Past exam of the physics of Nanoelectronic Systems course of the University of Cambridge / 2012 / Paper 1 / 3 / b / Solution

For a double quantum dot, write $(n_L,n_R)$ for occupation. Tune a bias window allowing the cycle $(0,1)\to (1,1)\to (0,2)\to (0,1)$. Two electrons in the same lowest orbital must form a singlet state by the Pauli exclusion principle. The $(0,2)$ triplet requires an excited orbital, with an energy cost ${\Delta }_{ST}$ above the accessible window. Approximately spin-conserving tunnelling transfers $S(1,1)$ into $S(0,2)$, but cannot transfer $T(1,1)$ into that singlet. Once a triplet loads, transport stops until spin relaxation or mixing occurs: this is Pauli spin blockade. The reverse bias can avoid trapping in this sequence, producing rectification. Coulomb charging sets the allowed occupancies, but the extra blockade is a spin-selection restriction, not merely an energetically forbidden charge state.

magnetically driven spin-blockade lifting uses electron paramagnetic resonance. A transverse oscillating magnetic field drives spin transitions atFlipping one spin in $|\uparrow \uparrow ⟩$ gives $|\downarrow \uparrow ⟩=(|T_0⟩-|S⟩)/\sqrt{2}$, which has a singlet component and can resume charge transfer. A perfectly uniform simultaneous rotation of two identical spins preserves their triplet character; effective relative rotation needs unequal local resonance conditions or another singlet–triplet mixing mechanism. In a real GaAs double dot, inhomogeneous nuclear fields can supply that difference. Hyperfine mixing, spin-orbit effects, cotunnelling and ordinary relaxation also produce leakage and must be distinguished from driven resonance.

A demonstrated device used surface Ti/Au gates to define two dots in a GaAs/AlGaAs electron gas $90\mathrm{nm}$ below the surface. An insulated on-chip gold coplanar stripline supplied the oscillating magnetic field. With the dots biased in spin blockade, RF excitation produced current peaks at the two field polarities satisfying resonance. Peak positions varied linearly with frequency, giving $|g^{\ast }|=0.35\pm 0.01$. RF applied to a side gate instead gave negligible response, testing magnetic rather than electric excitation. Gate pulses and RF bursts further produced coherent spin-rotation oscillations in the charge readout.^[ref]

An ideal isolated resonant spin has Rabi oscillation probability $P_{\mathrm{flip}}(t)={\sin }^2({\Omega }_{R}t/2)$. For a linearly oscillating field of peak amplitude $B_{\mathrm{ac}}$, the rotating-wave convention is ${\Omega }_R=|g^{\ast }|{\mu }_B{B}_{\mathrm{ac}}/(2\hslash )$. Repeated initialization, isolated manipulation and blockade-sensitive readout can therefore map a damped oscillatory singlet yield into an average current. Nuclear fluctuations, readout rates and dephasing change the envelope and contrast; an ideal sine squared is not a reconstruction of experimental data. Compare RF on/off, resonance tracking with frequency, drive-amplitude dependence and detuning controls. Current enhancement alone does not prove coherent rotations; the pulse-duration oscillations provide the further time-domain test.