Introduction

In modern semiconductor electronic devices, current-carrying electrons are locally driven far away from equilibrium (with an effective electron temperature Te largely exceeding the lattice temperature TL)1,2, and these hot electrons accelerate/decelerate frequently to fulfil intended functions. The excess energy of hot electrons typically dissipates locally to the lattice as the Joule heat, which not only leads to a major heat concern for post-Moore-era nanoelectronics3 but also exert a thermodynamic limitation to the energy efficiencies in all solid-sate electronic devices (such as the Shockley–Queisser limit4, being only ~30% for Si solar cells). To suppress net energy loss to lattice, the excess energy carried by hot electrons has to be transmitted along with the charge transport. Nevertheless, dissipationless charge transport is hitherto realized only in, e.g., superconductors5,6 or topological transistors7 under (quasi-)equilibrium limit (Te\(\backsimeq\)TL) or nanoscale vacuum transistors8 where phonon-emission process is eliminated. For electrons in the strong nonequilibrium conditions (Te\(\gg\)TL), however, dissipationless transport appears to be challenging and has never been addressed because electron cooling occurs spontaneously at an intrinsically fast speed due to rapid electron–phonon interactions (~ps).

Numerous works have been devoted to exploiting the exotic properties of the hot electrons in strong nonequilibrium conditions such as superdiffusion9, chaotic diffusion10, thermal oscillation11, and prototype devices have been attempted such as hot-electron transistors12, hot luminescent light sources13, highly efficient solar cells14 and plasmon-enhanced photochemistry15,16, etc. In all these works, however, the transport of the hot electrons remains to be highly dissipative which restricts severely the achievable device performance. Further improving the device performance requires a comprehensive understanding of nanoscale kinetics of how exactly the energy is carried by the nonequilibrium electrons, and how the energy dissipation to the lattice can be significantly suppressed. This, however, eluded direct nanothermometric observation in real space due to nonequilibrium nature of embedded electrons and their intrinsically small heat capacity (typically several orders of magnitude less than that of the lattice). A number of highly sensitive nanoscale mobile carrier17 or current imaging18 and scanning nanothermeometry techniques19,20,21,22 have recently been developed, revealing critical local information about charge transport and heat generation, but they are insensitive to the electron temperatures and do not necessarily provides straightforward access to the detailed interplay between the strongly nonequilibrium electrons and their host lattice systems.

Here, by using recently developed radiative electronic nanothermometry23 together with a conventional contact-type one, we separately imaged and compared heated electrons (Te) and lattice (TL) in GaAs/AlGaAs quantum well (QW) conducting channels. With these real-space measurements, we disclosed diffusive, but nearly dissipationless transport of hot electrons at room temperature: More than 90% of the electron energy passes unexpectedly through a channel of up to 1 μm length without substantial dissipation to the lattice (much longer than the mean free path). The dramatic suppression of the energy loss across this ultralong distance is attributed to overpopulated hot longitudinal–optical (LO) phonons that induce frequent LO-phonon reabsorption and thereby remarkably slow down the electron cooling. This hot-phonon-assisted electron transport is reminiscent of the previously reported “hot-phonon bottleneck effect” for photoexcited transient carriers (e.g., in perovskites)24,25,26,27,28,29. Noting that the hot-phonon bottleneck effect is exempted from any restriction of device operating temperatures, our observations may find promising applications in on-chip energy management for solid-state electronics and energy-harvesting technologies.

Results and discussion

Nanoscale thermometric imaging of both conduction electrons and the lattice

The non-contact electronic nanothermometry (top panel of Fig. 1a) is a scanning noise microscope (SNoiM) that has recently proven to sensitively detect the shot noise generated by hot electrons23,30: hot-electron distribution in real space is thereby visualized noninvasively. (see “Methods” and Supplementary Note 1). In this technique, a sharp metal tip scatters fluctuating electromagnetic evanescent fields at terahertz (THz) frequencies (about 20.7 ± 1.2 THz) that are generated on a sample surface by current fluctuation of conduction electrons, and the scattered THz wave is collected and detected by an ultrahigh sensitivity detector called a charge sensitive infrared phototransistor31. The second method is contact-type scanning thermal microscopy (SThM)19, where a thermistor is integrated with an atomic force microscope cantilever, which locally probes the lattice temperature (top panel of Fig. 1b) via contact thermal conduction rather than conduction electrons (see “Methods” and Supplementary Note 2).

Fig. 1: Nanoscale thermal probing of heated electrons and heated lattice.
figure 1

a SNoiM. Top panel: A non-contact probe tip, 10 nm above the sample surface, scatters fluctuating THz evanescent fields for detection. Middle panel: A 2D image of Te on a narrow conducting channel biased with Vb = 8 V. Bottom panel: A 1D profile of Te across the channel at x = 0. The signal averaging time is 300 ms/pixel for each pixel = 50 nm × 50 nm, where low-frequency fluctuation in the signal roughly corresponds to ΔTe ~±100 K in the region outside mesa (without conduction electrons) and ΔTe/Te = ±7% in the region of mesa (with conduction electrons). b SThM. Top panel: A nanothermometer (shown with yellow dots) is in thermal contact with the sample surface to measure TL. Middle panel: A 2D image of TL taken on the same device as that for SNoiM with Vb = 8 V. Bottom panel: A 1D profile of TL across the channel at x = 0. The dotted horizontal line shows the base temperature (TL = Tbase = 300.8 K). The signal averaging time is ~ 8 ms/pixel for each pixel = 40 nm × 40 nm, where relative precision of measurements is roughly ΔTL ~±100 mK. c AFM. Top panel: AFM tip. Middle panel: A 2D topographic image of the device studied, with a 400 nm wide and 870 nm long constriction channel of a quasi 2D electron system in a GaAs/AlGaAs quantum well (see “Methods”). Bottom panel: current–voltage trend.

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All the measurements are made at ambient temperature (TRoom ≈ 300 K). The middle panel of Fig. 1c shows the atomic force microscopy (AFM) topography of a narrow GaAs/AlGaAs QW heterostructure device with the current–voltage characteristics displayed underneath (see “Methods”). The middle panel of Fig. 1a displays a representative two-dimensional (2D) colour plot of the effective electron temperature Te, converted from the SNoiM signal (see “Methods”) at bias voltage Vb = 8.0 V (and the corresponding source-drain current is Ids = 0.24 mA). White broken lines are guides for the eyes and mark the edge that defines the constriction channel. The SNoiM signal exists only in the channel region, as elucidated explicitly in the bottom panel of Fig. 1a, with a one-dimensional (1D) profile of Te taken across the constriction channel (in the y-direction at x = 0). This indicates that electrons are probed and not the lattice. Notably, two distinct hot spots are recognized, one close to the entrance for electrons in the constricted channel and the other outside the channel close to the exit. The highest temperatures at the hot spots reach ΔTe ∼ 1700 K for Te = TRoom + ΔTe ≈ 2000 K.

The middle panel of Fig. 1b displays a 2D colour plot of the SThM signal and shows a profile of TL. In contrast to that of Te, the profile is featured by a broader structure with a single peak located outside the exit. No structure is discerned corresponding to the electron hot spot on the entrance side. In addition, the heated region is not confined in the conducting channel but smoothly spreads out of the channel, as explicitly shown in the bottom panel of Fig. 1b, with 1D profile across the channel (in the y-direction at x = 0) (see also Supplementary Fig. 4) The distribution of TL is smooth and spread. There is a small temperature rise such that ΔTL ~ 1 K at the maximum, where TL = Tbase + ΔTL and Tbase = 300.8 K (see “Methods” and Supplementary Fig. 3 and Supplementary Note 3), which is due to a large lattice specific heat and the fact that the heat spreads via lattice thermal conduction32.

The double-peak structure of Te, studied with SNoiM, contrasted with the single-peak structure of TL, is shown in Fig. 2. It shows how the Te distribution evolves with increasing Vb from 2.0 to 8.0 V in a similar but slightly shorter device than the one used for Fig. 1. A comparison of Fig. 2c, d demonstrates that the peaks at the entrance and at the exit of the constriction, also identified by different sizes, swap places when the bias is reversed. Additional experiments on other devices with differing channel lengths (200 nm–1.0 μm) make it clear that (i) the Te profile generally exhibits a double-peaked structure, with the first hot spot within the channel close to the entrance and the second one outside the channel 100–250 nm away from the exit and that (ii) the first hot spot does not produce the corresponding signature in the TL profile (see Supplementary Note 3).

Fig. 2: Evolution of the two-Te-peak structure in the real-space hot-electron distribution with increasing bias voltage, Vb.
figure 2

a–h 2D images of Te obtained with SNoiM, where d shows the image in the opposite bias polarity. The device studied is similar to that for Fig. 1, with a slightly different geometry of the constriction channel of 450 nm width and 640 nm length. White dotted lines are guides for the eyes to show the mesa edges that define the constriction. The current–voltage trend is shown in the inset.

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The one-dimensional profiles of Te and TL taken along the channel (in the x direction at y = 0) in the device of Fig. 1 are plotted together for comparison in Fig. 3a. The upper inset replots the 2D image of Te in the middle panel in Fig. 1a on the same x-axis scale as that of Fig. 3a–d. The second Te peak outside the channel exit coincides with the single TL peak, but the first Te peak does not have a corresponding structure in the TL profile. Without theoretical interpretation, this feature implies that hot electrons heated in the vicinity of the channel entrance are thermally isolated from the lattice and that electron heating does not lead to discernible lattice heating. In addition, if the excess energy from hot electrons is dissipated during the travel through the channel, it would increase the left-hand side of the TL-profile in Fig. 3a, making the curve asymmetric about the peak. The TL profile is, however, nearly symmetric around the peak outside the channel exit, indicating no discernible energy dissipation throughout the channel (see Supplementary Fig. 3 and Supplementary Note 3). Hence, the coincident peaks of Te and TL outside the channel exit suggest a surprising aspect that hot electrons are thermally isolated from the lattice throughout the channel and that the energy gained by the electrons passes through the channel without undergoing a significant loss and is eventually released to the lattice outside the channel exit. The length for which this quasiadiabatic transport takes place (roughly the channel length) largely exceeds the distance that hot electrons drift during the conventional energy relaxation time (see “Methods”). Hence, the phenomenon here requires explanation.

Fig. 3: Comparison between electron heating (Te) and lattice heating (TL).
figure 3

a 1D profiles of Te (red dots, taken with SNoiM) and TL (blue dots, taken with SThM) taken along the device of Fig. 1 at y = 0 with a bias voltage Vb = 8.0 V. The broken horizontal line shows the baseline for both Te and TL (ΔTe = ΔTL = 0 or Tbase = Te = TL = 300.8 K), where the vertical scales are given so that the peak heights agree to one another. b 1D profiles of Te and TL, similar to the data of a except that Vb = 3.0 V. c Theoretically derived curve of <Te> (red line) for Vb = 8.0 V reproduces the experimental Te-profile with the two-peak structure (red dots). The broken horizontal line marks the position of kBTbase with Tbase = 300.8 K. d Theoretically derived single peak structure of TL (blue line) reproducing the experimental profile (blue dots). The blue dotted curve represents the theoretically derived rate of energy loss due to LO-phonon scattering, PLO. The broken horizontal line marks the position of kBTbase with Tbase = 300.8 K for kBTL and zero for PLO.

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In the low-bias regime with Vb = 3.0 V, Fig. 3b shows that both Te and TL form roughly symmetric profiles centred at the middle of the channel (x = 0), reaching the peak values Te ≈ 1200 K and ΔTL ≈ 0.1 K without exhibiting peculiar features.

Hot LO-phonon bottleneck in the electron transport

It is established that hot electrons in GaAs primarily dissipate energy by emitting longitudinal–optical (LO) phonons with energy \(\hslash\)ωLO ≈ 37 meV at a typical rate of 1/τLO = (0.12 ps)−1 (ref. 33). The emitted LO phonons, in turn, are known to decay into two longitudinal acoustic (LA) phonons via the so-called Klemens channel after a decay lifetime of τl,LO ≈ 5.0 ps34. The two LA phonons eventually decay into thermalized longer-wavelength (λ) acoustic phonons, ending up with lattice heating. In the ordinary framework of hot-electron transport, possible rise in the effective LO-phonon temperature, TLO, caused by the emitted LO phonons is supposed to be insubstantial.

In the present work, TLO can be significantly elevated because (i) the electron density is relatively high, (ii) the LO phonons do not spatially diffuse due to nearly vanishing group velocity, and (iii) the emitted LO phonons are of small wave numbers being confined within a narrow q-space sphere around the Brillouin zone centre, containing only a limited number of states (NOS = 4.00 × 1018/cm3)35 (see Supplementary Fig. 5 and Supplementary Note 4). If the electrical input power, p = Ej, is dissipated utterly via the emission of LO phonons, the rate of LO-phonon emission is given by ∂NLO/∂t = p/ \(\hslash\)ωLO. The emitted LO phonons, in turn, pile up to the nonequilibrium density higher than the thermal equilibrium value by NLO =  (dNLO/dt) × τl,LO = (p/\(\hslash\)ωLO)τl,LO. One can hence estimate TLO through

$$(p/{\hbar }{\omega }_{{{{{{\rm{LO}}}}}}}){\tau }_{{{{{{\rm{l}}}}}},{{{{{\rm{LO}}}}}}}={\mathrm {NOS}}\{{n}_{{{{{{\rm{LO}}}}}}}({T}_{{{{{{\rm{LO}}}}}}})-{n}_{{{{{{\rm{LO}}}}}}}({T}_{{{{{{\rm{Room}}}}}}})\},$$

where nLO(T) ≡ {exp(\(\hslash\)ωLO/kBT) − 1}−1 is the LO-phonon occupation number at temperature T with the Boltzmann constant kB. From this relation, TLO is explicitly derived to be

$${T}_{{{{{{\rm{LO}}}}}}}=({\hbar }{\omega }_{{{{{{\rm{LO}}}}}}}/{k}_{\mathrm B})/\,{{{{\mathrm{ln}}}}}\,[1+{\{Ap+{n}_{{{{{{\rm{LO}}}}}}}({T}_{{{{{{\rm{Room}}}}}}})\}}^{-1}]$$
(1)

with A ≡ τl,LO/(NOS \(\hslash\)ωLO) ≈ 2.11 × 10−10 cm3/W. The blue line in Fig. 4 shows the values of TLO as a function of p according to Eq. (1), which converge to the linear asymptotic form TLO = {(τl,LO /kB)NOS}p in the higher p range.

Fig. 4: Criterion for determining the hot-phonon bottleneck effect.
figure 4

The blue line shows TLO estimated from Eq. (1) against the electrical input power density p. The two black circles mark the experimentally determined effective electron temperatures in the conductor channel under two bias conditions (Fig. 3a); viz., Te = 2000 K at Vb = 8.0 V and Te = 1200 K at Vb = 3.0 V. The white circle marks the experimental value, Te = 2100 K, outside the channel near the channel exit at Vb = 8.0 V. The criterion for the hot-phonon bottleneck regime is given by condition (2), Te < TLO. The thin black dotted line is a guide for eyes speculating possible variation of Te with p in the channel.

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Two black dots in Fig. 4 mark the experimental values of Te, respectively, in the higher- and the lower-bias conditions (Te = 2000 K at Vb = 8.0 V and Te = 1200 K at Vb = 3.0 V in Fig. 3a, b), where Vb = 8.0 V (I = 0.25 mA) and Vb = 3.0 V (I = 0.145 mA) correspond, respectively, to p = Ej ≈ 7.72 × 1010 W/cm3 (E ≈ 40 kV/cm, j ≈ 1.93 × 106 A/cm2) and p ≈ 5.06 × 109 W/cm3 (E ≈ 4.5 kV/cm, j ≈ 1.12 × 106 A/cm2) in the channel (see Supplementary Note 5). It is noted that in the high-bias condition (Vb = 8.0 V) the LO-phonon temperature, TLO = 7,300 K (blue triangle), estimated from Eq. (1) is distinctly higher than Te ≈ 2000 K. Differently, TLO = 780 K (blue triangle) estimated from Eq. (1) in the low-bias condition (Vb = 3.0 V) is lower than Te ≈ 1200 K.

In the high-bias condition, where

$${T}_{{{{{{\rm{e}}}}}}} < {T}_{{{{{{\rm{LO}}}}}}}$$
(2)

holds for TLO given by Eq. (1), the electron system would gain energy from the LO-phonon system through LO-phonon absorption. Since TLO is thereby reduced through the LO-phonon absorption, the value of TLO given by Eq. (1) is not physically realized. Self-consistent value of TLO has to be determined by considering both the emission and the absorption of LO phonons, and is shown to be nearly equal to Te, in fact, slightly lower than Te (see Supplementary Fig. 7 and Supplementary Note 6). If inequality relation (2) holds, therefore, the net energy loss is significantly suppressed with TLO ≈ Te, and the electron system is thereby adiabatically isolated from the lattice. This effect, which we call the hot-phonon bottleneck, accounts for the quasiadiabatic feature of the electron transport experimentally observed in the high-bias condition (Figs. 1a, b and 3a). In the lower-bias condition, where Te > TLO, the energy dissipation via LO-phonon emission is not significantly hindered, resulting in the ordinary hot-electron transport. This is consistent with the observed coincident broad symmetric peaks of Te and TL occurring in the middle of the channel (Fig. 3b).

Detailed profiles of Te and TL in the high-bias condition (Fig. 3a, Vb = 8.0 V) are interpreted below by considering spatial variation of E (or p) around the channel (see Supplementary Fig. 6). The first peak of Te near the channel entrance occurs within the channel where condition (2) holds. Physically, the peak is ascribed to the well-known velocity overshoot (or overheating) of electrons caused by abruptly increasing electric fields near the entrance36,37. Condition (2) is unaffected by the relatively small effect of electron overheating, which is roughly a 10% effect in amplitude. Hence the electron system remains adiabatically isolated from the lattice and the effect of electron overheating (or the first peak in Te) does not cause any discernible signature in the profile of TL. The other prominent peak of Te near the channel exit occurs outside the channel, where the electric field E is distinctly lower such that E ≈ 4.0 kV/cm, j ≈ 1.02 × 106 A/cm2 and p ≈ 4.08 × 109 W/cm3 (see Supplementary Fig. 6 and Supplementary Note 5). Equation (1) predicts TLO ≈ 690 Κ as marked by the white triangle in Fig. 4, while experimentally found Te ≈ 2100 Κ (white circle in Fig. 4, taken from Fig. 3a) is distinctly higher. Since condition (2) breaks down with Te >> TLO, the energy dissipation via LO-phonon emission becomes possible. The excess electron energy stored via the adiabatic passage through the channel is released outside the channel, causing the single peak of TL that is coincident with the second Te peak (Fig. 3a). The physical mechanism of Te taking the largest peak outside the channel is that hot electrons released from the channel outlet drift over the energy relaxation length, reaching the outside region of low electrostatic potential. (This effect was discussed in our previous work as the non-local energy dissipation23.) Unlike Te, the value of TLO drops rapidly outside the channel exit causing Te >> TLO.

The quasiadiabatic hot-electron transport discussed here can be expected to occur in a wide variety of materials because the linear increase in TLO with increasing p like Eq. (1) and the sublinear dependence of Te on p are supposed to be a general trend in high electric field transport in many conductors with appropriate interaction between charge carriers and LO phonons. There have been few reports, however, probably because of the lack of measurements so far. Aside from the electron transport, hot-phonon bottleneck effect has been extensively studied for the photoexcited transient state of III–V ionic crystals24,25 and perovskite compounds26,27,28,29, where the energy loss of photocarriers is found to significantly slow down at high excitation levels. In hot-electron transport phenomena, hot-phonon generation was reported experimentally in standard/exotic semiconductors38,39,40,41,42, but its effect on the hot-electron kinetics has been left unclear. In the transport phenomena, theoretical discussion has been limited to the drift velocity of electrons with major concern about the possible degradation of device performance due to reduction in the electron mobility1,2,43,44. In contrast to the earlier efforts, direct visualization of Te and TL in real space at the nanoscale in this work has disclosed a quasiadiabatic electron transport by clarifying the phenomena from the viewpoint of energy transport.

In graphene nanoconstrictions, asymmetric TL-profiles have been found in the measurements of SThM and interpreted in terms of Peltier effect45,46. In our experiments Peltier effect is ruled out because the local heating and the local cooling would take place, respectively, at the channel entrance and the channel exit, which is opposite to the observation in the present experiment (see Supplementary Fig. 3). It is, nevertheless, interesting to estimate the thermoelectric power due to Peltier effect in our experiment by assuming that the bulk Seebeck coefficient, Sbulk ≈ −100 μV/K, in the wide lead region in our n-GaAs device47 reduces to Schannel = 0 in the constriction channel. Since the heat flow \(\dot{Q}\) = SbulkTI = 7.5 μW at T = 300 K at I = 0.25 mA (Vb = 8.0 V) is blocked at the constriction, the heat power \(\dot{Q}\)Peltier = 7.5 μW is generated or annihilated at the entrance and the exit of the channel. This power is less than one per cent of the electrical input power PJoule = VchannelI = 2 mW (Vb = 8.0 V) in our experiments. Hence the small amplitude of Peltier effect with respect to the E-induced Joule heat power is consistent with the experimental observation. The ratio of the two powers is roughly given by \(\dot{Q}\)Peltier/PJoule ≈ SbulkT/Vchannel ∝ Sbulk/ρchannel with ρchannel the electrical resistivity of the constriction channel. While the amplitude of bulk Seebeck coefficient Sbulk is similar between the two material systems, the distinct difference is the resistivity ρchannel, which is by more than two orders of magnitude higher in the present GaAs constriction than in graphene constriction. In existing studies of nanoscale TL-distribution19,45,46, the TL-profile is often divided into symmetric and antisymmetric parts with respect to the bias current polarity, and the former and the latter are interpreted, respectively, as due to Joule heating effect and Peltier effect. Unlike those existing studies, the present experiments disclosed, by simultaneously measuring Te, that the antisymmetric part of the TL-profile (see Supplementary Fig. 3) is entirely dominated by the E-induced Joule heat effect in the hot-electron condition.

Hot-phonon bottleneck effect in the two-carrier transport

In the high E region exceeding Ec ≈ 10 kV/cm, two-carrier transport is involved because hot electrons in GaAs transfer to upper X valleys, lying ΔεΓX ≈ 550 meV above the Γ-valley (see Supplementary Note 7)48,49. Figure 5 illustrates schematically the kinetics of hot electrons interacting with hot phonons. While fundamental framework of the hot-phonon bottleneck effect is substantially unaffected, the electron–phonon kinetics are elaborated in more detail by explicitly considering the upper-valley transfer of electrons. The effective mass and the density of states of electrons in the X valleys are much larger than those in the Γ valley, so that the Γ → X transfer significantly reduces the electron mobility, introducing sublinear dependence in the current vs. voltage characteristics, as seen for Vb > 5.0 V/cm in the bottom panel in Fig. 1c and for Vb > 3.0 V/cm in the upper right inset in Fig. 2.

Fig. 5: Schematic representation of the kinetics of hot electrons and hot phonons.
figure 5

As electrons enter a narrow constriction channel, they are rapidly accelerated by intense electric field E and the effective electron temperature <Te> is elevated, as studied with SNoiM. It causes frequent LO-phonon emission in the Γ-valley (down-pointing wavy arrow in the left column) as well as Γ → X intervalley electron transfer (fat red arrow in the left column). The effective LO-phonon temperature, TLO, is significantly elevated, but X-valley electrons are not efficiently heated by E, so that the LO phonons emitted by Γ-valley electrons are strongly absorbed by X-valley electrons (wavy arrow pointing upward in the left column), suppressing the net energy loss in the channel and causing the “hot-phonon bottleneck effect”. As the electrons exit the channel, Γ-valley electrons are no longer rapidly accelerated by E, but energy is still fed via intervalley back-transfer from X valleys (fat yellow arrow in the right column). TLO drops faster than TX so that X-valley electrons change to emit LO phonons (wavy arrow pointing downward in the right column). The electrons dissipate net energy, causing prominent non-local energy dissipation near the exit. Emitted LO phonons quickly decay into two LA phonons (via Klemens channel), eventually thermalize slowly into heat, and are primarily composed of long wavelength acoustic phonons that are sensed with SThM as the lattice temperature TL.

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The electron temperature probed with SNoiM in the two-carrier condition is assumed to be the mean electron temperature defined by

$$ < {T}_{\mathrm e} > =\frac{{n}_{\Gamma }}{n}\cdot {T}_{\Gamma }+\frac{{n}_{\mathrm X}}{n}\cdot {T}_{\mathrm X},$$
(3)

where nΓ(x), nX(x), TΓ(x) and TX(x) are the fractional densities and the effective temperatures of the electrons in Γ- and X valleys, respectively. Here, we assume n = nΓ(x) + nX(x) to be a constant equal to the total electron density ignoring minor contribution from the L valleys48. The rate of net energy loss due to LO-phonon scattering is given by PLO(x) = PΓ(x) + PX(x), where

$${P}_{i}= \frac{\Delta \varepsilon }{{\tau }_{\mathrm {LO}}{D}_{0}}{\int }_{0}^{\infty }{\mathrm d}\varepsilon {D}_{i}(\varepsilon ){D}_{i}(\varepsilon +\Delta \varepsilon )\\ \, \times [{f}_{i}(\varepsilon +\Delta \varepsilon )\cdot \{1-{f}_{i}(\varepsilon )\}\cdot ({n}_{{\mathrm {LO}}}+1)-{f}_{i}(\varepsilon )\\ \cdot \{1-{f}_{i}(\varepsilon +\Delta \varepsilon )\}\cdot {n}_{{\mathrm {LO}}}]$$
(4)

for i = Γ or X is a function of x through nΓ(x) or nX(x), TΓ(x), ΤX(x) and TLO(x), and takes account of both the emission and the absorption of LO phonons (Δε = ħωLO) in each set of valleys. Here, \({D}_{i}\left(\varepsilon \right)\) is the density of states in each valley; viz., \({D}_{{\Gamma }}\left(\varepsilon \right)\)= (21/2/π2ħ3)(mΓd)3/2ε1/2, \({D}_{X}\left(\varepsilon \right)\)= (21/2/π2ħ3)(mXd)3/2(ε − ΔεΓX)1/2 for ε > ΔεΓX and \({D}_{X}\left(\varepsilon \right)\)= 0 for ε < ΔεΓX with the respective density-of-state effective masses mΓd = 0.067m0 and mXd = 1.09m0. D0 = 3.24 × 1025(m3eV)−1 is a constant describing the average density of states of hot electrons. The distribution function of electrons in each valley, fi (ε) = 1/[exp{(ε − μi)/kBTi} + 1], is characterized by Ti(x), where the local electrochemical potential μi(x) is determined by ni (x) = \({\int }_{0}^{\infty }f\)i(ε)Di(ε)dε. The LO-phonon occupation number, nLO ≡ {exp(\(\hslash\)ωLO/kBTLO) − 1}−1, is determined by TLO(x).

Values of <Te(x)> and PLO(x) in the device shown in Figs. 1 and 3 at Vb = 8.0 V can be estimated by speculating values of TΓ(x), TX(x), nΓ(x), nX(x) and TLO(x) based on the Monte Caro simulation, where relevant quantities are derived against E48 (see Supplementary Fig. 8 and Supplementary Note 3). For the estimation, we take into account the effect of finite channel length36,37 considering the spatial distribution of E(x) in the device (see Supplementary Fig. 6). We also note that nX, TΓ and TX in the channel take larger values due to hot-phonon bottleneck effect.

In the channel (E ≈ 40 V/cm), the majority of electrons are expected to transfer to X valleys with elevated electron temperatures in respective valleys; viz., nX/n ≈ 0.83 (nΓ/n ≈ 0.17), TΓ ∼ 3250 K (kBTΓ ∼ 280 meV) and TX ∼ 1740 K (kBTX ∼ 150 meV) (see Supplementary Figs. 8a, b). Here, TX is substantially lower than TΓ because the electron mobility in X valleys is much lower. As discussed in the last section, the hot-phonon bottleneck effect makes TLO in the channel close to but slightly lower than Te, which implies, in the two-carrier condition, that TLO is lower than TΓ but slightly higher than TX; viz., TLO ∼ 1750 K (see Supplementary Fig. 8c). The left panel of Fig. 5 schematically depicts the hot-phonon bottleneck effect in the two-carrier condition, where rapidly accelerated Γ-valley electrons frequently emit LO phonons (TΓ > TLO ⇒ PΓ > 0) elevating TLO, while less hot X-valley electrons absorb the emitted LO phonons (TX < TLO ⇒ Px < 0), nearly cancelling the loss of energy (PLO = PΓ + PX ∼ 0). Hence, quasiadiabatic electron transport through the channel is realized by storing the kinetic energy acquired by Γ-valley electrons in the upper X valleys (ΔεΓX ≈ 550 meV).

Detailed structures in the profile of <Te(x)> and PLO(x) arise in connection with nonstationary conditions of the transport caused by the rapidly varying E(x), as discussed in the next paragraph. In Fig. 3c, a solid red line shows that theoretical values of <Te(x)> reproduce well the experimentally observed profile of Te, including the double-peak structure. A dotted blue line in Fig. 3d shows that theoretical values of PLO(x) are suppressed in the channel but take a prominent peak outside the channel close to the exit (see Supplementary Fig. 8d), demonstrating the hot-phonon bottleneck effect. The profile of the lattice temperature TL(x) is broadened due to the lattice thermal conduction32, and is theoretically derived from PLO(x) by assuming a symmetric broadening parameter (see Supplementary Note 9). A solid blue line in Fig. 3d shows that the theoretically derived profile of TL(x) well reproduces the experimentally found single broad peak of TL outside the channel exit.

Detailed electron kinetics causing the profile of <Te(x)> and PLO(x) described in the last paragraph are discussed below (see Supplementary Figs. 8a–d). When Γ-valley electrons approach and enter the channel (from the left-hand side of the device depicted in the upper column of Fig. 5), the increase of the average kinetic energy of electrons is suppressed by the Γ → X transfer. Since the intervalley transfer slightly delays by the intervalley scattering time (roughly 40 fs) compared to the acceleration by E, the suppression of energy delays near the channel entrance where E rapidly increases (see Supplementary Fig. 5), resulting in an overheating/over-population of Γ-valley electrons; that is, values of TΓ and nΓ are slightly larger than the steady-state values expected from the local electric field E(x) near the entrance. This causes a peak of <Te(x)> near the entrance. After adiabatically transmitted through the channel, Γ-valley electrons (nΓ/n ≈ 0.17, TΓ ∼ 3250 K) and X-valley electrons (nX/n ≈ 0.83, TX ∼ 1740 K) are released from the channel exit to the wider lead region (Fig. 5), where hot-phonon bottleneck effect is extinguished with distinctly lower E and j (white arrow in Fig. 4). Hot Γ-valley electrons readily spread to the outside region close to the exit (x ≈ 600 nm, E ≈ 4 kV/cm) within the energy relaxation time (∼1 ps)23. Meantime, X-valley electrons rapidly back-transfer to the Γ-valley (∼40 fs) as schematically illustrated in the right panel of Fig. 5. It follows that hot electrons are efficiently supplied to the Γ-valley from the X valleys (right panel of Fig. 5), whereas Γ-valley electrons no longer rapidly gain energy from E so that TLO falls lower than TX. The rate of LO-phonon emission by Γ-valley electrons is thereby maintained to be high. Due to the back-transfer, X-valley electrons quickly disappear near the channel exit (nX/n → 0 and nΓ/n → 1), making the LO-phonon reabsorption insubstantial and thereby promoting the onset of net LO-phonon emission. The coincident peaks of Te and TL accordingly occur immediately outside the channel exit. Briefly, the hot-phonon bottleneck effect is lifted when the electrons leave the channel, the energy stored in the X valleys for adiabatic transmission is returned to the Γ-valley and dissipated to the lattice.

The suppression ratio of the energy loss rate, γsupp = PLO/P0, defined by the ratio of PLO to the fictitious loss rate P0 ≡ PLO(Tbase) expected in the absence of hot-phonon effect (TLO = Tbase = 300.8 K), is about 1% in the channel: Similarly, the profile of PLO(x) in Fig. 3d suggests that approximately 93% of the energy gained from E is transmitted through the channel without dissipation (see Supplementary Fig. 9 and the discussion in Supplementary Note 8).

This work has experimentally demonstrated a unique approach to access energy transport by probing different effective temperatures of nonequilibrium subsystems, which proved to be powerful for understanding the physics of current-carrying narrow conduction channels. In narrow GaAs constriction channels at high electric fields, conduction electrons generate LO phonons with a high density, while the emitted dense LO phonons prevent efficient cooling of hot electrons, giving rise to quasiadiabatic electron transport over a long distance around 1 μm at room temperature. The knowledge obtained here can serve as a building block for innovative on-chip energy management and energy-harvesting technologies.

Methods

SNoiM and estimation of T e

The instrument is a home-built microscope. The spatial resolution is ∼50 nm, which is primarily determined by the probe tip. The principle and the construction of SNoiM are described in refs. 23,30 (see also Supplementary Note 1). SNoiM exclusively senses evanescent radiation localized on the material surface, but does not sense the familiar THz photon emissions such as those due to the blackbody radiation50, externally induced coherent electron motion51, and the one-particle radiative transition between the initial and the final states52. This is because all those photon emissions do not yield evanescent field on the material surface. Detected with SNoiM is the charge/current fluctuation that generates intense evanescent waves but cancels out in the region away from the surface. In this work it is the hot-electron shot noise, the intensity of which is most simply characterized by the effective electron temperature Te. Absolute values of Te are derived from the signal intensity without using any adjustable parameter (see Supplementary Note 3).

SNoiM is thus far the only instrument that visualizes hot electrons in the steady-state transport condition, whereas in the photoexcited transient condition, hot electrons have been imaged by utilizing plasmonic techniques53.

SThM and estimation of T L

A commercial SThM (ANASYS INSTRUMENTS, NanoTA) is used to map the local lattice temperature distribution. A nanoscale temperature-sensitive resistive element is attached to the apex of an AFM tip, which is scanned across the sample surface in contact mode. The resistance change is measured with a Wheatstone bridge circuit, and the output voltage is referred to as the SThM signal. By scanning the surface of a well-calibrated pt100 planar resistive thermometer self-heated to a known temperature, we establish the SThM transfer characteristic linking its signal to the TL of the sample under study. We note that despite this SThM calibration procedure, the real local lattice temperature of a particular sample may differ from the readings due to a number of mechanisms19, particularly when operated in air, so the absolute values have a significant uncertainty, but the spatial distribution of the temperature is unaffected. The spatial resolution of the equipment is nominally 20 nm, but the realistic resolution is supposed to be ∼50 nm in the present experiment made in the ambient condition (see Supplementary Note 2).

The temperature measured with SThM is the lattice temperature TL because heat flow is dominated by the lattice that has a heat capacity several orders of magnitude larger than that of conduction electrons.

GaAs/AlGaAs QW structure, devices and transport coefficients

The GaAs/AlGaAs heterostructure used in this work is similar to the one described in ref. 23, which was grown with molecular beam epitaxy on the (100) plane. A quasi 2D electron gas (2DEG) layer with a density n2D = 1.16 × 1013 cm−2 or n = 3.3 × 1018 cm−3 (corresponding to the Fermi energy EF = 119 meV at absolute zero temperature T = 0 K) and Hall mobility μ = 0.167 m2/Vs is provided in a W = 35-nm-thick GaAs QW located 13 nm below the surface. The devices studied are fabricated with standard electron beam lithography and wet mesa etching with a depth ∼100 nm. The constriction channel is connected to the source and drain contacts through 2DEG leads with a typical width of ∼20 μm and a total length of ∼190 μm. The effective voltage applied to the short constriction channel, Vchannel = Vb−Vleads, is less than the bias voltage Vb by the voltage drop along the leads Vleads, which depends on the device-specific accurate dimensions of the leads. The electric field E in each device is evaluated by considering the known device-specific lead geometry (see Supplementary Note 5). Ohmic contacts of the source and the drain are prepared by alloying with AuGeNi. The drift velocity of electrons is experimentally estimated to be vd = I/(Wn2De) ≈ 3.5 × 104 ms−1 from I = 0.25 mA in the device with Vb = 8.0 V shown in Figs. 1 and 3. Hence, the distance the hot electrons drift in the high electric field during an event of LO-phonon scattering, Ld,LO = vdτLO ≈ 4.2 nm with τLO = 0.12 ps33, is far smaller than the length scales of the channel.