Structural analysis of semiconductor Bragg mirrors by acoustic phonon spectroscopyacoustic phonon spectroscopy

As described in the previous sections the phonon dispersion relation of a superlattice is influenced by the thicknesses of the single layers the superlattice consists of and the superlattice period. Hence, by comparing the detected acoustic modes with calculated modes information about the structure of the sample can be obtained.

By this technique thickness variations of a Si/Mo superlattice wafer were mapped [Geb10] and in [Bru12] the layers thicknesses of a complex quantum cascade laser structure could be determined.

However, the special feature about the samples we investigated is the long periodicity of the superlattice. Since the Bragg mirrors are optimized for near infrared, the thick-nesses of the AlAs and GaAs layers are in the range of 70-90 nm whereas the samples in the publications given above consist of layers with thicknesses in the range of just a few nm. The large superlattice period of the Bragg mirrors in the range of 150 nm leads to a small mini-Brillouin zone. Due to the Umklapp process at the edge of the mini-Brillouin zone the dispersion relation of a long-periodic superlattice exhibits more

6.1 Structural analysis by coherent acoustic phonon spectroscopy

Sample Manufacturer by optical spectroscopy by X-ray spectroscopy

no RTA 158.2 nm 156.3 nm 155.9 nm

RTA at 450^◦C 156.3 nm 155.9 nm

RTA at 650^◦C 156.3 nm 155.9 nm

RTA at 700^◦C 156.3 nm 155.9 nm

Table 6.1: Superlattice constants of different annealed Bragg mirrors. The nominal thicknesses given by the manufacturer differ from the thicknesses extracted by coherent acoustic phonon spectroscopy (optical spectroscopy) and X-ray spectroscopy.

detectable modes within a given frequency range than a short-periodic superlattice.

Hence, coherent acoustic phonon measurements of long-periodic superlattices require an experimental setup with a high frequency resolution. We used a pump-probe setup based on the ASOPS technique, as it is described in Section 4.3.2. Information about the measured signal and the data analysis can be found in Section 4.4.5.

Accuracy of the structural analysis via phonon spectroscopy

On the basis of the acoustic phonon measurements of an un-annealed Bragg mirror the precision of the layer thickness determination via coherent acoustic phonon spec-troscopy is demonstrated.

Fig. 6.3 shows the original time domain signal as well as the Fourier transform of the extracted oscillations, respectively the detected frequency spectrum, and the calculated phonon dispersion relation for the un-annealed Bragg mirror. The spectrum exhibits frequency triplets according to the modes for q = 0 and q = 2q_probe. Due to the long superlattice period of the sample q = 2q_probe had to be calculated using Eq. (6.5) for the Umklapp process resulting in a vector q⁰ close to 0.5·π/d_SL, so that the detected frequencies seem nearly equidistant.

The dispersion of the superlattice is calculated with Eq. (6.3). Since calculation of the dispersion relation is more sensitive to the superlattice period than to the thicknesses of the single layers, mainly the superlattice period can be extracted by acoustic phonon spectroscopy rather than the single layer thicknesses. Thus, analogous to [Bru12] we adjusted the layer thicknesses in the calculations for best agreement between the deter-mined active modes and the detected frequencies, which was achieved with a uniform reduction of 1.2 % of the nominal layer thicknesses given by the manufacturer. This method leads to a superlattice period of 156.3 nm, which differs from the nominal su-perlattice period about -1.9 nm. All susu-perlattice constants are listed in Table 6.1.

Since we could observe modes according to very high orders of the dispersion relation (up to 21st order,) which exhibit a higher structural sensitivity than lower order modes, the superlattice period could be adjusted with a precision better than 1 nm.

To proof the accuracy of our structural analysis we also performed X-ray measurements on the Bragg mirrors. As it can be seen in Table 6.1 the single layer thicknesses ex-tracted by X-ray measurements differ from the nominal thicknesses and the thicknesses

020040060080010001200 0 0.05 0.1

Time delay (ps)

∆ R/R

0

(a)

300500700900 0 1 2 x 10 −3

20040060080010001200 −2 0 2 x 10 −4(b)

∆R/R

0

Time delay (ps)

0 0.5 1(c)

050100150200250300350 (d)

Frequency (GHz)

q (π/d

SL)

∆ R/R

FFT Spectr. Ampl. (a.u.) Figure6.3:Coherentacousticphononmeasurementofun-annealedsemiconductorBraggmirror.DataanalysisisexplainedindetailinSection4.4.5.(a)Timedomainsignalobtainedbypump-probespectroscopy.Intheinsettheoscillationsoriginatingfromcoherentacousticphononsarevisible.(b)Numericallyextractedoscillations.(c)DispersionrelationofthesemiconductorBraggmirrorcalculatedaccordingtoEq.(6.3)withdifferentvaluesforthelayerthicknesses:Nominalthicknesses(orange),valuesmeasuredbyX-rayanalysis(lightgreen),andvaluesforbestagreementwiththedetectedphononmodes(blue).Thehorizontallightbluelinemarksq 0.Detailsaregiveninthetext.(d)FrequencyspectrumoftheactivemodesobtainedbyFouriertransformofthetimedomainsignalshownin(a).Theactivemodeswithq=0aremarkedwithdarkgreendashedlines,thedetectedmodescorrespondingtoq 0areindicatedwithbluedottedlines.ForbettervisibilitytheFFTamplitudesforhigherfrequenciesareenlarged.

6.1 Structural analysis by coherent acoustic phonon spectroscopy

based on the phonon spectroscopy. However, the superlattice period obtained by X-ray analysis is 155.9 nm. Hence, this value deviates from the superlattice period extracted from phonon spectroscopy by just 0.4 nm, which is within the structural resolution of the data acquisition that is 1 nm. This agreement is absolutely remarkable.

These measurements show that structural analysis by phonon spectroscopy is an ad-equate alternative to X-ray measurements. Since the phonon spectroscopy is an all-optical method, it is non-invasive so that thicknesses or even thickness changes of the sample due to any treatment can be measured, e.g. structural changes of the Bragg mirror before and after its use in a laser resonator. Furthermore, due to the compact setup for phonon spectroscopy without any special conditions concerning the sample preparation structural analysis by phonon spectroscopy can be done in addition to other experiments, i.e. heating or cooling or further pumping. Even two dimensional mapping of sample can be realized, as it is shown in [Geb10].

Hence, the structural analysis of the Bragg mirrors demonstrated that (i) coherent acoustic phonons of a long-periodic semiconductor superlattice can be detected and (ii) that the superlattice period can be extracted with accuracy better than 1 nm.

Consequently, this method is a precise, non-invasive method for structural analysis of semiconductor superlattices with accuracy in the nanometer range.

Thickness extraction of different annealed Bragg mirrors

Based on the excellent agreement between the superlattice constant extracted by X-ray spectroscopy and coherent acoustic phonon spectroscopy, we also investigated the different annealed Bragg mirrors considering the question, how annealing changes the superlattice structure. The same steps of data acquisition, data analysis, and thickness extraction as done in the experiment presented in the previous paragraph were used.

The frequency spectra of the four different annealed Bragg mirrors are plotted in Fig.

6.4. Apart from minor differences in the amplitudes and the broadening of the FFT peaks all four spectra are very similar. The experimentally observed frequencies do not vary due to the annealing process. This result was expected, since Raman studies on short-periodic GaAs/AlAs superlattices did also not show any changes in the phonon frequencies due to annealing [Lev87].

Since the phonon frequencies are mainly determined by the superlattice period, the unchanged frequencies indicate a constant superlattice period. As shown in [Fat93]

the lattice parameter of GaAs nearly remains constant due to annealing. Hence, the period of the superlattice should change neither.

The phonon measurements we performed on the Bragg mirrors confirm these expec-tations. The fact that all four Bragg mirror samples show similar phonon modes in-dicates that the structures of the Bragg mirrors, especially the superlattice constants, do not change due to annealing. Independent of the post-growth annealing process we extracted a superlattice constant of 156.3 nm. As it can be seen in Table 6.1 an

0 0.5 1

0 50 100 150 200

Frequency (GHz)

q (π/d SL)

∆ R/R FFT Spectr. Ampl. (a.u.)

x 5 x 20 no RTA

450°C 650°C 700°C

Figure 6.4: Coherent acoustic phonon measurements of semiconductor Bragg mirrors annealed at different temperatures. Top: Calculated dispersion relation of a Bragg mirror based on the Rytov model. The layer thickness is optimized for best agreement with the experimentally obtained modes. The horizontal light blue line marks the backfolded wave vector of the probe beam q⁰. Details concerning q⁰ are given in the text. Bottom: Fourier transform of coherent acoustic phonon measurements of different annealed Bragg mirrors. For reasons of clarity the amplitude of the spectra at higher frequencies are multiplied by the factors given in the figure. The vertical blue dotted lines indicate the modes corresponding to q =q⁰, the vertical dark green dashed lines correspond to modes with q= 0.

unchanged superlattice constant for all four samples was also confirmed in X-ray mea-surements. These results are a further confirmation for the high precision of thickness extraction by means of coherent acoustic phonon spectroscopy.

Based on the above mentioned experiment we can conclude that post-growth annealing of up to 750^◦C does not change the structure of semiconductor Bragg mirrors. As a further consequence high temperatures due to the exposure of the Bragg mirrors in a laser cavity should also not affect its structural properties. Thus, any changes in the behavior of the Bragg mirror due to annealing processes may not result from changes of the lattice constant. However, the structural analysis based on optical spectroscopy and X-ray measurements do not show any changes in the interface roughness or in-homogeneities of the layer structures. These effects may occur due to annealing and can change the properties of the Bragg mirror and the devices the Bragg mirror is incorporated in.