Transparency current (I_tr)

The transparency current $I_{\text{tr}}$ is the current at which the carrier density in a semiconductor laser's active region reaches the transparency density $N_{\text{tr}}$ (the density at which the material gain is zero). Below $N_{\text{tr}}$, the active region absorbs the lasing wavelength; above $N_{\text{tr}}$, it amplifies.

Transparency is a necessary but not sufficient condition for laser operation. The threshold current $I_{\text{th}}$ is always greater than $I_{\text{tr}}$. The active region must provide enough above-transparency gain to overcome the cavity losses (internal loss and mirror outcoupling):

$$g_{\text{th}}(I_{\text{th}})\cdot \mathrm{\Gamma }={\alpha }_i+{\alpha }_m,g_{\text{th}}>0.$$

The relationship can be inverted from the empirical $g(N)=g_0\ln (N\mathrm{/}{N}_{\text{tr}})$:

$$N_{\text{th}}=N_{\text{tr}}\exp \text{\hspace{0.17em}⁣}\left(\frac{{\alpha }_i+{\alpha }_m}{\mathrm{\Gamma }{g}_0}\right)\mathrm{.}$$

For typical telecom MQW lasers, $N_{\text{th}}\mathrm{/}{N}_{\text{tr}}$ is between 1.5 and 3. Because the recombination rate grows faster than linearly with carrier density, the ratio of threshold current to transparency current is larger than the carrier-density ratio.

Why transparency current is interesting.

Transparency current is the fundamental physics-limited lower bound on threshold:

• For zero-loss (idealized) cavity with ${\alpha }_i=0$ and $R=1$ (perfect reflection): $I_{\text{th}}\to I_{\text{tr}}$
• For real devices, the gap $I_{\text{th}}-I_{\text{tr}}$ measures the cavity-loss penalty

Comparing $I_{\text{tr}}$ between device generations isolates active-region material quality from cavity design quality. Long-cavity, low-loss devices have threshold carrier density approaching the material-limited $N_{\text{tr}}$; short cavities with low mirror reflectivity need much more gain, so their threshold sits well above transparency.

Typical transparency current densities (per unit active-region area; per quantum well for wells, per layer for bulk active regions and quantum dots):

These are representative ranges. Reported values depend on well width, strain and temperature, and on whether the internal loss was separated out in the extraction described below, so published figures for similar structures can differ by a factor of two or more.

For an $N$-well MQW: transparency density requires populating each well to $N_{\text{tr}}$, so $J_{\text{tr,total}}\approx N\cdot J_{\text{tr,per-well}}$.

Extraction. The usual method uses lasers of several cavity lengths $L$. Writing the gain as a function of current density, $g=g_{0J}\ln (J\mathrm{/}{J}_{\text{tr}})$, an empirical fit that does not need the current to be proportional to the carrier density, the threshold condition gives $J_{\text{th}}=J_{\text{tr}}\exp [({\alpha }_i+{\alpha }_m)\mathrm{/}(\mathrm{\Gamma }{g}_{0J})]$ with ${\alpha }_m=\ln (1\mathrm{/}R)\mathrm{/}L$, so $\ln J_{\text{th}}$ plotted against $1\mathrm{/}L$ is a straight line of slope $\ln (1\mathrm{/}R)\mathrm{/}(\mathrm{\Gamma }{g}_{0J})$. Its intercept at $1\mathrm{/}L=0$ is $\ln J_{\text{tr}}+{\alpha }_i\mathrm{/}(\mathrm{\Gamma }{g}_{0J})$, not $\ln J_{\text{tr}}$: the internal loss, found from the differential efficiency of the same lasers (the length method in internal quantum efficiency), has to be taken out before the intercept gives $J_{\text{tr}}$. The spectral alternative measures the net modal gain spectrum below threshold by the Hakki–Paoli method, from the fringe contrast of the ASE spectrum, and finds the current at which the net modal gain at the lasing wavelength equals $-{\alpha }_i$, where the material gain is zero; ${\alpha }_i$ is read from the long-wavelength tail of the same spectra, below the band edge, where the material neither absorbs nor amplifies.

References: L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), for the logarithmic gain model and the cavity-length method; B. W. Hakki and T. L. Paoli, "Gain spectra in GaAs double-heterostructure injection lasers," J. Appl. Phys. 46, 1299 (1975).