Threshold characteristics of semiconductor microdisk lasers

Whispering-gallery mode microdisk lasers using optical’ and electrical2 excitation have recently been demonstrated. The initial experiments* and results described here use continuous-wave optical excitation at liquid-nitrogen temperatures. The threshold characteristics of these microlasers are studied here as the small volume limit is approached where there is a single optical mode within the gain spectral region. We ftrst develop a model for the microdisk modes and measure photoluminescent spectra which yield mode Q values and spacings. This model along with the measured threshold characteristics allows us to estimate the fraction fi of the spontaneous emission that is emitted into the lasing mode. As this p parameter approaches its small volume limit of unity, the laser threshold pump powers are expected to decrease to very low values in the microwatt range and the nature of the threshold region is dramatically modified. In a first approximation, a microdisk laser may be viewed as a semiconductor disk of thickness L suspended in vacuum. In a slab with L &./3n, the lowest order transverse electric (TE) mode is dominant.3 The vacuum wavelength is il and n(w) is the frequency dependent bulk refractive index of the semiconductor material. Two dimensional propagation of the lowest order TE mode is described by an effective refractive index n,ff.4 Optical modes with trapped TE character within a dielectric disk may be approximated by solutions of the twodimensional Helmholtz equation ( V 2 + n&co2/2) $=O, where w =2?rc//z and c is the speed of light in vacuum. Because of the cylindrical symmetry it is convenient to choose, for a given mode, $=JM(x>eiMe, where JIK are Bessel functions of the tirst kind, x=n& w)wr/c, and r, 0 are polar coordinates. Radial mode numbers N= 1,2... are assigned where the corresponding mode frequencies, 0M.N are in ascending order for fixed M. Cylindrical symmetry specifies WM,,N=ir)-MN so that a twofold degeneracy exists for M>O. We abbreviate oM,i =w~. Boundary conditions are needed to determine the mode frequencies. Here we set $=O at the disk radius R, so that neff(m)Rdc=xg), i.e., the smallest positive root of J&x$‘) =O. Microwave frequency measurements of similar modes in sapphire disks show that this is a good approximation.’ Quality factors6 Q, describing radiation loss from an ideal disk for the N= 1 modes are expected, according to tunneling arguments, ’ to vary approximately exponentially, Q,,, = be2MJ, where J=tanh-‘(s) -s, and ‘S= . Comparison with exact results’ for a dielectric sphere of refractive index netf, for which identical tunneling arguments hold, leads to an estimate of the prefactor bl/7. In order to account for the relatively large dispersion4 of ner, the expression for Q, should include an additional multiplicative factor fg= 1 + (w/n,& (&&do). Numerically fg is near 1.5 1 assuming’ n(w) = [3.346 +0.333(&~-0.866 eV)] for the materials and disk dimensions used in the experiments. The frequency dependence of neff also results in a factor of l/f’ in the calculated mode spacing. Loss due to optical absorption in the cold cavity (pumping well below threshold) leads to Q;‘=aU 2rrn,s fg where a is the absorption coefficient obtained by a suitable spatial average. The observed Q values should be compared with those determined by Q-’ = Q;‘+ Q;‘+ Q; * , where Qi’ describes. losses due to surface roughness, etc. The total radiation losses are characterized by Qr where Q;‘=Q;‘+QZ:‘. Q, is typically limited by Qi to values less than 1000. Q, is a function of pump power and becomes a dominant limit to the Q value below threshold. Two microdisks are studied experimentally, a “small” disk with R = 1.1 ,um, and a “large” disk with R = 2.5 pm. Both disks haye designed L=O.15 pm (--/2/3n) and contain six 100 A thick In,,53Gac47As quantum wells separated by five 100 A thick InGaAsP barriers and enclosed by two 200 A thick InGaAsP barriers. The n,, for this conflguration4’8 is 2.35. The composition of the InGaAsP barrier material has a room temperature band gap Eg= 1.1 eV. The disk is supported by an InP pedestal of rhomboidal cross section. Microdisk photoluminescence and lasing spectra are measured by illuminating with a il=O.6328 pm HeNe laser that is focused to a waist wo=3.3(4.4)pm for the small (large) disk. Typical spectra for the broad photoluminescence and narrow laser radiation are shown in Fig. 1. Cold cavity Q values, &/A&, of 260( 500) are measured for the small(large) disk from the photoluminescent spectrum at low pump powers below threshold in the range where the dominant cavity resonance line [seen in Fig. 1 at & =1505(1553) nm for the small [large) disk] has a full width half-maximum value Ail,. At excitation levels well above threshold a second, shorter wavelength cavity mode appears in a few of the measured disks. As shown in the inset in Fig. 1, the measured mode spacing is 87 nm for the small disk and 37 nm for the large disk. Solutions to n,&w)wR/c=x$ for M= 1,2,... and N= 1 determine the dominant high Q mode frequencies We . Calculations of these mode wavelengths in the spectral region near 1500 nm yield ;1(M=7) = 1482 nm and A(M=6) = 1593 nm for the small disk and ,l(rW = 19) = 1516 nm and il(M= 18) = 1562 nm for the large