Electrically induced spin resonance fluorescence. II. Fluorescence spectra

We model the fluorescence spectra of planar spin oscillators to find conditions that maximize spin resonance fluorescence. Spin oscillators perform Rabi oscillations under the effect of a periodic effective magnetic field caused by the winding motion of an electron in a gradient of magnetic field. We show that, despite the weak coupling of the spin magnetic dipole to the vacuum, spin oscillators excited by a direct current output a few nanowatts of microwave power, which is comparable to the best microwave sources. The large quantum efficiency relies on the combination of two effects. On the one hand, the spontaneous emission rate is enhanced by the synchronization of spin oscillators, which interact through the microwave field that they emit. On the other hand, the huge Rabi frequencies experienced by spin oscillators promote spins into upper levels of Zeeman transitions, from which a radiative cascade is triggered. We demonstrate different regimes of fluorescence which correspond to different values of the Rabi period relative to the spontaneous decay time and to the oscillator dwell time in the gradient of magnetic field. We investigate the device parameters which make these regimes experimentally accessible and find conditions that optimize microwave output. We find that microwave emission is centered around the cutoff frequency of spin oscillators. This has the advantage that the peak emission frequency may be tuned from zero continuously up to a few hundred gigahertz using an electrostatic gate. Quite remarkably for a spintronics effect, electrically induced spin resonance fluorescence does not require the injection of a spin polarized current. In fact, we show that microwave spectra are mostly independent of the incoming spin polarization except for magnetic waveguides which are shorter than a certain critical length, which we will specify.

Figure 1

(Color online) Electromagnetic power radiated at resonance as a function of the electron dwell time in the gradient of magnetic field for different values of $r={\Omega }_1∕({\Gamma }_{sr}∕2)$, i.e., the Rabi frequency to the spontaneous decay rate. The power is expressed in units of the photon energy, $\hslash \omega$, times the rate $n$ at which electrons are injected in the device. The plot is for resonant conditions when the detuning of the photon frequency to the Larmor frequency is ${\omega }_{eg}=\omega -{\omega }_0=0$ ($e$, “excited state;” $g$, “ground state”). The direct current passing through the waveguide has no initial spin polarization.

Figure 2

(Color online) Fluorescence power plotted as a function of the oscillator dwell time in the magnetic field gradient and the detuning ${\omega }_{eg}$ of the photon frequency relative to the resonant frequency. The calculation is done in the Rabi regime: $r=5$.

Figure 3

(Color online) Fluorescence power plotted as a function of the oscillator dwell time in the magnetic field gradient and the detuning ${\omega }_{eg}$ of the photon frequency relative to the resonant frequency. The calculation is done at the threshold of the exponential regime: $r=0.5$.

Figure 4

(Color online) Microwave power emitted by individual open orbits $\left(\theta \right)$ as a function of the Larmor frequency $\left({\omega }_0\right)$. The Larmor frequency is expressed in units of the cutoff frequency of spin oscillators, which is ${\omega }_c=2.5\times {10}^{12}{\mathrm{s}}^{-1}$ or $400\mathrm{GHz}$. (a) and (b) show the same plot with different vertical scales and under different angles. The former gives the peak power and linewidth, while the latter shows the finer structure, which is discussed in the text. Parameters: current $I=10\mu \mathrm{A}$, electron effective mass $m^{*}=0.023m_0$ (InAs), Landé factor $g=-15$ (InAs), electron density $n_s=1\times {10}^{11}{\mathrm{cm}}^{-2}$, magnetic gradient $b=2\times {10}^6\mathrm{T}∕\mathrm{m}$, length of magnetic waveguide $L=1000\mu \mathrm{m}$, no incoming spin polarization $n_g(t=0)=n_e(t=0)=0.5$, and number of spins oscillating in phase $N={10}^{10}$.

Figure 5

(Color online) Microwave emission spectrum calculated as a function of the length of the magnetic waveguide. We have set ${\omega }_0={\omega }_c$ to show the dependence of the peak power. Other parameters are the same as those in Fig. 4.

Figure 6

(Color online) Dependence of fluorescence on the gradient of magnetic field. The magnetic field gradient is $b=1\times {10}^6\mathrm{T}{\mathrm{m}}^{-1}$ in (a) and $b=4\times {10}^6\mathrm{T}{\mathrm{m}}^{-1}$ in (b).

Figure 7

(Color online) Dependence of the fluorescence spectrum on the spin polarization of the incoming current. $n_e$ represents the fraction of spins prepared in the high energy level. (a) is calculated for a $1000\mu \mathrm{m}$ waveguide and (b) for a $10\mu \mathrm{m}$ waveguide.

Figure 8

(Color online) Microwave power emitted by the whole distribution of open orbits as a function of the Larmor frequency. Positive currents flow with open orbits for which $49.3°<\theta <180°$, whereas negative currents correspond to $0<\theta <49.3°$. Device parameters are the same as in Fig. 4.