Magnetic field dependence of the thermopower of Kondo-correlated quantum dots: Comparison with experiment

Signatures of the Kondo effect in the electrical conductance of strongly correlated quantum dots are well understood both experimentally and theoretically, while those in the thermopower have been the subject of recent interest, both theoretically and experimentally. Here, we extend theoretical work [T. A. Costi, Phys. Rev. B 100, 161106(R) (2019)] on the field-dependent thermopower of such systems to the mixed valence and empty orbital regimes, and carry out calculations in order to address a recent experiment on the field-dependent thermoelectric response of Kondo-correlated quantum dots [A. Svilans et al., Phys. Rev. Lett. 121, 206801 (2018)]. In addition to the sign changes in the thermopower at temperatures $T_1\left(B\right)$ and $T_2\left(B\right)$ (present also for $B=0$) in the Kondo regime, an additional sign change was found [T. A. Costi, Phys. Rev. B 100, 161106(R) (2019)] at a temperature $T_0\left(B\right)<T_1\left(B\right)<T_2\left(B\right)$ for fields exceeding a gate-voltage-dependent value $B_0$, where $B_0$ is comparable to, but larger than, the field $B_c$ at which the Kondo resonance splits. We describe the evolution of the Kondo-induced sign changes in the thermopower at temperatures $T_0\left(B\right),T_1\left(B\right)$, and $T_2\left(B\right)$ with magnetic field and gate voltage from the Kondo regime to the mixed valence and empty orbital regimes and show that these temperatures merge to the single temperature $T_0\left(B\right)$ upon entry into the mixed valence regime. By carrying out detailed numerical renormalization group calculations for the above quantities, using appropriate experimental parameters, we address a recent experiment which measures the field-dependent thermoelectric response of InAs quantum dots exhibiting the Kondo effect [A. Svilans et al., Phys. Rev. Lett. 121, 206801 (2018)]. This allows us to understand the overall trends in the measured field- and temperature-dependent thermoelectric response as a function of gate voltage. In addition, we determine which signatures of the Kondo effect [sign changes at $T_0\left(B\right),T_1\left(B\right)$, and $T_2\left(B\right)$] have been observed in this experiment, and find that while the Kondo-induced signature at $T_1\left(B\right)$ is indeed measured in the data, the signature at $T_0\left(B\right)$ can only be observed by carrying out further measurements at a lower temperature. In addition, the less interesting (high-temperature) signature at $T_2\left(B\right)\gtrsim \mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is the electron tunneling rate onto the dot, is found to lie above the highest temperature in the experiment, and was therefore not accessed. Our calculations provide a useful framework for interpreting future experiments on direct measurements of the thermopower of Kondo-correlated quantum dots in the presence of finite magnetic fields, e.g., by extending zero-field measurements of the thermopower [B. Dutta et al., Nano Lett. 19, 506 (2019)] to finite magnetic fields.

Figure 1

(a) Thermopower $S$ (in units of $k_{\mathrm{B}}/e=86.17\mu \mathrm{V}/\mathrm{K}$) vs temperature $T/\mathrm{\Gamma }$, and (b) $T=0$ total spectral function $A(\omega ,T=0)$ for a quantum dot with $U/\mathrm{\Gamma }=3.2$, as in the experiment [25], and for increasing values of the magnetic field $B/T_{\mathrm{K}}$ in the Kondo regime $[{\varepsilon }_0=-3\mathrm{\Gamma }({\mathrm{v}}_g=-0.1),T_{\mathrm{K}}/\mathrm{\Gamma }=9.192\times {10}^{-2}]$. For $B<B_0\approx 1.04T_{\mathrm{K}}$ (blue solid lines), two sign changes are found in $S\left(T\right)$ at $T_1\left(B\right)$ and $T_2\left(B\right)$, whereas for $B_0<B<B_1$ (green solid lines) an additional sign change occurs at a temperature $T_0\left(B\right)$ and, for $B>B_1$ (red solid lines), only the sign change at $T_2\left(B\right)$ is present in $S\left(T\right)$. In (b), ${\varepsilon }_0$ and ${\varepsilon }_0+U$ denote the bare Hubbard satellite excitations of the dot. NRG parameters: discretization parameter $\mathrm{\Lambda }=4$, $z$ averaging [64, 65] with $N_z=4$, retaining $N_{\mathrm{states}}=900$ states.

Figure 2

Thermopower $S$ (in units of $k_{\mathrm{B}}/e=86.17\mu \mathrm{V}/\mathrm{K}$) vs temperature $T/\mathrm{\Gamma }$ of a strongly correlated quantum dot ($U=3.2\mathrm{\Gamma }$) for increasing values of the magnetic field $B/T_{\mathrm{K}}$ and for local level position ${\varepsilon }_0=-0.4\mathrm{\Gamma }$ in the mixed valence regime (${\mathrm{v}}_g=-1.2$). Spin susceptibility $T_{\mathrm{K}}\approx 0.345\mathrm{\Gamma }$ is close to the mixed valence low-energy scale $\mathrm{\Delta }=0.5\mathrm{\Gamma }$. For $B<B_0\approx T_{\mathrm{K}}$, the thermopower exhibits no sign change as a function of $T$ (blue solid lines), while for $B\gtrsim B_0$ a single sign change at $T_0\left(B\right)$ occurs. NRG parameters as in Fig. 1.

Figure 3

$T_0\left(B\right)$ (solid lines), $T_1\left(B\right)$ (dotted lines), and $T_2\left(B\right)$ (dashed lines) in units of $\mathrm{\Gamma }$ vs $B/\mathrm{\Gamma }$ for different ${\varepsilon }_0/\mathrm{\Gamma }$ (${\mathrm{v}}_g$). Outside the Kondo regime ($|{\mathrm{v}}_g|\gtrsim 1.0$) only the sign change in $S\left(T\right)$ at $T=T_0\left(B\right)$ exists. $T_d$ (dotted-dashed line) is the highest temperature ($4.0\mathrm{K}$) in the experiment [25] and $T_{\mathrm{K}1}$ indicates the midvalley (${\mathrm{v}}_g=0$) Kondo scale. In detail, the local level positions ${\varepsilon }_0/\mathrm{\Gamma }$ are as follows. Mixed valence regime (green lines): $-0.4,-0.5,-0.55$, and, $-0.575$. Weak Kondo regime (orange lines): $-0.6,-0.625,-0.65,-0.675,-0.6875$, and, $-0.69375$. Kondo regime (red lines): $-0.7,-0.8,-0.9,-1.0,-1.1,-1.2,-1.3,-1.4$, and, $-1.5$. NRG parameters as in Fig. 1.

Figure 4

$S$ vs ${\mathrm{v}}_g$ at four temperatures $T_a<T_b<T_c<T_d$ for (a) $B<B_0$, (b) $B_0<B<B_1$, and (c) $B>B_1$, showing the possible sign changes (labeled ${\sigma }_0,{\sigma }_1$, and ${\sigma }_2$) of $S$ in the Kondo regime upon increasing temperature through $T_0,T_1$, and $T_2$ (for given fixed ${\mathrm{v}}_g$ in the Kondo regime). $U=8\mathrm{\Gamma }$ and $B_0({\mathrm{v}}_g\to 0)\approx 0.004\mathrm{\Gamma }$, $B_1({\mathrm{v}}_g\to 0)\approx 0.05\mathrm{\Gamma }$. Specifically, the sign changes involved are (a) ${\sigma }_1:T_{a,b}<T_1\left(B\right)<T_c$ and ${\sigma }_2:T_c<T_2\left(B\right)<T_d$, (b) as in (a) and the additional sign change ${\sigma }_0:T_a<T_0\left(B\right)<T_b$, (c) only the sign change ${\sigma }_2:T_c<T_2\left(B\right)<T_d$. Temperatures $T_{a,b,c,d}/\mathrm{\Gamma }=0.0038,0.0084,2.222$, and 3.62 chosen relative to midvalley estimates $T_1/\mathrm{\Gamma }=0.00426$ and $T_2/\mathrm{\Gamma }=2.69$ to observe the above sign changes. Vertical dashed lines delineate Kondo ($|{\mathrm{v}}_g|\lesssim 3.5$) from mixed valence and empty (full) orbital regimes at larger $|{\mathrm{v}}_g|$. NRG parameters as in Fig. 1.

Figure 5

$T_1$ and $T_2$ vs ${\mathrm{v}}_g$ in the Kondo regime for $U/\mathrm{\Gamma }=3.2$ and $B=0$. Also indicated is $T_1/T_{\mathrm{K}1}$ vs ${\mathrm{v}}_g$ and $T_{\mathrm{K}1}$ vs ${\mathrm{v}}_g$, where $T_{\mathrm{K}1}$ is the perturbative Kondo scale (11) as used in the experiment [25]. The dotted-dashed horizontal lines indicate the four experimental temperatures $T_{a,b,c,d}/\mathrm{\Gamma }=0.0725,0.123,0.181$, and 0.290. NRG parameters as in Fig. 1.

Figure 6

Left panels (a)–(d): calculated $GS$ vs ${\mathrm{v}}_g$ at the temperatures $T$ and the four magnetic fields $B$ of the experiment [25]. Vertical dashed lines delineate Kondo ($|{\mathrm{v}}_g|\lesssim 1.0$) from mixed valence and empty (full) orbital regimes at larger $|{\mathrm{v}}_g|$. Right panels (a)–(d): experimentally measured thermocurrent $I_{\mathrm{th}}/\mathrm{\Delta }T$ vs gate voltage $V_G$ at four magnetic fields and temperatures for device QD1a (adapted, with permission, from Ref. [25]). NRG parameters as in Fig. 1.

Figure 7

$B_0$ and $B_1$ (in units of $\mathrm{\Gamma }$) vs ${\mathrm{v}}_g$ for $U/\mathrm{\Gamma }=3.2$. Also indicated (in units of $\mathrm{\Gamma }$) are the minimum values of $B_0$ ($B_0^{\min }$) and $B_1$ ($B_1^{\min }$) at ${\mathrm{v}}_g=0$ (dashed horizontal lines) and the three lowest experimental fields $B_{a,b,c}=0.0\text{T},0.5\text{T}$, and $1.0\text{T}$ in Ref. [25] (solid horizontal lines). The highest experimental field $B_d=2.0\text{T}$ lies much above $B_0$ and $B_1$ for the indicated range of ${\mathrm{v}}_g$ and is not shown. NRG parameters as in Fig. 1.

Figure 8

$T_0\left(B\right)$ (solid lines) and $T_1\left(B\right)$ (dotted lines) in units of $\mathrm{\Gamma }$ vs $B/\mathrm{\Gamma }$ for different ${\mathrm{v}}_g$ [$T_2\left(B\right)>T_d$ and is not shown, being off the scale of the plot (except for ${\mathrm{v}}_g=-1.0$)]. Outside the Kondo regime (${\mathrm{v}}_g\gtrsim 1.0$) only the sign change in the thermopower at $T_0\left(B\right)$ exists. The horizontal dashed lines correspond to the four experimental temperatures $T_{a,b,c,d}=1.0\text{K},1.7\text{K},2.5\text{K}$, and $4.0\text{K}$, and the vertical dashed lines correspond to the experimental fields $B_{a,b,c,d}=0.0\text{T},0.5\text{T},1.0\text{T}$ in Ref. [25] (the highest field $B=2.0\text{T}$ is outside the scale of the plot). The midvalley Kondo scale $T_{\mathrm{K}1}$ (horizontal arrow) lies very close to the lowest experimental temperature $T_a$. NRG parameters as in Fig. 1.

Figure 9

Detailed view of Fig. 6 for the calculated $GS$ vs ${\mathrm{v}}_g$ for ${\mathrm{v}}_g<0$ at $B=0.5\text{T}$ and at each temperature $T_{a,b,c,d}$ of the experiment, and at one additional lower temperature $T_e=0.12\mathrm{K}$. Regions of gate voltage satisfying $B_0\left({\mathrm{v}}_g\right)<B<B_1\left({\mathrm{v}}_g\right)$ and $B_1\left({\mathrm{v}}_g\right)<B$ are indicated. While the sign change at $T_1$ (denoted ${\sigma }_1$) is observable for the experimentally used temperatures, e.g., on increasing temperature from $T_a$ to $T_c$, the sign change at $T_0\left(B\right)$ (denoted ${\sigma }_0$) in the range $-1.0\lesssim {\mathrm{v}}_g\lesssim -0.6$, is only observable by starting from the lower temperature $T_e<T_0\left(B\right)$, e.g., upon increasing temperature from $T_e$ to $T_a$. NRG parameters as in Fig. 1.

Figure 10

(a) Thermopower $S$ vs temperature $T/\mathrm{\Gamma }$ and (b) spectral function $\pi \mathrm{\Gamma }A(\omega ,T=0)$ vs $\omega /\mathrm{\Gamma }$ for increasing values of the magnetic field $B/T_{\mathrm{K}}$ in the mixed valence regime with ${\varepsilon }_0=-0.5\mathrm{\Gamma }$ and $U=8\mathrm{\Gamma }$. Increasing $B$ values in (a) indicated by black arrow. $T_{\mathrm{K}}\approx 0.32\mathrm{\Gamma }$ [from Eq. (9)] is close to the mixed valence low-energy scale $\mathrm{\Delta }=\mathrm{\Gamma }/2$. For $B<B_0\approx 0.9T_{\mathrm{K}}$, the thermopower exhibits no sign change as a function of $T$ (blue solid lines), while for $B\gtrsim B_0$ a single sign change at $T_0\left(B\right)$ occurs. Black vertical arrows in (b): bare Hubbard satellite peaks at $\omega ={\varepsilon }_0$ and $\omega ={\varepsilon }_0+U$ (for $B=0$). Inset (c): details of the low-energy mixed valence resonance for different magnetic fields. The resonance splits at a field $B_c\approx 0.8T_{\mathrm{K}}<B_0\approx 0.9T_{\mathrm{K}}$. Black dashed lines: schematic slope of the spectral function at the Fermi level for $B<B_0$ and $B>B_0$. NRG parameters as in Fig. 1.

Figure 11

(a) Thermopower $S$ vs temperature $T/\mathrm{\Gamma }$ and (b) spectral function $\pi \mathrm{\Gamma }A(\omega ,T=0)$ vs $\omega /\mathrm{\Gamma }$ for increasing values of the magnetic field $B/T_{\mathrm{K}}$ in the empty orbital regime with ${\varepsilon }_0=+\mathrm{\Gamma }$ and $U=8\mathrm{\Gamma }$. $T_{\mathrm{K}}\approx 5.5\mathrm{\Gamma }$ [from Eq. (9)], indicated by the vertical arrow in (a), is larger than the true low-energy scale for this regime, given by ${\varepsilon }_0=\mathrm{\Gamma }$ (nevertheless, for consistency in notation with the Kondo regime, we continue to use $T_{\mathrm{K}}$). For $B<B_0\approx 0.5T_{\mathrm{K}}\approx 2.75\mathrm{\Gamma }$ (blue solid lines), the thermopower exhibits no sign change as a function of $T$, while for $B\gtrsim B_0$ a single sign change at $T_0\left(B\right)$ occurs. Black vertical arrows in (b): bare Hubbard peaks at $\omega ={\varepsilon }_0$ and ${\varepsilon }_0+U$ (for $B=0$). The splitting of the resonance at $\omega ={\varepsilon }_0$, for fields $B>B_c\approx \mathrm{\Gamma }$, eventually leads to a change in slope of the spectral function at the Fermi level and hence a sign change in $S\left(T\right)$ at low $T$ for $B\gtrsim B_0>B_c$. NRG parameters as in Fig. 1.

Figure 12

$B_1$ (in units of $\mathrm{\Gamma }$) vs ${\mathrm{v}}_g$ for different $U/\mathrm{\Gamma }$. The horizontal dashed line corresponds to the field value $B=0.5\mathrm{T}$ of relevance in Sec. 5. NRG parameters as in Fig. 1.

Figure 13

Left panels (a)–(d): calculated $S$ vs ${\mathrm{v}}_g$ at the temperatures $T$ and the four magnetic fields $B$ of the experiment [25]. Vertical dashed lines delineate Kondo ($|{\mathrm{v}}_g|\lesssim 1.0$) from mixed valence and empty (full) orbital regimes at larger $|{\mathrm{v}}_g|$. Inset (e): lowest-temperature data ($T=1.0\mathrm{K}$) from (a)–(d) for increasing $B$ (red arrows). Right panels (a)–(d): the calculated linear-response thermocurrent $I_{\mathrm{th}}/\mathrm{\Delta }T=GS$ vs gate voltage ${\mathrm{v}}_g$ at the four magnetic fields and temperatures of the experiment [25]. Inset (e): calculated thermocurrent ($GS$) at the lowest temperature ($T=1.0\mathrm{K}$) from (a)–(d) for increasing $B$ (red arrows). Inset (f): experimental thermocurrent ($I_{th}/\mathrm{\Delta }T$) at the lowest temperature ($T=1.0\mathrm{K}$) with data taken from Fig. 6 (right panels) of the experiment [25]. NRG parameters as in Fig. 1.