scattering rate at the i-th subbandthrough Matthissen’s rule as follows: 1/τi = 1/τPOP + 1/τADP + 1/τRII + 1/τIFR, where 1/τPOP is polar optical phonon (POP) scattering rate, 1/τADP isacoustic deformation potential (ADP) scattering rate, 1/τRII is remoteionized impurity (RII) scattering rate, 1/τIFR_rec is interfacial roughness(IFR) scattering rate in RQW, and 1/τIFR_tri is IFR scattering rate in TQW.Inter-subband and intra-subband scatterings are both considered bychoosing proper wave functions in POP and ADP scattering calcula-tions. Each scattering rate is described as follows39,47–50:1τPOP=e2m_ωLO8π2_3ε01κ1� 1κ0� �11� f 0 Eð ÞXj1� f 0 E + _ωLO� �� �NqZ jI qz� �j2q2+ +q2zdqz + 1� f 0 E � _ωLO� �� �u E � _ωLO� �ðNq + 1ÞZ jI qz� �j2q2� + q2zdqz!ð5Þ1τADP=D2AkBT2_cLXjZ 1�1φi zð Þφj zð Þdz�  Dj Eð Þ ð6Þ1τRII=Z tsp + tdopetsp12nimpmπ_3� �e22κ0ε0� �2Z 2π0exp �2q � zð Þq+ qTF� �2 1� cosθð Þdθ" #dzð7Þ1τIFR rec=4πmE2i Δ2Λ2t +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2_2m V0�E ið Þr� �2_3� 12πexp �Λ2 2m E�E ið Þ_2� �1� cosθð Þ20B@1CA 1� cosθð Þ qq+ 2aBð8Þ1τIFR tri=mΔ2Λ2e2 e n2 +ndepl� �� �22_3 κ0ε0 +1q � e2m2π_2� F qð Þ� �2 exp �Λ2q24 !1� cosθð Þ ð9ÞwhereℏωLO is the longitudinal optical phonon energy, ε0 is the vacuumdielectric constant, κ∞ is the relative high-frequency dielectricconstant, κ0 is the relative static dielectric constant, Nq is thedistribution function of optical phonon. |I(qz)|2 is the form factor dueto the quantized wave function; I qz� �=R1�1φi zð Þφj zð Þ expðiqzzÞdz,whereφi(z) is the wave function at i-th subband, z is the distance alongthe direction perpendicular to the sample surface (z =0 is defined asthe interface position of undoped GaAs/AlGaAs spacer.), qz is thescattering wave vector in the z direction. q described as q =k2-k1 is atwo-dimensional scattering wave vector from initial state k1 to the finalstate k2 in the elastic collisions. q+ and q- are two-dimensionalscattering wave vectors in the phonon absorption and the phononemission, respectively, as follows:q+ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � 2m E � E i� �_2� �+2m_ωLO_2� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m E � E i� �_2s�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m E � E i� �_2+2m_ωLO_2scosθvuutð10Þq� =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 � 2m E � E i� �_2� �� 2m_ωLO_2� 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m E � E i� �_2s�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m E � E i� �_2� 2m_ωLO_2scosθvuutð11Þq= 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m E � Ei� �_2ssinθ2ð12Þwhere θ is the scattering angle between k1 and k2. DA is the defor-mation potential constant, kB is Boltzmann constant, cL is the long-itudinal elastic constant, tdope is the thickness of Si-dopedAl0.3Ga0.7As layer, nimp is the concentration of impurity atoms, qTFis Thomas-Fermi wave number, V0 is the energy barrier height, Δ andΛ are the mean interface roughness values at the z direction and atthe perpendicular direction to z, respectively (in this calculation,these parameters are fixed at Δ=0.5 nm and Λ=5 nm), aB is theeffective Bohr radius, ndepl is the charge density of the depletionlayer, and F(q) = ∫dz∫dz′|φ(z)|2|φ(z′)|2exp(−q|z − z′|). In the calculation,m value shown in Table 1 was used for each subband under theassumption that the non-parabolicity effect on m is negligible51.The energy band diagram was computed using 1D Poissonsolver developed byG. Snider;wavefunction and carrier concentrationdistribution were self-consistently computed using thePoisson–Schrodinger equation. 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C., Perrin, R. E., Messham, R. L. & Yen, M. Y. Two-dimensional electron gas in GaAs/Al1-xGaxAs heterostructures:effective mass. Phys. Rev. B 43, 11787–11790 (1991).AcknowledgementsThisworkwas supported byGrant-in-Aid for Scientific ResearchA (GrantNo. 19H00853, and 23H00258) (Y.N.) and JSPS Fellows (T22KJ2052)(Y.U.). We would like to thank Dr. T. Noda (NIMS) for some discussion.Author contributionsY.N. conceived the idea. Y.U. measured thermoelectric properties of thesamples. Y.U. and T.I. performed numerical calculation. Y.U., T.I., andY.N. discussed the physics and wrote the paper. Y.N. is the principalinvestigator of this work. T.M. and A.O. formed the samples. H.T.M. andT.K. formed the electrode. All authors discussed the results and con-tributed to the revision of the final manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-44165-3.Correspondence and requests for materials should be addressed toYoshiaki Nakamura.Peer review information Nature Communications thanks Je-HyeongBahk, Yue Lin and the other, anonymous, reviewer(s) for theircontribution to the peer review of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-023-44165-3Nature Communications |          (2024) 15:322 8https://doi.org/10.1038/s41467-023-44165-3http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Anomalous enhancement of thermoelectric power factor in multiple two-dimensional electron gas�system Results Sample structures and calculated energy band diagrams Theoretical demonstration of�M2DE Thermoelectric properties Experimentally observed anomalous power factor enhancement in M-2DEG Discussion Methods Sample preparation Thermoelectric property measurements Numerical calculation Data availability References Acknowledgements Author contributions Competing interests Additional information