Introduction to the quantum confined Stark effect

The modulators developed in this work rely for their operation upon a voltage dependent absorption effect, occurring within QWs, known as the quantum confined Stark effect [Miller et al. ‘85]. At zero applied electric field the absorption spectrum of a quantum well (QW), assuming no cavity effects, comprises a number of peaks due to the formation of excitons (electron-hole pairs). An underlying staircase structure is due to the density of states (DOS) function of the QW. The QWs energy levels, dependent upon the well and barrier composition and width, dictate the presence and position (in wavelength) of the peaks within the absorption spectrum. Selection rules, which relax under the application of a field, further modify the absorption spectrum such that only ‘allowed’ transitions are observed.

Importantly the absorption peaks are modified by broadening mechanisms. Essentially, the absorption peaks from a set of ideal QWs are reduced (in magnitude) if the QWs contain defects. These defects may be in the form of QW well-width or composition fluctuations. Furthermore, within strained QWs any degree of relaxation will also reduce the absorption. Finally the presence of phonons (or other interacting particles, for example free carriers) will also reduce the peak absorption.

Zero bias Reverse bias

el-

hhl; s:-hhl

Figure 5.7 Schematic band structure o f quantum well, shown without

and with an applied electric field to demonstrate the QCSE..

It is interesting to note that the linewidth of a given peak, dependent upon the broadening mechanisms present, scales (inversely) with the absorption such that the area, within the

absorption envelope, remains constant [Miller et al. ‘8 6]. Thus, by theoretically calculating the area under a given peak and actually measuring the linewidth of that peak we may determine the magnitude of the absorption, this method is used when measured data is not available.

The absorption peak (transition) of greatest interest to us is that due to the exciton associated with the el-hhl QW levels. In high quality QW structures this exciton is generally, well resolved. This feature aids considerably within the design process. Figure 5.7 schematically shows the el and hhl levels within a QW. In the unbiased case the wavefunctions for the electrons and holes are centred within the QW and a strong absorption peak results (corresponding to a large spatial overlap between the electron and hole wavefunctions). However, when an electric field is applied, perpendicular to the plane of the QWs, the potential barriers are altered (skewed) and the electron and hole wavefunctions become spatially polarised. The energy levels within the QWs also move to lower (closer) energies. The result, corresponding to a reduced overlap of the electron and hole wavefunctions, is a reduction in the absorption peak and an attendant shift in its wavelength. Collectively these two effects are called the quantum confined Stark effect (QCSE).

30000 0.06 OV _{« 0.04 : 4V 6V} 25000 2V _8V . 2 0 . 0 2 : I 0.00 ! . lOV 20000 - 2V 2 -0.02 : a 15000 -■ _,4V k -0.04 : ,6V 1 0 0 0 0 - _{-0.06 :} 8V 3 -0.08 : u -0 . 1 0 : 5000 - lOV -0.12 820 840 860 880 _{800 820 840 860 880 900} Wavelength (nm) Wavelength (nm)

Figure 5.8 Absorption and refractive index change fo r GaAs QWs. Absorption curves are derived from photocurrent measurements.

Figure 5.8a shows the absorption curves, with applied bias, for an 85Â GaAs QW, with 60Â Alo.slGaoôçAs barriers. The curves are inferred from photocurrent measurements of a PIN device containing 50 QWs, the intrinsic region width is buffered to be -Ipm. The unbiased spectrum clearly shows a well defined exciton (highest peak) attributed to the el-hhl resonance, the second peak is the el-lhl exciton (other peaks are higher order excitons). Under the application of a bias the spectrum clearly exhibits the QCSE. The el-hhl peak decreases in magnitude and also shifts to lower energies (longer wavelengths). A shift exceeding 20nm is

observed with a corresponding decrease in absorption of approximately 2 0 , 0 0 0 cm'^ (for the QW). It is important to remember that within a device these absorption values will be reduced as the QW barrier must be taken into account.

Two regions of interest are apparent within figure 5.8a, marked and respectively. At the wavelength the QW absorption can be seen to decrease with application of a field. At this wavelength there is an attendant background (continuum) absorption of ~10,000cm'\ At X<z the reverse happens and an increasing field increases the absorption, from a low initial value. It is noted that both of these wavelengths may thus be modulated, although the device characteristics will be different in terms of insertion loss, modulation depth and chirp.

Accompanying the changes in the absorption, the imaginary part of the refractive index, of the QWs (figure 5.8a) are changes in the real part of the refractive index. The magnitude of these changes may be calculated by application of the Kramers-Kronig transformation. This relates changes in the absorption spectrum of a structure to changes in the real refractive indices, it is given in equation 5.4.1 [Weiner et al. ‘87, see also Yariv ‘89].

' “ lower ' ' ' '

In this equation An represents the absolute change in refractive index at angular frequency CO (wavelength X), Aa is the QCSE induced absorption change, co‘ is the scanning variable and coiower and cOupper are the limits of the integration. Given the absorption spectrum over all wavelengths it would, in principal, be possible to calculate the absolute value of the real refractive index of the QWs. Unfortunately limits must be imposed, for practical reasons, upon the wavelength range of the absorption spectrum. Thus only changes in the refractive index may be calculated. In practise the absorption changes for a QW, with bias, are negligible as we move, in wavelength, far away from the el-hhl exciton. It is therefore reasonable to use these distant wavelengths as the integration limits.

Figure 5.8b shows the changes in refractive index (per QW) accompanying the QCSE induced absorption changes of figure 5.8a. Large index changes, of order 0.1, can be seen to occur at wavelengths close to the unbiased el-hhl exciton wavelength. Much smaller changes occur at longer wavelengths. These changes in refractive index will lead to changes in the phase of light interacting with the QWs and this will lead, in both active and passive devices, to a chirping (shift) of the operational wavelength. Importantly the amount of chirp is dependent upon the absorption change occurring within the QW, this is itself dependent upon the QW structure (composition and material system).