Quantum transport in carbon nanotubes

Carbon nanotubes are a versatile material in which many aspects of condensed matter physics come together. Recent discoveries have uncovered new phenomena that completely change our understanding of transport in these devices, especially the role of the spin and valley degrees of freedom. This review describes the modern understanding of transport through nanotube devices. Unlike in conventional semiconductors, electrons in nanotubes have two angular momentum quantum numbers, arising from spin and valley freedom. The interplay between the two is the focus of this review. The energy levels associated with each degree of freedom, and the spin-orbit coupling between them, are explained, together with their consequences for transport measurements through nanotube quantum dots. In double quantum dots, the combination of quantum numbers modifies the selection rules of Pauli blockade. This can be exploited to read out spin and valley qubits and to measure the decay of these states through coupling to nuclear spins and phonons. A second unique property of carbon nanotubes is that the combination of valley freedom and electron-electron interactions in one dimension strongly modifies their transport behavior. Interaction between electrons inside and outside a quantum dot is manifested in SU(4) Kondo behavior and level renormalization. Interaction within a dot leads to Wigner molecules and more complex correlated states. This review takes an experimental perspective informed by recent advances in theory. As well as the well-understood overall picture, open questions for the field are also clearly stated. These advances position nanotubes as a leading system for the study of spin and valley physics in one dimension where electronic disorder and hyperfine interaction can both be reduced to a low level.

Figure 1

Atomic structure of carbon nanotubes. (a) Derivation of nanotube structure from graphene. A single-wall nanotube is equivalent to a rolled-up graphene strip (shaded, with the direction of rolling chosen so that the printed pattern faces outward). The chiral vector $\mathbf{C}$ spans the nanotube circumference (inset) and connects lattice sites that are brought together by rolling up. Chiral indices $(n,m)$ completely define the nanotube structure. The unit cell of the nanotube (which is much larger than the unit cell of graphene) is outlined by dashed lines, and the unit vector $\mathbf{T}$ is indicated. Graphene coordinates $(x,y,z)$, nanotube coordinates $(t,c,r)$, and the chiral angle $\theta$ are also marked. In this example, $(n,m)=(6,2)$ and $\theta =13.9°$. (b) Nanotubes are divided into three classes according to their chiral indices: zigzag, armchair, or chiral. Zigzag and armchair nanotubes are so called because of the shape of the edge formed by a cut perpendicular to the nanotube axis [see highlighted lines in (a)]. These three nanotubes are (12,0), (6,6), and (6,4). (c), (d) Nanotubes directly imaged by transmission electron microscopy (TEM) [(c), a (28,0) zigzag nanotube] and scanning tunneling microscopy [(d), an unidentified chiral nanotube]. Adapted from [264], [40], [268], and [44].

Figure 2

Schematic of a basic quantum dot device. The device consists of a nanotube contacted by source and drain electrodes and capacitively coupled to a gate. Tunnel barriers to the source and drain, imposed through the combination of the gate potential and Schottky barriers, define a quantum dot. The current $I$ through the device is measured as a function of bias voltage $V_{\mathrm{SD}}$ and gate voltage $V_{\mathrm{G}}$. Both the number of electrons $N$ on the island and the dot energy levels can be adjusted by tuning $V_{\mathrm{G}}$.

Figure 3

(a), (c), and (e) Schematics and (b), (d), and (f) scanning electron micrographs of devices fabricated by different methods. (a), (b) Top gating: Nanotubes are located on a growth chip, and electrodes fabricated afterward. Here a nanotube (not visible) is contacted by metal electrodes (▪) and covered by a thin gate oxide. Five gates (•) control a double quantum dot, while a floating antenna (F) allows charge sensing via a separate dot on the same nanotube. Other electrodes (○) are helper gates. Adapted from [46]. (c), (d) Bottom gating: Trench, contacts, and gates are fabricated from inert materials before synthesis, and nanotubes grown across. Adapted from [249]. (e), (f) Mechanical transfer: Suspended nanotubes are synthesized on a growth chip, while electrodes are patterned on a device chip. By stamping the chips together, a nanotube is transferred to the device. Electrical current can be used to cut the nanotube at specific places. In this complex two-nanotube device, five gates define a single or double quantum dot in the upper nanotube, while a pair of dots in the lower nanotube serve as independent charge sensors. Adapted from [266].

Figure 4

(a) Electron orbitals of atomic carbon. Lighter (darker) colors denote regions where the $p$-orbital wave functions are positive (negative). Bond directions in graphene are indicated by gray lines. (b) Schematic energy levels of atomic (left) and $s{p}^2$ hybridized (right) carbon. Energies are referenced to $E_{\mathrm{F}}$, approximated as equal to the negative of the work function. (c) Segment of graphene with the unit cell shaded and the $A$ and $B$ sublattices marked. (d) First Brillouin zone of graphene in reciprocal space, showing the six symmetry points, labeled $K$ or $K^{\prime }$. (e) Energy bands ($\sigma$ bands omitted) of graphene close to the Fermi level, showing the six Dirac cones where $\pi$ and ${\pi }^{*}$ bands touch.

Figure 5

The effect of periodic boundary conditions. (a), (b) Requiring that the electron wave function be single valued constrains $\mathbf{k}$ to lie on one of the quantization lines in reciprocal space corresponding to integer values of $k_{c}D/2$. If quantization lines intersect the Dirac points the nanotube is (a) metallic; otherwise it is (b) semiconducting, with minimum quantization line offset $|\mathrm{\Delta }{\kappa }_{\perp }|=2/3D$. The $\{{\kappa }_{\perp },{\kappa }_{\parallel }\}$ axes in reciprocal space, corresponding to motion around or along the nanotube, are indicated. (c), (d) Dispersion relations (in the lowest-energy one-dimensional band) close to a Dirac point for the two kinds of nanotubes, showing how the offset gives rise to a band gap. The Fermi level for undoped nanotubes is indicated. (e) Examples of quantization lines for several metallic and semiconducting structures of the three types shown in Fig. 1. Of the six combinations, armchair semiconducting nanotubes do not exist.

Figure 6

Robustness of the valley index in nanotubes. All subbands of the 1D dispersion relation (corresponding to different quantization lines in Fig. 5) are plotted in the first longitudinal Brillouin zone vs longitudinal wave vector $k_t$. All nominally metallic nanotubes can be classified as armchairlike or zigzaglike. For armchairlike nanotubes, the two Dirac points are separated in $k_t$; for zigzaglike nanotubes, they are separated in crystal angular momentum. Since all metallic nanotubes fall into one of these classes, valley is a good quantum number in a slowly varying potential. (a) Armchairlike (4,1) nanotube. (b) Zigzaglike (6,3) nanotube. Each spin-degenerate band is calculated using a graphene tight-binding model taking account of nearest-neighbor overlap integrals but without spin-orbit coupling. Only states near $E=0$ participate in transport. Note that the Brillouin zone in (a) has been plotted wider than in (b), to reflect the different longitudinal length $|T|$ of the unit cell in real space.

Figure 7

Signatures of the band gap in transport. (a)–(c) Room temperature conductance measurements as a function of gate voltage. (a) A semiconducting nanotube has $E_{\mathrm{G}}\gg k_{\mathrm{B}}T$ and can be tuned between a conducting state (Fermi level in the valence band) and an insulating state (Fermi level in the band gap). Transport via the conduction band is not observed, because it would require a much higher gate voltage. (b) A narrow-gap nanotube ($E_{\mathrm{G}}\sim k_{\mathrm{B}}T$) shows transport via both conduction and valence bands. Tuning the Fermi level into the band gap does not completely suppress current at room temperature. (c) A few nanotube devices ($<1\%$) show no gate dependence of conduction. These could be truly metallic, although it cannot be excluded that this device in fact contains a bundle of nanotubes that screens the gate. (d) Conductance of a single narrow-gap nanotube at 300 mK. Transport is now completely suppressed in the band gap, and the device can be tuned into electron or hole configurations by tuning $V_{\mathrm{G}}$. Adapted from [36] and [44].

Figure 8

Perturbation of the graphene band structure by the curvature in nanotubes. (a) Displacement of the Dirac points away from the corners of the Brillouin zone due to curvature in a (4,1) nanotube. For visibility, the shift has been exaggerated by a factor of 15. Top inset: Decomposition of the displacement vector $\mathbf{\Delta }{\mathit{\kappa }}^{\mathrm{cv}}$ near the $K$ point into components parallel and perpendicular to the nanotube axis. The shift is at an angle $3\theta$ to the nanotube circumference. Bottom inset: Shift for an armchair nanotube. Because the shift is along the quantization lines, curvature does not lead to a gap in these structures. (b) Dirac cones close to $K$ and $K^{\prime }$ valleys with (solid) and without (dotted) curvature effects, showing how horizontal shifts by $\mathrm{\Delta }{\kappa }_{\perp }^{\mathrm{cv}}$ open a band gap in a nominally metallic tube.

Figure 9

Contribution of strain to the band gap. (a) Distortion of the lattice by tensile strain. (b) The experimental setup. An AFM tip is used simultaneously to gate and tension a suspended nanotube. (c) Conductance of an initially metallic nanotube as a function of gate voltage measured for several values of strain $\epsilon$, showing an increasing band gap with $\epsilon$. Inset: Maximum resistance as a function of $\epsilon$, fitted assuming thermally activated conductance across a band gap proportional to $\epsilon$. The fit yields the strain sensitivity $d{E}_{\mathrm{G}}/d\epsilon =35\mathrm{meV}/\%$, where $\epsilon$ is expressed as a percentage. (d) Similar data for a semiconducting nanotube. In this case, the band gap is found to decrease linearly with $\epsilon$, with fitted $d{E}_{\mathrm{G}}/d\epsilon =-53\mathrm{meV}/\%$. Adapted from [173].

Figure 10

(a) Schematic energy levels in the same quantum dot potential for two limits of dispersion and confinement discussed in the text. (Electron correlations beyond the constant-interaction model, which change the potential depending on occupation, are not taken into account.) Superimposed on the level diagram are wave function envelopes $\psi (t)$ for several electron shells. Mode index $\nu$ is indicated for the first three shells. (b) Sequence of Coulomb blockade peaks measured in a quantum dot of length $L\approx 760\mathrm{nm}$. Peaks corresponding to three successive shells are colored to illustrate the connection to different wave functions. Although the four electrons that fill each shell occupy states of similar energy, extra energy $\mathrm{\Delta }E$ is needed to populate a higher shell, leading to fourfold periodic peak spacing. (The connection between colored shells and particular longitudinal modes is only schematic, because absolute shell numbers cannot be deduced from these data.) Adapted from [233].

Figure 11

Dependence of the band structure on a parallel magnetic field. (a), (b) Left: Dirac cones and quantization lines for a nanotube that is metallic at zero field, (a) without and (b) with magnetic field. Arrows in (b) mark the shift from zero-field (dotted) to finite-field (solid) Dirac cones. A field-induced horizontal shift $\mathrm{\Delta }{\kappa }_{\perp }^B$ opens a band gap between the conduction band (darker circles) and valence band (lighter circles). Right: Corresponding one-dimensional electron dispersion relations. (c), (d) The same plots for a nanotube with zero-field gap $E_{\mathrm{G}}$. A magnetic field shifts one Dirac point toward the quantization line and one away, lifting valley degeneracy. (e) Conduction-band (darker line) and valence-band (lighter line) edges as a function of magnetic field for a metallic nanotube. In the conduction band both valleys increase in energy with field, corresponding to a negative magnetic moment ${\mu }_{\text{orb}}=-{Dev}_{\mathrm{F}}/4$ or counterclockwise circulation. The valence band decreases in energy, corresponding to a positive magnetic moment and clockwise circulation. (f) Shift of the Dirac points perpendicular to the quantization lines for a zero-gap nanotube. (g) Band edges for a gapped nanotube. In the conduction band, $K(K^{\prime })$ states move with positive (negative) magnetic moments. In the valence band, $K^{\prime }(K)$ states move with positive (negative) magnetic moments. (h) Shift of the Dirac points for $B_{\parallel }=B_{\text{Dirac}}$, showing how one set of Dirac points is shifted onto the quantization lines. (i) Electron circulation directions corresponding to upmoving and downmoving states in (e) and (g).

Figure 12

Conductance of a nanotube quantum dot as a function of gate voltage and magnetic field, allowing ground-state spectroscopy of the first 12 hole states. (The first two peaks of the $\nu =1$ shell are not visible on this color scale.) Occupation numbers $N$ for holes 1–12 and shell numbers $\nu$ are indicated. The orbital magnetic moment leads to a field-dependent shift of the conductance peaks, marked by dashed lines. From the slope of these lines, ${\mu }_{\text{orb}}$ values of 0.90, 0.80, and $0.88\mathrm{meV}/\mathrm{T}$ for shells 1–3 can be deduced, an order of magnitude larger than the spin magnetic moment. As well as the linear shift of peak positions, other features are seen: the complex network of lines above 2 T on the right-hand side reflects energy crossings between different shells, while the barely resolved low-field anticrossing at $V_{\mathrm{G}}\sim 3\mathrm{V}$ probably reflects spin-orbit coupling not recognized at the time (Sec. 3f). Adapted from [102].

Figure 13

Magnetoconductance of a semiconducting nanotube near the charge neutrality point measured at 3.1 K. The observed maximum is slightly below the expected value (indicated by the bar labeled $B_0/3$). The inset shows a calculated energy gap for a (95,15) semiconducting nanotube ($D=8.1\mathrm{nm}$) vs parallel magnetic field. Solid and dashed lines are without and with Zeeman effect. Adapted from [109].

Figure 14

Effect of confinement on magnetic moment. Main plot: Experimentally determined $g_{\text{orb}}$ and ${\mu }_{\text{orb}}$ (points) as a function of gate voltage, together with a fit by Eq. (12). The fit assumes a constant dot length and linear dependence of $E_{\text{conf}}$ on $V_{\mathrm{G}}$. Left inset: Dirac cone showing quantization line and points corresponding to ${\kappa }_{\parallel }$ values for three longitudinal energy levels. Right inset: View of the Dirac cone from above, giving a physical explanation for the reduction of ${\mu }_{\text{orb}}$. Regardless of ${\kappa }_{\parallel },{\kappa }_{\perp }$, the velocity (black arrow) of the electron is $v_{\mathrm{F}}$ directed away from the origin. Higher energy states, with larger ${\kappa }_{\parallel }$, have smaller perpendicular components $v_{\perp }$ and therefore smaller $g_{\text{orb}}$. From [107].

Figure 15

How curvature enhances spin-orbit coupling. (a) Atomic spin-orbit coupling and interatomic hopping in flat graphene. The $p_z$ and $p_x$ orbitals of two adjacent atoms are shown, with the sign of the wave function in each lobe marked. Intra-atomic spin-orbit coupling mixes opposite-spin states involving different $p$ orbitals in the same atom, e.g., $|p_z^A\uparrow ⟩$ and $|p_x^A\downarrow ⟩$. However, this does not mix spin states in the band structure, because hopping between different $p$ orbitals is forbidden by symmetry. (For example, the hopping contributions between $p_x^A$ and $p_z^B$ marked by a double arrow exactly cancel.) Consequently, there is no coupling between states such as $|p_z^A\uparrow ⟩$ and $|p_z^B\downarrow ⟩$ and therefore no bulk spin-orbit coupling. (b) In a nanotube, curvature breaks the up-down symmetry, meaning that direct hopping between different orbitals on adjacent atoms becomes possible. The combination of atomic spin-orbit coupling and interatomic hopping therefore mixes opposite-spin states on adjacent atoms (e.g., $|p_r^A\uparrow ⟩$ and $|p_r^B\downarrow ⟩$), leading to a spin-orbit splitting of the $\pi$ band.

Figure 16

Observation of spin-orbit coupling in a nanotube. (a), (b) Expected spectra without and with spin-orbit coupling. (c) Conductance as a function of magnetic field and gate voltage across the $0\text{−}1e$ transition at $V_{\mathrm{SD}}=2\mathrm{mV}$, allowing high-bias spectroscopy of the lowest four one-electron states. From line slopes (inset), the spin and orbital magnetic moments can be deduced, allowing assignment of spin and valley quantum numbers. The data are clearly consistent with (b) rather than (a). Adapted from [135].

Figure 17

Two types of spin-orbit coupling. (a), (b) Zeeman-like coupling ([100]) leads to a spin-dependent vertical shift of the band structure, equivalent to a Zeeman splitting (a) that is opposite in the two valleys. (b) Energy levels as a function of $B_{\parallel }$. Going beyond Fig. 18, a residual band gap (induced, for example, by level quantization) is assumed, leading to rounding of the band minima. (c), (d) Orbital-like coupling ([5]) leads to a horizontal shift of the band structure (c) equivalent to a spin-dependent magnetic field coupling to ${\mu }_{\text{orb}}$. (d) Corresponding energy levels as a function of $B_{\parallel }$. A signature to distinguish these two couplings comes from the minima in (b) and (d); whereas Zeeman-like coupling leads to minima in the first and second energy levels, orbital-like coupling leads to two minima in the first level. In (a) and (c), thick arrows indicate evolution of the state energies in the Dirac cones as $B_{\parallel }$ is increased. Spectra (b) and (d) are calculated from Eq. (20) using $E_{\mathrm{G}}=4\mathrm{meV}$, $E_{\text{conf}}=1\mathrm{meV}$, ${\mu }_{\text{orb}}^0=0.9\mathrm{meV}/\mathrm{T}$, and ${\mathrm{\Delta }}_{\mathrm{SO}}^0,{\mathrm{\Delta }}_{\mathrm{SO}}^1=0$ or 2 meV.

Figure 18

Dependence of band edges on $B_{\parallel }$ for different types of spin-orbit coupling for (a) Zeeman-like, (b) orbital-like, and (c) mixed spin-orbit. At finite field, electron-hole symmetry is broken by orbital-like but not by Zeeman-like coupling; at zero field, a combination of both is required.

Figure 19

Experimentally distinguishing orbital-type and Zeeman-type spin-orbit coupling contributions. (a) Measured magnetic field dependence of the first 12 electron energy levels, split across three shells, in a narrow-gap nanotube, derived from low-bias spectroscopy. (The vertical offsets of each trace, which this experimental technique leaves undetermined, have been chosen so that adjacent levels align.) The levels correspond more closely to Fig. 17 than Fig. 17 (and the colors have been chosen accordingly), indicating predominantly Zeeman-like coupling in this device. (b) ${\mathrm{\Delta }}_{\mathrm{SO}}$ as a function of $V_{\mathrm{G}}$ in a separate device. The dashed lines are fits to theory taking account of the different dependence of the two contributions on orbital moment. Inset: Expected dependence of ${\mathrm{\Delta }}_{\mathrm{SO}}$ on $E_{\text{conf}}$, showing the strengths of the two contributions. ${\mathrm{\Delta }}_{\mathrm{SO}}^0$ and ${\mathrm{\Delta }}_{\mathrm{SO}}^1$ parametrize the Zeeman-like and orbital-like contributions, respectively. Adapted from [108] and [251].

Figure 20

(a) Example of a pair of states mixed by intervalley scattering, which couples states with equal spin in opposite valleys. (b) Close-up of Fig. 16. Intervalley scattering is manifest as an anticrossing between $|K\uparrow ⟩$ and $|K^{\prime }\uparrow ⟩$ levels, with strength ${\mathrm{\Delta }}_{K{K}^{\prime }}$. Adapted from [135].

Figure 21

(a)–(f) Single-particle energy levels as a function of magnetic field in perpendicular and parallel directions for various strengths of spin-orbit coupling and intervalley scattering, taking ${\mu }_{\text{orb}}=0.2\mathrm{meV}/\mathrm{T}$. (a) No spin-orbit or intervalley scattering, (b) spin-orbit only, (c) intervalley scattering only, and (d)–(f) combined spin-orbit and intervalley scattering. In selected panels, insets show the same spectra for field axes misaligned by $\mathrm{\Theta }=10°$ from the nanotube; this introduces an anticrossing (marked by arrows) between the two highest levels. Selected energy splittings mentioned in the text are marked. Double-thickness lines indicate degenerate levels.

Figure 22

Energy levels for comparable spin-orbit coupling and intervalley scattering. (a) Measurements of the four states of an electron shell by cotunneling spectroscopy (see text). The second derivative $d^{2}I/d{V}_{\mathrm{SD}}^2$ is plotted as a function of $V_{\mathrm{SD}}$ and the magnetic field. (b) Calculated transition energies to the first three excited states assuming ${\mathrm{\Delta }}_{K{K}^{\prime }}=3{\mathrm{\Delta }}_{\mathrm{SO}}$, showing good agreement with the peaks and dips in (a). (c) Energy levels corresponding to the transitions (marked with arrows) in (b). Adapted from [108].

Figure 23

Transport through double quantum dots involves a dot-to-dot transition with a probability reflecting selection rules. The strictness of spin and valley selection rules depends on interactions that mix the spin or valley states, e.g., spin-orbit coupling, hyperfine interactions, and disorder.

Figure 24

Charge stability with electrons and holes (a) Circuit model of a series double quantum dot. Each of the three tunnel barriers is characterized by a tunnel rate ${\mathrm{\Gamma }}_i$ and a capacitance $C_i$. (b) Stability diagram for a weakly tunnel-coupled double dot. Pairs of triple points define a honeycomb pattern. The neutral Coulomb valley is largest, which can be understood by considering the $(1h,0)\to (1e,0)$ transition: adding two electrons to the left dot requires lowering its potential by an electrostatic energy (charging energy) and a kinetic energy (band gap), $\mathrm{\Delta }{V}_{\mathrm{L}}\propto 2E_{\mathrm{C}}+E_{\mathrm{G}}$. Axes of approximate spatial and electron-hole symmetry are denoted by dashed and dotted lines, respectively. The top right shaded region is most similar to conventional double dots, in which transport and occupation involve only electronlike carriers. The dashed circle marks a transition with particularly strong spin-valley blockade (cf. Figs. 32 and 33).

Figure 25

Conductance through a narrow-gap nanotube suspended over five gate electrodes that allow independent control over $N_{\mathrm{L}}$, $N_{\mathrm{R}}$, ${\mathrm{\Gamma }}_{\mathrm{L}}$, ${\mathrm{\Gamma }}_{\mathrm{M}}$, and ${\mathrm{\Gamma }}_{\mathrm{R}}$. For the gate configuration shown, the nanotube is neutral in the center of the plot $(N_{\mathrm{L}},N_{\mathrm{R}})=(0,0)$, forms a double quantum dot in the heteropolar regions (top left, bottom right), and forms a single quantum dot in the homopolar regions (top right for electrons, bottom left for holes). Insets visualize the charge distribution along the nanotube for electrons and holes. Adapted from [266], device shown in Fig. 3.

Figure 26

(a)–(c) Dependence of bias triangles on $V_{\mathrm{SD}}$ (assumed negative). Dashed (dotted) lines indicate alignment of the electrochemical potential of the right (left) dot with the Fermi level in the right (left) lead. Sequential tunneling from source to drain is allowed only in the shaded regions, which expand with increasing $|V_{\mathrm{SD}}|$. The discrete density of states within each dot additionally restricts the current (not shown). (d)–(f) Representative bias triangles for increasing $V_{\mathrm{SD}}$. Excited-state lines in (e) and (f) identify ${\mathrm{\Gamma }}_{\mathrm{M}}$ as the rate-limiting tunnel barrier. Adapted from [235], [44], and [113].

Figure 27

Pauli rectification in conventional double dots and carbon nanotubes (theory). (a) In a conventional double dot without spin-orbit coupling, the spin-triplet (0,2) state involves an excited single-particle orbital of the right dot and hence is higher than the spin-singlet (0,2) state by energy ${\mathrm{\Delta }}_{S,T}$. Near the (1,1)-(0,2) degeneracy, one of the spin-triplet states Triplet (1,1) can by chance become occupied. For sufficiently large bias (${\mu }_{\mathrm{S}}-{\mu }_{\mathrm{D}}\gg k_{\mathrm{B}}T$), it is long lived, resulting in a suppression of current compared with under opposite bias (not shown). (b) If each dot participates with one longitudinal single-particle shell, then 16 different (1,1) states are possible in carbon nanotubes, colored here according to their longitudinal symmetry ($S$ or $AS$). If the right dot participates with the lowest longitudinal single-particle shell, then only six different (0,2) states are possible. Other (0,2) states are higher in energy by ${\mathrm{\Delta }}_{S,AS}$ (see Fig. 28). Similar to (a), sufficiently large bias ${\mu }_{\mathrm{S}}-{\mu }_{\mathrm{D}}$ can populate the long-lived (1,1) states [denoted $AS(1,1)$], leading to a suppression of current.

Figure 28

Lowest quantum states of (0,2) for (a) conventional semiconductors and (b) nanotubes in the presence of interactions (INT) and spin-orbit coupling (SO). For weak electron interactions (${\mathrm{\Delta }}_{AS,S^{\prime }}\ll {\mathrm{\Delta }}_{\mathrm{ls}}$), the multiplet splitting ${\mathrm{\Delta }}_{S,T}$ or ${\mathrm{\Delta }}_{S,AS}$ is approximately the single-particle level spacing ${\mathrm{\Delta }}_{\mathrm{ls}}$, and Pauli blockade between Triplet (1,1) and Singlet (0,2) or $AS(1,1)$ and $S(0,2)$ can be expected. The states $S$, $AS$, and $S^{\prime }$ are part of the basis states of the matrix shown in Fig. 29. Spin-orbit interaction splits the two-electron multiplets into three. This is shown for $S$ and $AS$ and further discussed in Fig. 30. Additional effects due to intervalley exchange are discussed in Fig. 51.

Figure 29

(a) Matrix representing the two-electron Hamiltonian for a conventional semiconductor double dot, assuming a single orbital on the left and two orbitals on the right. The basis states are spin-singlet and spin-triplet states as indicated, and symbols indicate nonzero matrix elements. The submatrices in the (1,1) and (0,2) subspaces are all diagonal, and interdot tunneling ($+$) leaves singlet and triplet states uncoupled. To lowest order, interactions modify the diagonal elements between Triplet (0,2) and ${\text{Singlet}}^{\prime }$ (0,2) eigenstates (▪ and •), corresponding to the interaction energy ${\mathrm{\Delta }}_{AS,S^{\prime }}$ discussed in the text. For completeness, we include the higher-lying ${\text{Singlet}}^{\prime \prime }$ (0,2). (b) The analogous matrix for a nanotube double quantum dot with spin-orbit interaction, assuming a single longitudinal shell on the left and two shells on the right. The basis states are those of Table 4; hence longitudinally symmetric $S(1,1)$ and antisymmetric $AS(0,2)$ states are uncoupled. Most submatrices are still diagonal, but interactions are expected to result in significant off-diagonal matrix elements (○) within the (0,2) sectors. For example, Sec. 7 shows that valley backscattering leads to off-diagonal elements between spin-valley-unpolarized states within the $S(0,2)$ subspace. Without going into details of the interactions, all off-diagonal elements marked ○ are in principle allowed, as drawn here. In some cases, a detailed analysis shows that particular interaction matrix elements vanish. (c) Energy vs detuning of the shaded states in (a), neglecting interactions. Zero detuning corresponds to the degeneracy between the ground states of (0,2) and (1,1). Avoided crossings due to interdot tunneling occur for states with the same symmetry. The degeneracy of the levels is indicated. Numerical parameters in this plot were chosen for illustration and differ in actual devices. (d) The same plot for a nanotube, assuming identical spin-orbit coupling (${\mathrm{\Delta }}_{\mathrm{SO}}$) in all shells. Interdot tunneling now leads to anticrossings between states with the same longitudinal symmetry. The effect of interactions on $S(0,2)$, and $AS(0,2)$ is neglected in this plot; see the discussion of Figs. 30, 31, and 51 for details.

Figure 30

Magnetic field dependence of the 16 lowest (0,2) states [cf. Fig. 28]. (a) Spectrum of the two lowest single-particle shells of the right dot, vs parallel magnetic field $B_{\parallel }$. There are six ways to fill the lowest shell with two electrons, each corresponding to a symmetric state in (b). The magnetic moment of each combination is simply the sum of the involved single-particle spin and orbital magnetic moments, shown here for two examples (colored dots). (b) Magnetic field dependence of the ten antisymmetric and six symmetric states. States that involve the ground state of the lowest shell $K\downarrow$ are plotted with solid lines.

Figure 31

Addition spectroscopy of two-electron states in a nanotube single quantum dot, assuming that the one-electron dot starts in its ground state $K\downarrow$. (a) Two-electron spectrum $E(2e)$ from Fig. 30 as a function of the parallel magnetic field. States that do not involve $K\downarrow$ are omitted. (b) Expected $1e\text{−}2e$ addition spectrum from (a). (c) Measured addition spectrum, with lines from (b) superimposed. The topmost conductance feature ($w$) results from the finite-bias window of 1.7 mV used during measurement (see Fig. 54). The separation between solid and dashed lines is surprisingly small (0.85 meV) compared to the single-particle level spacing (7.8 meV), indicating that ${\mathrm{\Delta }}_{S,AS}\ll {\mathrm{\Delta }}_{\mathrm{ls}}$ due to electron-electron interactions. From [201].

Figure 32

Pauli rectification (experiment). Current in the bias triangles measured in (a) forward and (b) reverse directions near the $(3h,1e)\text{−}(2h,0)$ transition. The asymmetry due to Pauli rectification is clearly observed and persists up to $\pm 10\mathrm{meV}$, i.e., detunings much larger than ${\mathrm{\Delta }}_{\mathrm{SO}}$ ($=1.6\mathrm{meV}$ in this device). The arrow defines the detuning axis used in Fig. 33. Adapted from [202].

Figure 33

Different levels of Pauli blockade involving spin and valley quantum numbers for the charge transition $(3h,1e)\to (2h,0)$. (a) Single-particle electron levels vs the magnetic field, sketched for the left and right sides of a $pn$ double dot. (b) At low field [marked $B_1$ in (a)] the ground-state-to-ground-state transition requires a spin flip. At higher field (marked $B_2$) the ground-state transition requires a spin flip and a valley flip. The detuning $\delta$ between left and right dots is marked. Because of the different magnetic moments of the single-particle states, the value of $\delta$ at which each transition becomes resonant depends on magnetic field. (c) Measurement of the leakage current vs $B_{\parallel }$. Some of the features can be identified within the single-particle representation of (b) (lines 1 and 2). The slight reduction of the ground-state leakage current above 1 T indicates that a spin and valley flip (line 2) is slightly less frequent than a spin-only flip (line 1), and is evidence for a spin-valley blockade. The absence of a simple charge relaxation $(RK\uparrow )\to (LK\uparrow )$ (line 3) and the onset of leakage current at $\delta \sim 2{\mathrm{\Delta }}_{\mathrm{SO}}$ (dashed line) are currently not understood. Adapted from [202].

Figure 34

(a) Two localized electronic wave functions with different valley, longitudinal, and spin quantum numbers. Hyperfine interaction (HF) with a ${}^{13}\mathrm{C}$ nuclear spin (fat arrow) can scatter between them, because of the local nature of the scattering vertex (b). This mechanism allows lifting of spin- and/or valley-based Pauli blockade, if conditions of energy conservation are fulfilled.

Figure 35

(a) Current through a top-gated multielectron double dot formed in a narrow-gap ${}^{13}\mathrm{C}$ nanotube. Near the base of the bias triangles (ground-state-to-ground-state transition) no current is observed for reverse bias. (b) Current at forward bias. (c) Magnetic field dependence of current in reverse and forward bias measured near the base of a triangle [dots in (a) and (b)]. The peak in reverse current at $B_{\parallel }=0$ is attributed to hyperfine-mediated relaxation. (d) In a similar device formed in a predominantly ${}^{12}\mathrm{C}$ nanotube, the opposite magnetic field dependence is observed: The reverse current shows a minimum at $B_{\parallel }=0$, presumably a consequence of Van Vleck cancellation. (e) Leakage current in a different device with natural abundance of ${}^{13}\mathrm{C}$ vs detuning and $B_{\parallel }$. This peak, measured in the spin-valley blockade regime of Fig. 33, is narrower than in (c), possibly due to the smaller concentration of ${}^{13}\mathrm{C}$. (f) Horizontal cut through (e) at location of the arrow, showing a small dip at $B_{\parallel }=0$. (a)–(d) Adapted from [45]. (e), (f) From [202].

Figure 36

Measurement of spin-valley $T_1$. (a) Pulse cycle for measuring relaxation from $AS$ to $S$ states (see text). (b) Stability diagram close to the (1,1)-(0,2) transition with pulse cycle $E\to R\to M\to E$ and ${\tau }_{\mathrm{M}}=0.5\mu \mathrm{s}$. Color represents time-average charge-sensor conductance. Inside the “pulse triangle” (marked), Pauli blockade leads to an occupancy of (1,1) for metastable $AS$ states, yielding a reduced sensor conductance. (c) Decay of pulse triangle visibility (points) as a function of ${\tau }_{\mathrm{M}}$, measured for three magnetic field values. Lines are fits from which $T_1$ values (inset) are extracted. (d) Points: $T_1$ as a function of $B$ close to the upper spin-orbit anticrossing ($K^{\prime }\uparrow -K^{\prime }\downarrow$). Line: fit of the form $T_1\propto \sqrt{\mathrm{\Delta }E}$. Adapted from [46].

Figure 37

Measurement of $T_2^{*}$ in a ${}^{13}\mathrm{C}$ double quantum dot. (a) Pulse cycle to measure mixing between $S$ and $AS$ states (see text). (b) Stability diagram close to the (1,1)-(0,2) transition, measured via time-averaged charge-sensor conductance. Gate settings at the three steps of the pulse cycle are indicated $P$, $S$, and $M$. (The step at $P^{\prime }$, not discussed here, was inserted to reduce pulse overshoot.) The triangle marks the region where $AS$ states are Pauli blocked. (c) Stability diagram with pulses applied and ${\tau }_S=50\mathrm{ns}$. The triangle of reduced conductance indicates mixing of $S$ and $AS$ states leading to higher probability of (1,1) during the measurement part of the cycle. (d) Points: Return probability as a function of ${\tau }_S$, deduced from $g_C$. Line: Gaussian fit giving $T_2^{*}=3.2\mathrm{ns}$. Adapted from [46].

Figure 38

(a) Bloch sphere representation of a generic qubit. The qubit state $|\psi ⟩$ is represented by a point with polar coordinates $(\vartheta ,\phi )$, so that north and south poles correspond to the basis states $|0⟩$ and $|1⟩$. In the rotating frame, with Cartesian coordinates $(X,Y,Z)$, the Rabi rotations driven by microwave bursts with phase $\varphi =0,\pi /2$ are marked. (b) Four possible qubits in the spin-orbit coupled energy levels. (c) Qubit device, driven by an oscillating gate voltage. The applied and effective magnetic fields are indicated. Adapted from [69].

Figure 39

(a) Principle of bend-mediated EDSR. An electron driven across a bend experiences a time-varying effective magnetic field which induces coherent qubit precession. (b) Schematic of bent nanotube double quantum dot. Gates G1–G5 define the confinement potential and carry the time-dependent manipulation voltages. (c) Pauli blockade leakage current in a highly disordered device (${\mathrm{\Delta }}_{K{K}^{\prime }}\gg {\mathrm{\Delta }}_{\mathrm{SO}}$), showing resonance lines at $g\approx 2$ and at subharmonics. To make the resonance clearer, the mean current at each frequency is subtracted. Adapted from [202] and [138].

Figure 40

(a) Pulse sequence used for coherent qubit manipulation. After initialization in a Pauli-blocked qubit state, the device is pulsed into Coulomb blockade to allow qubit manipulation and returned to Pauli blockade for readout. Electron tunneling occurs only if a qubit was flipped during the manipulation step. Repeated with a period $\sim 1\mu \mathrm{s}$, this leads to a current proportional to the qubit flip probability during the manipulation step. (b) Current (points) as a function of microwave burst duration for various applied powers, showing coherent Rabi rotations. The curves are fits to a model assuming a slowly varying random qubit detuning due to, e.g., charge noise. Traces are offset for clarity. (c) Rabi frequency as a function of microwave amplitude, with fit showing the expected proportionality. (d) Rabi frequency at constant microwave amplitude and power, showing a dependence on field angle $\mathrm{\Theta }$ consistent with bend-mediated EDSR. Adapted from [138].

Figure 41

Hahn echo amplitude (symbols) as a function of $\tau$ plotted for two phases of the central burst, together with fits (lines) of the form $e^{-(\tau /T_{\text{echo}}{)}^{\alpha }}$, yielding $T_{\text{echo}}=65\mathrm{ns}$. Inset: example fringes as a function of $\varphi$. The manipulation sequence is sketched at bottom right, and Bloch spheres along the top illustrate the resulting state evolution. Adapted from [138].

Figure 42

Quantum transport from closed to open devices. (a)–(d) Plots of $dI/d{V}_{\mathrm{SD}}$ as a function of gate voltage $V_{\mathrm{G}}$ and bias $V_{\mathrm{SD}}$ for four nanotube devices with average conductances $g\sim 0.01,0.5,1.5,3e^2/h$ [schematic in (e)]. All data were taken at 1.5 K and zero field. High (low) conductance is shown by light (dark) colors. Black bars indicate the gate voltage range for the addition of four additional electrons. (e) Device schematic. Dashed paths indicate the origin of Fabry-Pérot resonances induced by reflections at the contacts. Adapted from [145].

Figure 43

Kondo physics in nanotubes due to spin and valley degrees of freedom. (a) Schematics of an elastic cotunneling process resulting in a spin flip on the quantum dot. Such higher-order processes give rise to the spin Kondo effect. (b) The Kondo effect can also occur if another quantum number is present in the leads and the dot, e.g., the valley quantum number as shown here. (c) In nanotubes valley and spin quantum numbers may additionally give rise to the SU(4) Kondo effect involving both degrees of freedom. (d) Device schematic showing valley mixing in the lead $({\mathrm{\Delta }}_{K{K}^{\prime }}^{L,R})$ and the dot $({\mathrm{\Delta }}_{K{K}^{\prime }}^{\text{disorder}})$. For SU(4) Kondo effects to be observed these should be small and valley-conserving tunneling $(t)$ from nanotube leads is required. (e) Energy vs parallel and perpendicular magnetic fields for two conduction-band shells with spin-orbit interaction (${\mathrm{\Delta }}_{\mathrm{SO}}>0$). Dashed ellipses and related $B$ fields indicate degeneracies resulting in Kondo phenomena, e.g., SU(4) and various SU(2) Kondo effects. The SU(4) Kondo effect can be observed (lower shell) when the related energy scale is much larger than the zero-field (spin-orbit) splitting. (f) Flow diagram at $B=0$ for fillings $N=1-3$ showing that valley mixing reduces the SU(4) Kondo effect to two-level ($2L$), singlet-triplet ($S\text{−}T$), or no Kondo effect. Similarly spin-orbit interaction reduces the SU(4) Kondo effect to SU(2) or no Kondo effect. The intervalley mixing parameter ${\mathrm{\Delta }}_{K{K}^{\prime }}$ represents all relevant mixing terms. (a)–(c) Adapted from [103].

Figure 44

Kondo physics near zero field. (a) Conductance vs gate voltage of a narrow-gap nanotube for four electron shells I–IV. For shells I and II the occupancy in each diamond is indicated. More positive $V_{\mathrm{G}}$ increases the tunnel couplings, tuning the dot from the Kondo to the mixed-valence regime. Different traces correspond to temperatures from 1.3 to 15 K. (b) Differential conductance in the same device as a function of $V_{\mathrm{G}}$ and $V_{\mathrm{SD}}$ at 3.3 K. (c) Similar data for a different device showing spin-orbit split SU(2) and SU(4) Kondo physics at low and high hole fillings, respectively. (d) Magnetic field splitting of $N=1$ Kondo resonance into four peaks, indicating SU(4) Kondo physics. (a), (b) Adapted from [155]. (c) From [47]. (d) From [103].

Figure 45

Kondo physics at finite field. (a) Differential conductance vs gate voltage and parallel magnetic field for the first three electrons in the conduction band. Dashed lines indicate faint Kondo resonances: Kramers (for $N=1,3$ at $B_{\parallel }=0$), valley (for $N=2$ at $B_{\parallel }=B_{\mathrm{o}1}$), and spin SU(2) (for $N=1$ at $B_{\parallel }=B_{\mathrm{s}}$). (b) Theoretical linear conductance vs gate voltage filling (expressed as effective filling $N_{\mathrm{G}}$) and normalized parallel field ${\tilde{B}}_{\parallel }$. The calculation assumed $|{\mathrm{\Delta }}_{\mathrm{SO}}|>T_{\mathrm{K}}^{\mathrm{SU}(4)}$ as in the top shell of Fig. 43. (c) Differential conductance as a function of gate voltage $V_{\mathrm{G}}$ and parallel magnetic field $B$ at 30 mK. Boxed numbers indicate the occupation of the topmost shell and unboxed numbers indicate the ground-state spin $(0,1/2,1)$. Solid lines highlight the motion of the Coulomb blockade peaks. Dashed lines mark valley Kondo effects [compare white arrows to Fig. 43]. (a)–(c) Adapted from [47], [71], and [103], respectively.

Figure 46

Level renormalization. (a) Energy vs parallel magnetic field of two valley states with coupling ${\mathrm{\Delta }}_{K{K}^{\prime }}$ that splits the states at zero field. Insets show the electron density distributions at zero and high fields. (b) Lead couplings vs parallel magnetic field of the two eigenstates in (a). (c)–(f) Cotunneling processes for the two doublets of a singly occupied shell near the 1 to 0 (c), (d) and 1 to 2 (e), (f) charge degeneracy points. The thickness of the arrows indicates the tunnel couplings. (g) Electrochemical potentials ${\mu }_{1g,0}=E_{1g}-E_0$ (solid lines) and renormalized versions $\tilde{\mu }$ (dashed lines). Here $E_{1g}$ and $E_0$ are the energies of the one-electron ground state ($1g$) and the empty-dot state (0). The lead potentials are set at the threshold for inelastic cotunneling; renormalization enhances the threshold energy ${\delta }^{\prime }$ compared to the original doublet splitting $\delta$. (h) Schematic charge stability diagram for the one-electron diamond. The resulting level renormalizations induce a gate-dependent inelastic cotunneling threshold (dashed line) in contrast to the single-particle prediction $\delta$ (solid line). Adapted from [78].

Figure 47

Tunnel renormalization in nanotube stability diagrams. (a) Stability diagram for a shell with similar lead couplings to ground and excited states (${\mathrm{\Gamma }}_g\sim {\mathrm{\Gamma }}_e$). Kondo ridges are observed for one and three electrons and inelastic cotunneling thresholds are gate independent. Occupation numbers are marked above the plot. (b) Stability diagram for a different shell. Unlike in (a), the Kondo ridge appears only for the $N+1$ diamond, and the cotunneling thresholds in the $N+2$ and $N+3$ diamonds are gate dependent. This is consistent with cotunneling provided that the doublet lead couplings are asymmetric (${\mathrm{\Gamma }}_g\gg {\mathrm{\Gamma }}_e$ deduced from thresholds for $N+2$ and $N+3$). (c) Diagram at zero field for a different device. Again, a gate-dependent cotunneling threshold (arrow) indicates asymmetric couplings. (d) At $B_{\parallel }=2\mathrm{T}$ the threshold is gate independent, consistent with equalization of the lead couplings by the magnetic field as discussed in the text. Adapted from [88] and [78].

Figure 48

Toy model for the two-electron Wigner molecule in one dimension. (a) Two-electron potential $V(x_1,x_2)$ (left panels), symmetric ground-state wave function ${\mathrm{\Psi }}_S(x_1,x_2)$ (middle panels), and antisymmetric excited-state wave function ${\mathrm{\Psi }}_{AS}(x_1,x_2)$ (right panels). Upper panels: no Coulomb interaction ($ϵ=\infty$); lower panels: $ϵ=1$. (b) Line cuts of $V(x_1,x_2)$ along $x_1=-x_2$ with (dashed) and without (dotted) Coulomb interactions. (c) Line cut of the ground-state two-electron probability density $|\mathrm{\Psi }(x,-x){|}^2$ along $x_1=-x_2$ in the single-particle model (dotted line) and the Wigner molecule (solid line). Dashed line: $V(x,-x)$. The suppression of $|\mathrm{\Psi }(x,-x){|}^2$ near $x=0$ is an indication of strong correlations. (d) Electron density $\rho (x)=\int |\psi (x,x_2){|}^{2}d{x}_2$ in the Wigner molecule limit.

Figure 49

Exact calculation for two electrons in a nanotube quantum dot. Vertical axis: energies of the antisymmetric and symmetric excited multiplets relative to the symmetric ground multiplet. Along the horizontal axis, the dielectric constant is changed to tune the interaction strength. Numbers (6, 10, and 6) indicate the degeneracy of each multiplet. Spin-orbit coupling is not included (${\mathrm{\Delta }}_{\text{SO}}=0$) to illustrate only effects from the long-range interactions. Increasing $\alpha =2.2/ϵ$, there is a transition from the single-particle limit (left) to the Wigner molecule limit (right). In the Wigner molecule, the splitting ${\mathrm{\Delta }}_{S,AS}$ becomes exponentially small. Adapted from [277].

Figure 50

Observation of a Wigner molecule. (a) Coulomb blockade spectroscopy of the 0-1 electron transition from which a single-particle level spacing ${\mathrm{\Delta }}_{\mathrm{ls}}=8\mathrm{meV}$ is extracted. (b) Spectroscopy of the 1-2 electron transition: analysis of the magnetic field dependence of the levels allows identification of the excited state corresponding to splitting ${\mathrm{\Delta }}_{S,AS}$. The value ${\mathrm{\Delta }}_{S,AS}=0.85\mathrm{meV}$ is extracted, 10 times smaller than ${\mathrm{\Delta }}_{\mathrm{ls}}$, indicating that the two-electron quantum dot is in the Wigner molecule regime with $r_{\mathrm{s}}\approx 1.6$ ($\alpha \approx 0.5$ in Fig. 49). Adapted from [201].

Figure 51

Calculated two-electron quantum dot energies including short-range interactions. (a) Energies relative to the ground state of the spatially symmetric multiplet (solid lines) and the spatially antisymmetric multiplet (dashed lines). The parameter $\alpha$ characterizes the strength of the long-range interactions: $\alpha =1$, Wigner molecule regime; $\alpha =0$: no long-range correlations. The valley-polarized, spin-polarized, and unpolarized (ground and excited) levels are labeled, and the degeneracy of each set of lines is indicated. For $\alpha =1$, the dashed lines become degenerate with the solid lines, as in Fig. 49 without spin-orbit coupling and short-range interactions. For $\alpha =0$, long-range correlations are suppressed, the single-particle wave functions begin to overlap, and short-range interactions become stronger. Here new splittings ${\mathrm{\Delta }}_{\mathrm{VBS}}$ and ${\mathrm{\Delta }}_{\mathrm{SO}}^{*}$ appear in the valley-spin structure of the multiplet. (b), (c) Magnetic field dependence of the ground-state multiplet for (b) $\alpha =0$ and (c) $\alpha =1$. Short-range interactions in (b) break the degeneracy between the valley-polarized states and spin-polarized states at $B=0$. For $\alpha =1$, the valley and spin structure of the multiplet is the same as in the single-particle model, even though the longitudinal wave functions $\mathrm{\Psi }(x_1,x_2)$ are highly correlated. States are labeled as in Table 4, except that the states used in the calculation are simplified such that they are independent of longitudinal $\mathrm{\Psi }(x_1,x_2)$. (a) Adapted from [277].

Figure 52

Short-range Coulomb interactions in a many-hole quantum dot (estimated as containing 10–40 holes). (a)–(c) Cotunneling spectroscopy of different charge configurations. (d)–(f) Expected spectra from a model with short-range Coulomb scattering. Quantum numbers of occupied electronic states in the valence band are indicated (cf. Fig. 18). Lines in (a)–(c) measure length of the arrows in (d)–(f). Colors in (e) correspond to those used in Fig. 51. The signature of interactions can be seen by comparing the spectra of different charge configurations: the valley and spin spectra of the $4N+1$ and $4N+3$ configurations are consistent with those predicted by the single-particle model, while the spectrum of the $4N+2$ configuration is modified by short-range Coulomb interaction. Adapted from [47].

Figure 53

Electrical model of a single quantum dot: a conducting island is coupled with tunnel rates ${\mathrm{\Gamma }}_{\mathrm{S}}$ and ${\mathrm{\Gamma }}_{\mathrm{D}}$ to source and drain, and with capacitance $C_{\mathrm{S}}$, $C_{\mathrm{D}}$, and $C_{\mathrm{G}}$ to source, drain, and gates.

Figure 54

Two techniques of electron transport spectroscopy. In low-bias spectroscopy (a)–(d), at most a single quantum dot level falls within the bias window set by the lead electrochemical potentials. (a) With no ground-state level in the bias window, levels are filled up to ${\mu }_{\mathrm{D}}$, and the electron number is fixed (in this case, at $N-1$) due to Coulomb blockade. (b) With one of the levels $\mu (N)$ in the bias window, the electron number fluctuates between $N$ and $N-1$, leading to conductance through the dot. (c) Sweeping $V_{\mathrm{G}}$ moves each level in turn through the bias window, leading to a series of current peaks. From the peak spacing, the addition energy $E_{\mathrm{add}}$ for each electron number can be deduced, allowing the ground-state energy to be studied. (d) Transport peak evolution measured using this technique, where the energy levels $E_i$ are assumed to vary with magnetic field $B$ as shown in (e). In high-bias spectroscopy (f)–(j), more than one level can fall within the bias window, allowing excited-state energies to be measured. Sweeping $V_{\mathrm{G}}$ through the $N-1\leftrightarrow N$ transition, a series of transport peaks appear in $dI/d{V}_{\mathrm{SD}}$. (f) The current first appears when $\mu (N)$ crosses ${\mu }_{\mathrm{S}}$. (g) When ${\mu }^{\prime }(N)$ crosses ${\mu }_{\mathrm{S}}$, additional current can flow through an excited state of the dot. (h) The last peak in the series occurs when $\mu (N)$ crosses ${\mu }_{\mathrm{D}}$, leaving the bias window. (i) The corresponding series of transport peaks, with the spacings corresponding to $\mathrm{\Delta }E$ and $V_{\mathrm{SD}}$ marked. When $\mu (N)<{\mu }_{\mathrm{D}}$, Coulomb blockade is reestablished with the dot in its $N$-electron ground state; there are therefore no peaks corresponding to states with excitation energies larger than $e{V}_{\mathrm{SD}}$, such as that marked by the dotted line in (d)–(f). (j) Sketch of the peak evolution measured using this technique for the same underlying energy levels as shown in (e). The $V_{\mathrm{G}}$ trace in (i) corresponds to the vertical dashed line in (j).

Figure 55

Examples of orthonormal spatial eigenfunctions satisfying both Eqs. (b1) and (b3) at the $K$ point for the minimal tight-binding model. The site coefficients for a one-orbital description of the eigenfunctions are shown, with $\chi =e^{i2\pi /3}$. In the general case, the ${\mathrm{\Psi }}_A^{\mathbf{K}}$ and ${\mathrm{\Psi }}_B^{\mathbf{K}}$ wave functions have the same rotational symmetry as in the figure. The eigenfunctions at the $K^{\prime }$ point are the same but with $\chi$ replaced by ${\chi }^{*}$.