Measurements of Kondo and Spin Splitting in Single-Electron Transistors

We measure the spin splitting in a magnetic field $B$ of localized states in single-electron transistors using a new method, inelastic spin-flip cotunneling. Because it involves only internal excitations, this technique gives the most precise value of the Zeeman energy $\Delta =|g|{\mu }_{B}B$. In the same devices we also measure the splitting with $B$ of the Kondo peak in differential conductance. The Kondo splitting appears only above a threshold field as predicted by theory. However, the magnitude of the Kondo splitting at high fields exceeds $2|g|{\mu }_{B}B$ in disagreement with theory.

Figure 1

(a) Differential conductance as a function of $V_{\mathrm{d}\mathrm{s}}$ and $V_{\mathrm{g}}$ at $B=7.25\mathrm{T}$ for SET 2. The odd and even Coulomb valleys are labeled. (b) Differential conductance as a function of drain-source voltage at the gate voltage marked by the dashed line in (a). The dots denote data while the solid line shows a fit to the sum of two ${cosh}^{-2}$ functions. (c) Zeeman splitting as a function of magnetic field determined by fitting curves like that shown in (b). The solid line shows a fit to the data to determine $|g|$. The inset shows an electron micrograph of an SET similar to those we studied.

Figure 2

Cotunneling processes through a Zeeman split orbital, which can occur when the dot is in the Coulomb-blockade regime with an odd number of electrons as in (a). We take spin up to be the lower energy state. A spin-down electron from the lead can tunnel onto the dot forming the virtual intermediate state shown in (b). For any value of $V_{\mathrm{d}\mathrm{s}}$, the spin-down electron can tunnel off the dot as shown in (c), resulting in elastic cotunneling. If $|e{V}_{\mathrm{d}\mathrm{s}}|\ge \Delta =|g|{\mu }_{B}B$, the spin-up electron can also tunnel off the dot resulting in inelastic cotunneling as shown in (d).

Figure 3

(a) Differential conductance as a function of $V_{\mathrm{d}\mathrm{s}}$ and $V_{\mathrm{g}}$ at $B=8\mathrm{T}$ for SET 1. The inelastic spin-flip cotunneling gap is in the middle of the center diamond and is marked by arrows. (b) Differential conductance as a function of drain-source voltage in the middle of the Coulomb valley for the same sample as in (a), but with different voltages on the electrodes and $B=11\mathrm{T}$. The dots are the data and the solid line is a fit described in the text. (c) Temperatures extracted from fitting the cotunneling gap of SET 2 at $B=7.65\mathrm{T}$. The horizontal axis is the mixing chamber temperature $T_{\mathrm{M}\mathrm{C}}$ and the vertical axis is the temperature $T_{\mathrm{F}\mathrm{i}\mathrm{t}}$ extracted from fitting the gap as discussed in the text. The line shows $T_{\mathrm{M}\mathrm{C}}=T_{\mathrm{F}\mathrm{i}\mathrm{t}}$.

Figure 4

(a) Evolution of a cotunneling gap (dots) and a Kondo peak (solid line) with increasing magnetic field in SET 2. The vertical scale on the left (right) is for the cotunneling data (Kondo data). For the cotunneling data, the coupling to the leads was reduced so that no Kondo peak was seen at zero field at the measurement temperature $T\approx 110\mathrm{m}\mathrm{K}$. The solid lines through the cotunneling data at high fields are examples of fits to Eq. (1). The dashed lines at the three highest fields mark the center of the cotunneling edge; the lines are $2\Delta$ apart. (b) Splittings from cotunneling gaps (open circles) and Kondo peaks (solid squares) from SET 2. The solid line through the cotunneling data is a linear fit. The line through the Kondo data is the result of fixing the slope at the value obtained from the cotunneling data and fitting the data above 3 T to extract the intercept. (c) Evolution of a Kondo peak into a cotunneling gap with increasing magnetic field in SET 1. The solid line in the 5 T trace shows a fit as described in the text. (d) Splittings from the cotunneling gaps and Kondo peaks for SET 1. The Kondo splitting is determined at low field and the cotunneling gap at high field, but all other parameters are unchanged. We fit the data with lines the same way as for (b).