doi:10.1038/nature12036 We provide in the following additional experimental data and details on our demonstration of an electrically pumped exciton-polariton laser by supplementing optical and electrical characterization. DC electrical characteristic In Fig. S1 the current-voltage characteristics of our device with a diameter of 20 µm is depicted, operated under direct current conditions at 10 K. From the slope of the curve, we can calculate a nearly constant differential resistance du/di ~5 kω at ~5-7 V. The onset diode voltage amounts to 1.5 V. To avoid heating at high current densities, quasi-dc electrical excitation is employed by injecting high frequency rectangular current pulses into the diodes with pulse durations of 200 ns (duty cycle 1:67) and an impedance-matched input-circuit. Figure S1. Current-voltage characteristics for a 20 µm diameter polariton diode. WWW.NATURE.COM/NATURE 1
RESEARCH SUPPLEMENTARY INFORMATION Q-factor determination In order to accurately extract the Q-factor of our wafer, circular micropillar cavities were fabricated. The micropillars were defined by electron-beam lithography and electron-cyclotron-resonance reactive-ion etching 1. Due to the relatively small diameters in the micron range, the linewidth broadening caused by a change of detuning over a single device area (which limits the minimum extractable linewidth in the case of a probed circular area with 20 µm diameter in this range) is negligible compared to the cavity lifetime limited linewidth. Furthermore, the strong photonic localization spectrally isolates the fundamental mode from the excited photonic states. It is worth noting, that the extracted Q-factor of 6320 does not represent the empty cavity limit, which can have values in excess of 10000 also for doped structures with an equivalent amount of Bragg mirros 2, but is, most likely, limited by the weak absorption tail of the QWs in the cavity. Figure S2. Scanning electron microscope image of a fabricated micropillar cavity with a diameter of 2 µm for Q-Factor determination via micro-photoluminescence. Lower polariton parameters and exciton-photon detuning Since the upper polariton branch in our devices is fully thermalized due to the high Q- factor for all pump currents, the polaritonic character of the electroluminescence 2 WWW.NATURE.COM/NATURE
RESEARCH emission has to be assessed by analyzing the emission from the lower polariton branch. Compared to a purely parabolic dispersion with a light effective mass of ~(3.2±0.2) 10-5 m e (m e denotes the free electron mass), which is characteristic for a device operating in the weak coupling regime, the emission from the LP branch exhibits a larger effective mass of ~(4.2±0.2) 10-5 m e for 0 T in Fig. 2a and ~(3.9±0.2) 10-5 m e for 5 T in Fig. 2d. Due to the enhanced exciton-oscillator strength in the magnetic field, the Rabi-splitting increases slightly from (5.5±0.2) mev at 0 T to (6.0±0.2) mev 5 T in agreement with theory, as we assess below in this supplementary information material, and with literature 3. The exciton-photon detuning parameters of our polariton laser diode device were carefully extracted by studying a number of devices at differing positions (hence detunings) fabricated on the wafer. At first, the uncoupled exciton energy was determined by photoluminescence measurements after removing the top DBR of a sample piece, and the Rabi-splitting is extracted by position dependent reflexion measurements such as depicted in Fig. 1b in the main body text. The photon energies were then approximated in the high excitation regime of our diodes. Since it is well known that these values do not necessarily represent the exact photon energy due to the possible presence of a refractive index and according blueshift change in the cavity mediated lasing regime, they serve as a starting point of our fitting procedure. With these parameters, we are able to iteratively fit our polariton dispersions and accurately reproduce them, as seen in Fig. S3 for three different detunings. WWW.NATURE.COM/NATURE 3
RESEARCH SUPPLEMENTARY INFORMATION Figure S3. Energy-momentum dispersions of three polariton-microcavities recorded in the linear regime (in the left side of each panel) and the weak-coupling lasing regime (right side of each panel). Lower polariton occupancy Fig. S4 depicts the particle occupancy as a function of the energy. The occupancy was extracted by multiplying the measured luminescence intensity with the polariton lifetime at a given k-value and by the momentum space extension. In each spectrum, the occupancy was normalized to its maximum value for the sake of visibility. Fig. S4a depicts the extracted occupancies from the energy-momentum dispersion graphs shown in Fig. 2a-2c of the main text. Below the assigned polariton laser threshold, the occupancy of the excited states can be reproduced by a Maxwell-Boltzmann distribution with a temperature of 30K, whereas the ground state occupation at E=0 is clearly supressed. Above the assigned polariton laser threshold, an enhanced population of this polariton ground state can be observed, deviating from the Maxwell-Boltzmann distribution of a thermalized polariton gas. This strongly indicates, that polariton lasing is observed in this regime. In Fig. S4b, the occupancies are shown for the 5 T case. Here, a pronounced bottleneck ~ 1.7 mev above the weakly populated polariton ground state dominates the spectrum below threshold. Above the polariton laser threshold, this 4 WWW.NATURE.COM/NATURE
RESEARCH polariton ground state is strongly populated as expected for a polariton laser. Above the second threshold (right column), both graphs are dominated by a series of photonic resonances corresponding to the fundamental mode and the higher lateral modes of a circular micropillar with a diameter of 20 µm. Figure S4. Occupancy plotted as a function of the energy in the three different regimes of the device. The upper row a depicts the 0T case, whereas in b the 5 T case is plotted. The first column represents data below the assigned threshold for a polariton laser, the second column shows data recorded in the polariton laser regime and the third column depicts the behaviour in the cavity mediated lasing regime. Exciton Density Estimate We have estimated the exciton (electron-hole pair) density per QW at the two observed thresholds using the measured current values with the assumption that all injected carriers are captured in the four QWs. The estimated exciton density per QW at the first threshold current density j th = (77 ) A/cm² is jth rel = (6 10 9 1/cm 2 and the en QW WWW.NATURE.COM/NATURE 5
RESEARCH SUPPLEMENTARY INFORMATION photon dominated laser regime attributed second threshold (j/j th 2.5) occurs at a density of (1.5 ) 10 10 1/cm 2, where e is the elementary charge. In order to assess the relaxation time of the excitons which are scattered into polariton states, we have carried out streak camera measurements without an applied magnetic field on a 100 µm diameter, quasi-planar pillar from the same wafer under short pspulsed optical excitation (80 µm spot size) in a comparable exciton-cavity detuning regime of ~ -4 mev. Figure S2 shows representative time-resolved measurement data at different optical excitation powers. The relaxation time of ~180 ps in the sub-threshold regime significantly reduces with higher excitation power and converges on the order of 50 ps. Relaxation times on the order of 10-20 ps have been reported in a sample with GaAs quantum wells 3. Taking into account the slightly different QW species used in our device, these relaxation rates are in general agreement. In order to estimate the density values at the first and the second lasing threshold, we assumed a relaxation time of (50 ps. The relatively large error was chosen given the fact, that these data were acquired under optical pumping, whereas our polariton laser diode device was operated in the electrical injection mode. 6 WWW.NATURE.COM/NATURE
RESEARCH Figure S5. Time-dependent intensities of photoluminescence signals pumped by a pulsed Ti:Sa laser are plotted at P= 27 mw (a), 40 mw (b) and 74 mw (c) at 80 μm spot size. From the exponential fit to the decay time, the exciton lifetime is estimated as ~180 ps at 27 mw and is reduced to around 30-40 ps (31 ps at P/P th = 1 and 34 ps at P/P th = 1.87). Polariton diamagnetic shift By increasing the magnetic field from 0 T to 5 T, the LP energy experiences a 0.16 mev blueshift below j th1 (compared to the 0 T case) which is in agreement with its excitonic fraction of 13% and an independently measured diamagnetic shift of 1.7-1.8 mev determined for the uncoupled InGaAs QW exciton mode between 0 to 5 T. The obtained exciton shift corresponds to a diamagnetic coefficient κ X 7x10-5 ev/t², well agreeing with literature for such a QW composition 5. Magnetic field measurements Due to the magnetic field dependence of the polariton lasing threshold, one needs to adjust the injection current density for every magnetic field to explore the system for comparable conditions (i.e. comparable polariton densities). Namely, for each magnetic field, we determined the excitation current which delivers a characteristic blueshift of 60% between the polariton emission below the first threshold and the bare photon mode. WWW.NATURE.COM/NATURE 7
RESEARCH SUPPLEMENTARY INFORMATION Magnetic-field induced effects In this section, we briefly explain the magnetic field dependent increase of the Rabisplitting in our system. The dependency of the Rabi splitting on the exciton oscillator strength reads:, where is the electron mass and is the effective cavity length. The oscillator strength (in case of small and intermediate magnetic fields) can be written as a function of the Bohr radius:. Hence, in good approximation, we obtain:. The magnetic field - dependent Bohr radius can be found from the variational solution of the Schrödinger equation (minimization of the average energy of the QW exciton with respect to the Bohr radius), which results in the transcendental equation, [ ( ) ] ( ), where is the magnetic length, - the coordinate of the exciton center of mass, is the reduced effective mass, and the Coulomb potential and hole confined in the QW reads between the electron. 8 WWW.NATURE.COM/NATURE
RESEARCH Here and are -coordinates of the electron and hole, and the wave-functions can be approximated by the solutions of the 1D problem with the infinite-barrier QW, given that the confinement of the particles in the QW is sufficiently strong. Summarizing, we can estimate an increase of the Rabi-splitting from Ω R (0) = 5.5 mev at B = 0 T, to at B=5 T. References 1. Reitzenstein, S. et al. AlAs/GaAs micropillar cavities with quality factors exceeding 150000. Appl. Phys. Lett 90, 251109 (2007) 2. Kistner, K. et al. Demonstration of strong coupling via electro-optical tuning in high-quality QD-micropillar systems. Optics Express 16, 15006-15012 (2008). 3. Fisher, T.A. Afshar, A.M. Skolnick, M.S. & Whittaker, D.M. Vacuum Rabi coupling enhancement and Zeeman splitting in semiconductor quantum microcavity structures in a high magnetic field. Phys. Rev. B 53, 10469 (1996). 4. Tempel, J.S. et al. Characterization of two-threshold behaviour of the emission from a GaAs microcavity. Phys. Rev. B 85, 075218 (2012) 5. Rahimi-Iman, A. et al. Zeeman splitting and diamagnetic shift of spatially confined quantum-well exciton polaritons in an external magnetic field. Phys. Rev. B 84, 165325 (2011) WWW.NATURE.COM/NATURE 9