We study in experiment and theory thermal energy and charge transfer close to the quantum limit in a ballistic nanodevice, consisting of multiply connected one-dimensional electron waveguides. The fabricated device is based on an AlGaAs/GaAs heterostructure and is covered by a global top-gate to steer the thermal energy and charge transfer in the presence of a temperature gradient, which is established by a heating current. The estimate of the heat transfer by means of thermal noise measurements shows the device acting as a switch for charge and thermal energy transfer. The wave-packet simulations are based on the multi-terminal Landauer-Büttiker approach and confirm the experimental finding of a mode-dependent redistribution of the thermal energy current, if a scatterer breaks the device symmetry.

Quantized conductance steps of an electric current through an electron waveguide with a constriction are a hallmark of ballistic transport1–3 and have been successfully modelled within the Landauer-Büttiker approach.4–6 In the ballistic transport theory, the scattering and transmission coefficients of the geometry regulate the flux through the system of non-interacting electrons. Injected electrons inherit the properties of the reservoirs from which they come from and no dephasing or dissipation occurs within the ballistic system. Ballistic, non-dephasing, electrons are essential for interferometric devices which probe the phase differences along different pathways.7–9 

Here, we investigate to what extent the ballistic description reproduces experimental data in multi-terminal devices, where the branching ratio of the thermal energy current is varied by applying a top-gate voltage.10 In previous quasi-one-dimensional transport studies, thermal conductance in multiples of the theoretical quantum limit11  κ = k B 2 π 3 ħ T has been observed.12,13 and linked to the electrical conductance by a Wiedemann-Franz relation.14 A suitable layout for studying ballistic transport in a multi-terminal device is shown in Fig. 1(a) and is designed such that the two-dimensional (2D) electron reservoirs are separated from each other by a network of one-dimensional (1D) electron waveguides. The design and operation of the device is discussed in Ref.10.

FIG. 1.
FIG. 1. Device geometry. a) Scanning electron micrograph of an identically processed sample. The 1D waveguides with a lithographic width of 170 nm form a half-ring connected to reservoirs A-F. A global top-gate is present. Heating of reservoirs A, B is generated by applying a current Ih, thermal noise measurements are performed at contacts E, F. The reservoirs C and D are left floating. b) Device potential for the ballistic transport model with labels A∗ and E∗ denoting the joined reservoirs A+B and E+F. Harmonic waveguide network with Gaussian scatterer Vsc(x, y) = V0 e−[(x−ox)2+(y−oy)2]/a2, V0 = 50 meV, a = 80 nm, ox = 590 nm, oy = − 90 nm (indicated by arrow). Mode spacing is ħ ω = 5 meV.
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Device geometry. a) Scanning electron micrograph of an identically processed sample. The 1D waveguides with a lithographic width of 170 nm form a half-ring connected to reservoirs A-F. A global top-gate is present. Heating of reservoirs A, B is generated by applying a current Ih, thermal noise measurements are performed at contacts E, F. The reservoirs C and D are left floating. b) Device potential for the ballistic transport model with labels A∗ and E∗ denoting the joined reservoirs A+B and E+F. Harmonic waveguide network with Gaussian scatterer Vsc(x, y) = V0 e−[(x−ox)2+(y−oy)2]/a2, V0 = 50 meV, a = 80 nm, ox = 590 nm, oy = − 90 nm (indicated by arrow). Mode spacing is ħ ω = 5 meV.

FIG. 1.
FIG. 1. Device geometry. a) Scanning electron micrograph of an identically processed sample. The 1D waveguides with a lithographic width of 170 nm form a half-ring connected to reservoirs A-F. A global top-gate is present. Heating of reservoirs A, B is generated by applying a current Ih, thermal noise measurements are performed at contacts E, F. The reservoirs C and D are left floating. b) Device potential for the ballistic transport model with labels A∗ and E∗ denoting the joined reservoirs A+B and E+F. Harmonic waveguide network with Gaussian scatterer Vsc(x, y) = V0 e−[(x−ox)2+(y−oy)2]/a2, V0 = 50 meV, a = 80 nm, ox = 590 nm, oy = − 90 nm (indicated by arrow). Mode spacing is ħ ω = 5 meV.
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Device geometry. a) Scanning electron micrograph of an identically processed sample. The 1D waveguides with a lithographic width of 170 nm form a half-ring connected to reservoirs A-F. A global top-gate is present. Heating of reservoirs A, B is generated by applying a current Ih, thermal noise measurements are performed at contacts E, F. The reservoirs C and D are left floating. b) Device potential for the ballistic transport model with labels A∗ and E∗ denoting the joined reservoirs A+B and E+F. Harmonic waveguide network with Gaussian scatterer Vsc(x, y) = V0 e−[(x−ox)2+(y−oy)2]/a2, V0 = 50 meV, a = 80 nm, ox = 590 nm, oy = − 90 nm (indicated by arrow). Mode spacing is ħ ω = 5 meV.

Close modal

If a heating current Ih is sent through reservoirs A and B, electrons with elevated temperature are injected into the waveguides and transmitted to terminals C to F or backscattered into AB. The amount of injected thermal energy current depends on the number of open modes in the waveguides, controlled by global top-gate voltage Vg. At terminals E and F the electron temperature is determined from the frequency-independent part of the power spectral density of the thermal noise.10,15 The phase coherence of interfering electrons in a similar device has been characterized before in theory and experiment16,17 and thermal transport data measured in the same device have been reported.10,15

The presence of junctions and crossings in a multi-terminal device requires to calculate the electric and thermal transport for the specific sample geometry across a range of Fermi energies. This task is facilitated by switching to a time-dependent intermediate representation of the quantum-mechanical transport, using wave packets which are decomposed into plane wave components pointing to the different terminals.18,19 The theoretical model elucidates the origin of the observed mode-dependent branching ratio of the device10 by connecting it to specific device imperfections.

For an unbent 1D waveguide analytic expressions are available for the thermal energy current.11,14,20,21 The thermal conductance κ is expressed as the thermal energy current Q divided by the temperature difference of the reservoirs ΔT = (T2 − T1),

κ = − Q Δ T I = 0 ,
(1)

evaluated at zero electric current I. For the 1D waveguide and at low temperatures the Sommerfeld expansion yields κ in terms of the Lorenz number L 0 = k B 2 π 2 / ( 3 e 2 ) and the electric conductance G14 

κ = L 0 T G .
(2)

For an unbent waveguide with unit transmission t ( E ) = 1 at all energies E the electric conductance becomes G = 2je2/h, where j denotes the number of transverse modes below the Fermi energy and 2 accounts for spin degeneracy. Then, after introducing the average temperature T = (T1 + T2)/2, the maximum thermal energy current transported by one mode is given by

Q max = π 2 k B 2 3 h ( T 2 2 − T 1 2 ) .
(3)

For the bent waveguide (Fig. 1) the transmission t ( E ) is reduced by backscattering and the thermal energy current is limited to values Q < Qmax. Only for a single waveguide Eq. (2) implies a linear relation between thermal and electric conductance. In a multi-terminal setup this relation becomes more involved and is described by the Landauer-Büttiker formalism, where the thermal energy current Qi in lead i is given by

Q i = 2 h ∫ 0 ∞ d E E ∑ j ≠ i , n i , n j [ f ( E , μ i , T i ) − f ( E , μ j , T j ) ] | t i , n i , j , n j ( E ) | 2
(4)

and related to the scattering matrix elements ti,ni,j,nj, with lead modes ni and nj. The Fermi functions f ( E , μ i , T i ) = ( e ( E − μ i ) / k B T i + 1 ) − 1 characterize the temperatures Ti and chemical potentials μi of the macroscopic reservoir connected to lead i. For the electric current, a similar equation holds

I i = 2 e h ∫ 0 ∞ d E ∑ j ≠ i , n i , n j [ f ( E , μ i , T i ) − f ( E , μ j , T j ) ] | t i , n i , j , n j ( E ) | 2 .
(5)

For the theoretical model, we treat the heated contacts A, B as one common reservoir A∗ at temperature TA∗. Similarly we combine leads E, F into the reservoir E∗, see Fig. 1(b). The device is probed over a large range of gate-voltages to vary the population of waveguide modes. This requires to calculate the transmission t ( E ) for a range of energies E ⁠. In addition, we solve Eqs. (4) and (5) at finite, non-zero voltage and temperature differences. We use a wave-packet approach well-suited for the simultaneous evaluation of the transmission across the complete transport window.17–19 This approach has been used before to accurately describe Aharonov-Bohm oscillations in a similar device probed over several transverse mode openings and magnetic fields.17 The channels of the interferometer are modelled with a harmonic confinement.5,17 The harmonic confinement results in an equidistant spacing of mode-openings as a function of Fermi energy. For a linear relationship between Fermi energy and applied gate-voltage, the equidistant spacing is in line with the experimental data probing the mode-openings as a function of applied gate-voltage, Fig. 2. We do not include electron-electron interactions explicitly in the calculation and the harmonic potential is viewed as an effective potential including screening effects.

FIG. 2.
FIG. 2. Measured electrical 1D conductances GAF, GAC, GAD and GCD at 4.2 K, reproduced from Ref. 10. The indices refer to the 2D electron reservoirs A, C, D and F that are connected by the corresponding 1D waveguide. Conductance measurements between reservoirs B and E yield the same results as the measurements between reservoirs A and F and are therefore not shown.
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Measured electrical 1D conductances GAF, GAC, GAD and GCD at 4.2 K, reproduced from Ref. 10. The indices refer to the 2D electron reservoirs A, C, D and F that are connected by the corresponding 1D waveguide. Conductance measurements between reservoirs B and E yield the same results as the measurements between reservoirs A and F and are therefore not shown.

FIG. 2.
FIG. 2. Measured electrical 1D conductances GAF, GAC, GAD and GCD at 4.2 K, reproduced from Ref. 10. The indices refer to the 2D electron reservoirs A, C, D and F that are connected by the corresponding 1D waveguide. Conductance measurements between reservoirs B and E yield the same results as the measurements between reservoirs A and F and are therefore not shown.
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Measured electrical 1D conductances GAF, GAC, GAD and GCD at 4.2 K, reproduced from Ref. 10. The indices refer to the 2D electron reservoirs A, C, D and F that are connected by the corresponding 1D waveguide. Conductance measurements between reservoirs B and E yield the same results as the measurements between reservoirs A and F and are therefore not shown.

Close modal

The electric conductance of the device is experimentally recorded by two-point AC measurements, with the other leads kept floating. The measured conductances G are shown in Fig. 2 for four different probe configurations. The conductance GAF has a smaller threshold-voltage compared to the conductances GAD, GAC, and GCD, which points to an asymmetry in the device geometry. Further inspection of the transmission curves pins the location of the asymmetry down to the junction connecting reservoirs C and D, since only there it is possible to affect all remaining conductances in a similar fashion. The theoretical model thus includes a Gaussian obstacle located at the corner marked by the arrow in Fig. 1(b).

Fig. 3(b) shows that the computed electrical conductances closely match the experimental data in Fig. 2. This is not the case for a symmetric device model with identical junctions, shown for reference in Fig. 3(a).

FIG. 3.
FIG. 3. Calculated electric conductances GA∗E∗, GCD, GA∗D and GA∗C. a) Symmetric device model, b) with Gaussian obstacle. The indices refer to the conductances between reservoirs A∗, C, D, and E∗.
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Calculated electric conductances GA∗E∗, GCD, GA∗D and GA∗C. a) Symmetric device model, b) with Gaussian obstacle. The indices refer to the conductances between reservoirs A∗, C, D, and E∗.

FIG. 3.
FIG. 3. Calculated electric conductances GA∗E∗, GCD, GA∗D and GA∗C. a) Symmetric device model, b) with Gaussian obstacle. The indices refer to the conductances between reservoirs A∗, C, D, and E∗.
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Calculated electric conductances GA∗E∗, GCD, GA∗D and GA∗C. a) Symmetric device model, b) with Gaussian obstacle. The indices refer to the conductances between reservoirs A∗, C, D, and E∗.

Close modal

Having reproduced the electric conductance by the multi-terminal Landauer-Büttiker approach, we investigate next the propagation of ballistic thermal energy currents through the device. The basic assumption underlying the Landauer-Büttiker model is that the injected electrons inherit all thermal properties from the respective reservoirs.5 Neglecting any temperature equilibration and heat transfer within the waveguides and junctions, the geometry of the system completely determines the transmission coefficients and thus the distribution of the thermal energy current, Eq. (4). In the experiment the temperature of the 2D electron reservoir AB is increased relative to the lattice temperature by the heating current Ih according to Joule’s law Δ T e AB ∝ I h 2 ⁠. The increase of the electron temperature in reservoir EF, Δ T e EF ( I h ) ⁠, is determined from the thermal noise.10,15 The measurements are performed at gate-voltages supporting n open modes in the 1D waveguide connecting the reservoirs AB and EF, see Fig. 4(a). Without any open mode (n = 0), no increase in temperature of reservoirs EF is detected within the measurement uncertainty (square symbols, Fig. 4(a)). Therefore heat transfer mediated by electron-phonon coupling through the lattice is negligible.

FIG. 4.
FIG. 4. a) Measured electron temperature increase Δ T e EF as a function of the square of the heating current I h 2 applied to the reservoir AB, reproduced from Ref. 10. The measurement is repeated for different numbers of populated subbands n = 0, 1, 2, 3 of the 1D-waveguide that connects the reservoirs AB and EF. b) Slopes of the linear dependence Δ T e EF ( I h 2 ) in a) yield the thermal conductance as a function of n.
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a) Measured electron temperature increase Δ T e EF as a function of the square of the heating current I h 2 applied to the reservoir AB, reproduced from Ref. 10. The measurement is repeated for different numbers of populated subbands n = 0, 1, 2, 3 of the 1D-waveguide that connects the reservoirs AB and EF. b) Slopes of the linear dependence Δ T e EF ( I h 2 ) in a) yield the thermal conductance as a function of n.

FIG. 4.
FIG. 4. a) Measured electron temperature increase Δ T e EF as a function of the square of the heating current I h 2 applied to the reservoir AB, reproduced from Ref. 10. The measurement is repeated for different numbers of populated subbands n = 0, 1, 2, 3 of the 1D-waveguide that connects the reservoirs AB and EF. b) Slopes of the linear dependence Δ T e EF ( I h 2 ) in a) yield the thermal conductance as a function of n.
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a) Measured electron temperature increase Δ T e EF as a function of the square of the heating current I h 2 applied to the reservoir AB, reproduced from Ref. 10. The measurement is repeated for different numbers of populated subbands n = 0, 1, 2, 3 of the 1D-waveguide that connects the reservoirs AB and EF. b) Slopes of the linear dependence Δ T e EF ( I h 2 ) in a) yield the thermal conductance as a function of n.

Close modal

For a 1D waveguide, the Wiedemann-Franz relation (2) predicts a proportionality between the electric and thermal conductance. In a multi-terminal setup, Eq. (4) has to be solved for specific boundary conditions. The heated reservoir A∗ is at an elevated temperature with respect to all other reservoirs at temperature Tbase. In addition, we enforce the zero-current condition in each lead. This corresponds to Eq. (2) and the experimental conditions. First, we calculate the thermal energy current QA∗E∗ from reservoir A∗ to E∗ as a function of the temperature difference TA∗ − Tbase. To compare the calculated thermal energy current with the experimental results Δ T e EF ( I h 2 ) we divide QA∗E∗ by the heat conductance that is carried by one energy mode κmax = L0(TA∗ + Tbase) e2/h following Eqs. (1),(3). If we trace this relationship at the Fermi energy corresponding to the center of the nth conductance plateau of GA∗E∗ (⁠ E F = n ħ ω ⁠), we obtain a linear relation between the thermal energy current and the increase in temperature for a given n, Fig. 5(a).

FIG. 5.
FIG. 5. a) Calculated thermal current from A∗ to E∗ divided by κmax = L0(TA∗ + Tbase) e2/h. b) Slopes of curves in a) (squares) yield the thermal conductance as a function of open modes n for the asymmetric device. For the symmetric model the thermal conductance has a delayed onset and different relation to the number of open modes (circles).
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a) Calculated thermal current from A∗ to E∗ divided by κmax = L0(TA∗ + Tbase) e2/h. b) Slopes of curves in a) (squares) yield the thermal conductance as a function of open modes n for the asymmetric device. For the symmetric model the thermal conductance has a delayed onset and different relation to the number of open modes (circles).

FIG. 5.
FIG. 5. a) Calculated thermal current from A∗ to E∗ divided by κmax = L0(TA∗ + Tbase) e2/h. b) Slopes of curves in a) (squares) yield the thermal conductance as a function of open modes n for the asymmetric device. For the symmetric model the thermal conductance has a delayed onset and different relation to the number of open modes (circles).
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a) Calculated thermal current from A∗ to E∗ divided by κmax = L0(TA∗ + Tbase) e2/h. b) Slopes of curves in a) (squares) yield the thermal conductance as a function of open modes n for the asymmetric device. For the symmetric model the thermal conductance has a delayed onset and different relation to the number of open modes (circles).

Close modal

This linear relationship is an indication of a Wiedemann-Franz relation valid for n open modes, with the thermal conductance given by the slope

κ ( n ) ≡ κ ( E F = n ħ ω ) = Δ Q A ∗ E ∗ ( n ) / ( T A ∗ − T base ) .
(6)

Only for a perfectly transmitting 1D waveguide κ is proportional to the number of open modes. For the multi-terminal device investigated here, a non-linear relation between κ(n) and the number of open modes n in the path from A∗ to E∗ results. For the symmetric device model, both the electric conductance (Fig. 3) and the thermal current (Fig. 5(b), circles) differ from the measured quantities. Only the placement of the Gaussian scatterer in the corner between terminals ED causes a delayed mode-opening for the electric conductance involving these terminals and leads to a similar non-linear relationship of the measured and calculated thermal currents (squares in Figs. 4(b), 5(b)).

We have demonstrated that the ballistic transport theory reproduces the measured electric and thermal properties of a multi-terminal nanodevice. The calculation takes into account the waveguide profile and the precise scattering properties of the junctions.

The non-linear relationship between number of open modes (controlled by the gate-voltage) and the amount of thermal energy transmitted from reservoirs AB to EF demonstrate a mode-influenced energy distribution. Thus, the device can be understood as a splitter for both, charge and heat flow, steered by the global top-gate. This mode-dependent behaviour can be explained by a device asymmetry located at the crossing CD and is verified by matching both, electric and thermal conductances, to the experimental data. The theoretical Landauer-Büttiker descriptions neglects inelastic scattering within the device. Inelastic scattering reduces the visibility of Aharonov-Bohm interference effects upon placing the device in a magnetic field.17 This points to considerably different dephasing and relaxation length-scales within the device considered here. To incorporate inelastic effects would require to move beyond the Landauer-Büttiker ballistic theory and to consider a bath of interacting electrons and phonons.22 

We gratefully acknowledge financial support by the priority programme ‘Nanostructured thermoelectrics’ of the German Science Foundation (DFG) SPP 1386 grant Fi932/2-2. TK was supported by the Heisenberg programme of the DFG, grant Kr2889/5. A. D. Wieck acknowledges gratefully support of BMBF-Q.com-H 16KIS0109. We further thank Dr. Rüdiger Mitdank for fruitful discussions.

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© 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
2016
Author(s)