- Email: [email protected]
Statistical analysis of micro-luminescence maps of implanted optical centers in diamond
Journal of Luminescence 131 (2011) 489–493
Contents lists available at ScienceDirect
Journal of Luminescence journal homepage: www.elsevier.com/locate/jlumin
Statistical analysis of micro-luminescence maps of implanted optical centers in diamond Y. Deshko a, Mengbing Huang b, A.A. Gorokhovsky a,n a b
The College of Staten Island and The Graduate Center of CUNY, 2800 Victory Boulevard, Staten Island, NY 10314, USA The University at Albany, The State University of New York, 1400 Washington Avenue, Albany, NY 12222, USA
a r t i c l e in f o
abstract
Available online 3 November 2010
We report on an approach to determine the number of optical emitters based on establishing the proper relationship between the statistics of implanted ions and the signal from optical centers (e.g. photoluminescence) collected at different points in the sample. Knowing the number of implanted ions, one can estimate the probability for an ion to create an optical center—the conversion efficiency. The micro-luminescence mapping and statistical analyses were performed on a model Xe optical center in diamond and the conversion efficiency was estimated to be about 30%. & 2010 Elsevier B.V. All rights reserved.
Keywords: Ion implantation Diamond Photoluminescence Compound Poisson
1. Introduction One of the techniques to create optical centers in solids is ion implantation followed by thermal annealing. This method gives control over the accuracy of the spatial distribution of implanted ions as well as the dose of implantation and the type of the implanted ion. Due to a variety of combinations of host materials and implanted ions, the search for optical centers with suitable properties is in progress, in particular, for creating single optical centers in diamond for quantum optics applications [1]. One of the important parameters to characterize the creation of optical centers by ion implantation is conversion efficiency—the probability to form optical centers as a result of implantation and thermal annealing. It is defined as the ratio of the density of emitting centers Nemit to the total density of implanted ions Nimpl: q¼Nemit/Nimpl. There are two basic ways to measure this parameter: direct counting of single emitting centers and collecting the signal from an ensemble of such centers. The former method, while being conceptually easy, is hard to use in most experiments, since it requires a signal-collection system with high spatial resolution as well as a controlled implantation process that allows one to implant a fixed number of ions [2]. To use the signal from an ensemble of optical centers, one has to establish the connection between the statistics of the signal (e.g. photoluminescence) and the statistics of the emitting centers. In this paper, we discuss an approach to relate the statistics of the signal and the number of emitting centers. The relationship found is non-trivial and, as shown further in the work, is determined by the experimental conditions. We apply this statistical method to estimate
n
Corresponding author. Tel.: + 1 718 982 2815; fax: + 1 718 982 2830. E-mail address: [email protected] (A.A. Gorokhovsky).
0022-2313/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2010.10.039
the conversion efficiency of the model center—Xe optical center in diamond—one of the few centers having sharp emission lines in the infrared spectral region, specifically at 813 and 794 nm.
2. Theoretical background Experimental conditions, described in Section 3, allow us to use the following model assumptions. All optical centers are positioned in a plane close to the sample surface with surface density r0 and their spatial distribution is uniformly random and follows Poisson distribution. We will discuss two types of laser excitation: an easy to understand case of ‘‘flat-top’’ beam and a more realistic Gaussian 2 2 beam with the intensity distribution I ¼ I0 er =w . The photoluminescence signal is collected from a circular-shaped ‘‘useful’’ area with radius R. The radius of the useful area R can be written in terms of the effective size of the laser beam w as R¼kw, where k is the dimensionless size of the useful area. The optical saturation is taken into account by means of the intensity dependent absorption coefficient:
aðIÞ ¼
a0 1 þ I=Is
,
ð1Þ
where Is is the saturation intensity and a0 the unsaturated absorption coefficient. The expression (1) actually describes saturation for a twolevel system [3]. In our consideration it will be used for an arbitrary system as long as it gives a reasonable approximation of the experimental results. The signal from the useful area around a given point on the sample is given by S¼
N X i
Xi ,
ð2Þ
490
Y. Deshko et al. / Journal of Luminescence 131 (2011) 489–493
where Xi is the signal from an individual center and N the number (random) of optical centers inside the useful area. The total signal S is a random quantity described by the compound Poisson distribution with the following properties [4]:
s2S ¼ N:X 2 :
S ¼ N:X,
ð3Þ
The difference between regular and compound Poisson is in the expression for the variance s2S . For the former, the variance is proportional to the square of the average signal, while for the compound Poisson it is proportional to the average of the squared signal. The average number of optical centers is therefore 2
S
N¼b
s2S
,
b
X2 X
ð4Þ
2
2.1. Flat-top excitation 2
Flat-top excitation is the only case when X equals X 2 , and therefore, b ¼1. In this case the total signal is just the product of the number of emitters inside the laser spot and the signal from one emitter: S¼ NS1. The connection between the statistics of the optical centers and the signal is linear. Since the former is assumed to be Poissonian, s2S ¼NS21, one can find the average number of emitters inside the useful area and the signal from a single emitter: N¼
S
12
2
s2S
,
S1 ¼ s2S S:
ð5Þ
2.2. Gaussian excitation In the case of Gaussian excitation (see Fig. 1b), the total signal from an ensemble of emitters is given by the sum of the signal from single emitters: N X
10 8
Comparing Eqs. (4) and (5), one can see that b has the meaning of the correction factor, which upgrades the Poisson distribution of the total signal from regular to compound. The signal from a single center S1 depends on the excitation power as well as on the quantum yield of luminescence and the sensitivity of detection. In contrast, the average number of optical centers does not depend on the quantum yield as long as it is the same for all centers. What really matters is whether the optical center emits with detectable intensity, regardless of its quantum yield.
S¼
14
2
2
S1 eri =w :
ð6Þ
i
Here S1 is the signal from the single emitter, when it is positioned in the center of the laser spot. The absorption coefficient 2 2 for the Gaussian excitation I ¼ I0 er =w can be written as
aðr, gÞ ¼
a0 1þ ger2 =w2
:
6 4 2 0 0
1
2
3
4
5
Fig. 1. Intensity distribution in the laser spot is shown for case of (a) flat-top laser beam that corresponds to the Poissonian statistics of the total signal and (b) Gaussian laser beam that corresponds to the compound Poissonian statistics of the signal. The correction factor between both cases is shown in (c) as a function of the dimensionless focal spot size k.
saturations: low g ¼0.01, medium g ¼1.0 and high g ¼100. As can be seen, the larger the saturation parameter, the closer the flat-top case. Two saturation limits can be recovered from Eq. (8): (i) g-N, flat-top case 2 with b ¼1 and (ii) g-0, non-saturated case with 2 b ¼ k2 coth k2 . For medium and low saturation levels the correction factor depends on the size of the useful area k; therefore we need a criterion for its determination.
ð7Þ
The dimensionless ratio g I0/Is is the saturation parameter. A special feature of Gaussian excitation is the absence of complete saturation of the total signal. The signal grows as ln(1 + g) due to the increase in size of the useful area. This is different from flat-top excitation, when the signal saturates as g/(1+ g). Calculating the average of the signal and the average of the squared signal, the expression for the correction factor can be presented in the following form: 2 2 2 ln ð1þ gÞ=ð1 þ gek Þ g 1ek =ð1 þ gÞ 1 þ gek 2 : ð8Þ b¼k 2 ln ð1þ gÞ=ð1 þ gek2 Þ The behavior of the correction factor as a function of dimensionless radius of the useful area k is shown in Fig. 1c for three levels of
2.3. Useful area size and conversion efficiency One possible way to choose k is based on the ratio of the signal from the optical centers to noise. The sample surface around the focal spot can be divided into the inside of the circle of radius R¼kw (useful area), and the outside. Since the intensity in the Gaussian beam quickly decreases with r, the signal from the outside of the circle is a small fraction of the total signal: 2
Sout ¼ S0 lnð1 þ gek Þ,
ð9aÞ
Stotal ¼ S0 lnð1 þ gÞ:
ð9bÞ
We have neglected Sout since it is equal to or less than the noise. It may even be totally lost if the spectra are smoothed to remove the noise. If SNR denotes the signal to noise ratio, the expression for
Y. Deshko et al. / Journal of Luminescence 131 (2011) 489–493
491
dimensionless radius of the useful area k is given by the formula ! k2 ¼ ln
g
ð1 þ gÞ1=SNR 1
:
ð10Þ
As mentioned above, the size of the useful area grows as the saturation level increases. To estimate the conversion efficiency, one has to calculate the average number of emitting centers inside the useful area Nemit, and the average number of implanted ions Nimpl determined by the implantation dose r0 and the size of the useful area. The expression for the quantum efficiency is then given by 2
q¼
bðS =s2S Þ Nemit ¼ : Nimpl pk2 w2 r0
ð11Þ
Experimentally, one has to establish the size of the focal spot w, the saturation parameter g and collect the signal at different points on the sample to determine the statistics of the total signal. In the discussed model, we assumed that all centers are equivalent with respect to the excitation conditions (i.e. polarization). This is consistent with our experiment (see Section 3.1). In the case of two or more non-equivalent groups of emitters (i.e. due to non-equivalent orientations), the total signal S is the sum of the signals from different groups of centers each one being described by the compound Poisson. In this case, there is no simple relationship between the statistics of the total signal and the average numbers of emitters. Nevertheless, if the population and the signal between centers in each group are linearly related and known, the model can be reduced to the all equivalent centers description.
3. Experimental The model presented above has been applied to Xe + ion implanted diamond. Our interest in the Xe center originated from the fact that this center is one of a few (Ni, Si, Cr, Xe) centers in diamond having sharp emission lines in the infrared spectral region. At low temperatures, the photoluminescence spectra features the single zero phonon lines (ZPL) at 811.7 nm and a weak phonon sideband. Room temperature luminescence consists of a zero phonon line at 813 nm and a weaker line at 794 nm (see Fig. 4a). Our previous studies proved that vacancies are involved in the formation of this defect [5], it contains a single Xe ion [6] and is a /1 1 1S oriented defect [7]. These results are in agreement with the previous calculations [8], which concluded that due to its large size, the Xe ion generates stresses and yields a stable configuration in the semi-divancy site V–Xe–V. 3.1. Sample and experimental setup The sample studied was a single crystal CVD film grown along the (0 0 1) crystallographic plane and implanted with Xe + with energy 180 keV at doses of 1010, 1011 and 5 1012 ion/cm2. Following Xe + implantation, annealing at 1400 1C was applied for 1 h. SRIM simulation [9] shows that the Xe centers form a thin (9.6 nm) layer close to the surface of the sample. Spectra were collected in a backscattering geometry (Fig. 2) using a Horiba-JY T64000 spectrometer equipped with a confocal microscope and the cover glass corrected objective 60, N.A. ¼0.7, a XYZ scanning stage, a half-wave plate to control polarization of the incident light at 514.5 nm and a LiN2 cooled CCD detector. In the cubic diamond lattice there are four equivalent /1 1 1S directions; therefore the crystal and laser light polarization were oriented in a way such that Xe centers along each equivalent direction were excited uniformly. To increase the signal/noise ratio, the sample was placed in an optical cryostat and cooled to T¼80 K.
Fig. 2. Experimental setup for micro-luminescence mapping.
3.2. Focal spot size and saturation parameter To measure the size of the focal spot w, we used the knife-edge method [10] based on Raman signal intensity variation during scanning across the border between graphitic and diamond phases of the sample. These two phases have different Raman spectra (see Fig. 3a): a graphite-like phase shows broad band centered at 1380 and 1600 cm 1, while diamond exhibits a sharp line at 1331 cm 1. A thin layer of graphite-like phase was introduced in the diamond sample by irradiation with a high dose of implantation. For a Gaussian beam, the intensity of the 1600 cm 1 band during linear scanning changes according to the expression Imax xX0 1erf , ð12Þ I1600 ðxÞ ¼ 2 w where x is the coordinate of the center of the laser beam and X0 the position of the border. One can find w by the fitting formula (12) to the experimental data (Fig. 3b). The measurements revealed a slight ellipticity of the focal spot, with the approximate values of wx ¼0.4770.04 mm and wy ¼0.3370.04 mm. The 20% error in the determination of the product wx and wy turned out to be one of the major sources of uncertainty in the conversion efficiency. Luminescence spectra of our sample are shown in Fig. 4a. The saturation parameter g was determined from the dependence of 811.7 nm line intensity on laser power Il. The measured signal together with its fit according to formula (9b) is shown in Fig. 4b. The saturation parameter was found to be g ¼0.6Il, it varied in the experiments from 0.47 up to 60.
3.3. Micro-luminescence maps, results and discussion Having determined w and g, we performed a series of mappings. A single mapping implies measurements of Xe center photoluminescence spectra at different points on the sample’s surface, arranged in a periodic 2D array. A regular map consists of 20 20 points, separated by 2–4 mm from each other. The representative map is shown in Fig. 5a and the respective histogram is shown in Fig. 5b. Each pixel is visualized with pseudo-color, corresponding to the ZPL intensity at 811.7 nm. After accumulation of the map, the statistics of the peak’s intensity was extracted and analyzed; the average of the signal, its variance and the signal to noise ratio were determined. The correction factor b was calculated and the average number of 2 emitting centers was estimated as N emit ¼ b sS 2 . The number of S
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Y. Deshko et al. / Journal of Luminescence 131 (2011) 489–493
4000 T = 80 K
3500
Intensity, c/s
Intensity, c/s
40 30
2000 1500
500
10
0 785 790 795 800 805 810 815 820 825 830 Wavelength, nm
0 1200
1300
1400
1500
1600
1700
1800
Wavenumber, 1/cm
600
30
500
PL signal, c/s
25 20 I1600
2500
1000
20
15 10 5
400 300 200 100
0
0 30
31
32
33
34
Fig. 3. (a) Raman spectrum of graphitic phase with bands at 1380 and 1600 cm 1 and of pure diamond with the sharp line at 1332 cm 1 and (b) variation of 1600 cm 1 band intensity during a linear scan across the border between two phases. Experimental data and fit to formula (12) are shown.
implanted ions is given by Nimpl ¼ pk2wxwyr0; thus one can estimate the conversion efficiency q¼Nemit/Nimpl. Results are shown in the table below for three accumulated maps measured under different conditions. The signal to noise ratio was of the order 102. Based on the data gathered from these and other mappings, we conclude that the conversion efficiency is close to 0.30 with 20% relative error.
q0 (1/cm2) S (c/s) r (c/s) c 1010 1010 1011
T = 300 K
3000
50
92.5 218 43.8
12 24.5 2.52
k 7.8 2.49 60 2.73 0.96 2.21
Nest 1.71 102 1.4 111 2.01 607
b
Nimpl 302 363 2380
q 0.34 0.31 0.26
Our analysis shows that the parameters w and k are the main sources of uncertainty in q. To increase accuracy, one has to use a method that is less sensitive to these parameters. One possible way to accomplish this is to ‘‘force’’ the size of the useful area. In a confocal microscope, this can be done by changing the size of the confocal aperture. Making it comparable to the size of the laser spot image in the aperture plane and working at high saturation levels is
0
25
50 Il, a. u.
75
100
Fig. 4. (a) Photoluminescence spectra of the Xe optical center in diamond (implantation dose of 5 1012 ion/cm2) at T¼300 K (top) and 80 K (bottom) and (b) intensity of the 811.7 nm luminescence line as a function of power of the exciting laser beam Il. The experimental data is fitted with Eq. (9b); Il ¼100 a.u. corresponds approximately to 4 mW.
likely to be a more accurate way to measure conversion efficiency. However, in this approach one has to take into account diffraction effects.
4. Summary We proposed a method for the determination of the probability of implanted ions to form optical centers. The method is based on statistical analysis of the measured signal from many ensembles of emitters. The method is applicable to any type of optical center and is independent of their quantum yield. The analysis revealed that the method is rather sensitive to errors in the size of the focal spot of the laser used to excite the centers, and requires an unambiguous criterion for choosing the useful (working) area. The approach was applied to Xe optical centers in a diamond crystal, and the conversion efficiency was estimated as 0.30 with 20% relative error. An alternative more accurate approach, based on the ability through confocal aperture to change the useful area of the sample, is in progress.
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493
Acknowledgments The authors thank Dr. A. Zaitsev for thermal annealing of the samples and for useful discussions. This work was supported in part by PSC/CUNY. References [1] F. Jelezko, J. Wrachtrup, Phys. Status Solidi A 203 (2006) 3207. [2] J. Meijer, B. Burchard, M. Domhan, C. Wittmann, T. Gaebel, I. Popa, F. Jelezko, J. Wrachtrup, J. Appl. Phys. 87 (2005) 261909. ¨ [3] W. Demtroder, Laser Spectroscopy, Springer, 1998 Section 3.6.1. [4] E.H. Lloyd, Probability, Handbook of Applicable Mathematics, John Wiley & Sons, 1980 Chapter 14. [5] V.A. Martinovich, A.V. Turukhin, A.M. Zaitsev, A.A. Gorokhovsky, J. Lumin. 102–103 (2003) 785. [6] A.A. Bergman, A.M. Zaitsev, Mengbing Huang, A.A. Gorokhovsky, J. Lumin. 129 (2009) 1524. [7] A.A. Bergman, A.M. Zaitsev, A.A. Gorokhovsky, J. Lumin. 125 (2007) 92. [8] A.B. Anderson, E.J. Grantscharova, Phys. Rev. B 54 (1996) 14341. [9] L.G. Jacobson, J.-K. Lee, B.L. Bennett, R.E. Muenchausen, M. Nastasi D.W. Cooke, J. Lumin. 124 (2007) 5–9. [10] A.H. Firester, M.E. Heller, P. Sheng, Appl. Opt. 16 (1977) 1971.
Fig. 5. (a) Map of luminescence signal of ZPL at 811.7 nm from a 40 40 mm2 (20 20 points) region at T¼ 80 K and (b) histogram of the peak luminescence signal for the map (a). Statistical analysis of the map gives the average intensity S¼ 218.1 c/s and standard deviation sS ¼24.5 c/s.
Contents lists available at ScienceDirect
Journal of Luminescence journal homepage: www.elsevier.com/locate/jlumin
Statistical analysis of micro-luminescence maps of implanted optical centers in diamond Y. Deshko a, Mengbing Huang b, A.A. Gorokhovsky a,n a b
The College of Staten Island and The Graduate Center of CUNY, 2800 Victory Boulevard, Staten Island, NY 10314, USA The University at Albany, The State University of New York, 1400 Washington Avenue, Albany, NY 12222, USA
a r t i c l e in f o
abstract
Available online 3 November 2010
We report on an approach to determine the number of optical emitters based on establishing the proper relationship between the statistics of implanted ions and the signal from optical centers (e.g. photoluminescence) collected at different points in the sample. Knowing the number of implanted ions, one can estimate the probability for an ion to create an optical center—the conversion efficiency. The micro-luminescence mapping and statistical analyses were performed on a model Xe optical center in diamond and the conversion efficiency was estimated to be about 30%. & 2010 Elsevier B.V. All rights reserved.
Keywords: Ion implantation Diamond Photoluminescence Compound Poisson
1. Introduction One of the techniques to create optical centers in solids is ion implantation followed by thermal annealing. This method gives control over the accuracy of the spatial distribution of implanted ions as well as the dose of implantation and the type of the implanted ion. Due to a variety of combinations of host materials and implanted ions, the search for optical centers with suitable properties is in progress, in particular, for creating single optical centers in diamond for quantum optics applications [1]. One of the important parameters to characterize the creation of optical centers by ion implantation is conversion efficiency—the probability to form optical centers as a result of implantation and thermal annealing. It is defined as the ratio of the density of emitting centers Nemit to the total density of implanted ions Nimpl: q¼Nemit/Nimpl. There are two basic ways to measure this parameter: direct counting of single emitting centers and collecting the signal from an ensemble of such centers. The former method, while being conceptually easy, is hard to use in most experiments, since it requires a signal-collection system with high spatial resolution as well as a controlled implantation process that allows one to implant a fixed number of ions [2]. To use the signal from an ensemble of optical centers, one has to establish the connection between the statistics of the signal (e.g. photoluminescence) and the statistics of the emitting centers. In this paper, we discuss an approach to relate the statistics of the signal and the number of emitting centers. The relationship found is non-trivial and, as shown further in the work, is determined by the experimental conditions. We apply this statistical method to estimate
n
Corresponding author. Tel.: + 1 718 982 2815; fax: + 1 718 982 2830. E-mail address: [email protected] (A.A. Gorokhovsky).
0022-2313/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jlumin.2010.10.039
the conversion efficiency of the model center—Xe optical center in diamond—one of the few centers having sharp emission lines in the infrared spectral region, specifically at 813 and 794 nm.
2. Theoretical background Experimental conditions, described in Section 3, allow us to use the following model assumptions. All optical centers are positioned in a plane close to the sample surface with surface density r0 and their spatial distribution is uniformly random and follows Poisson distribution. We will discuss two types of laser excitation: an easy to understand case of ‘‘flat-top’’ beam and a more realistic Gaussian 2 2 beam with the intensity distribution I ¼ I0 er =w . The photoluminescence signal is collected from a circular-shaped ‘‘useful’’ area with radius R. The radius of the useful area R can be written in terms of the effective size of the laser beam w as R¼kw, where k is the dimensionless size of the useful area. The optical saturation is taken into account by means of the intensity dependent absorption coefficient:
aðIÞ ¼
a0 1 þ I=Is
,
ð1Þ
where Is is the saturation intensity and a0 the unsaturated absorption coefficient. The expression (1) actually describes saturation for a twolevel system [3]. In our consideration it will be used for an arbitrary system as long as it gives a reasonable approximation of the experimental results. The signal from the useful area around a given point on the sample is given by S¼
N X i
Xi ,
ð2Þ
490
Y. Deshko et al. / Journal of Luminescence 131 (2011) 489–493
where Xi is the signal from an individual center and N the number (random) of optical centers inside the useful area. The total signal S is a random quantity described by the compound Poisson distribution with the following properties [4]:
s2S ¼ N:X 2 :
S ¼ N:X,
ð3Þ
The difference between regular and compound Poisson is in the expression for the variance s2S . For the former, the variance is proportional to the square of the average signal, while for the compound Poisson it is proportional to the average of the squared signal. The average number of optical centers is therefore 2
S
N¼b
s2S
,
b
X2 X
ð4Þ
2
2.1. Flat-top excitation 2
Flat-top excitation is the only case when X equals X 2 , and therefore, b ¼1. In this case the total signal is just the product of the number of emitters inside the laser spot and the signal from one emitter: S¼ NS1. The connection between the statistics of the optical centers and the signal is linear. Since the former is assumed to be Poissonian, s2S ¼NS21, one can find the average number of emitters inside the useful area and the signal from a single emitter: N¼
S
12
2
s2S
,
S1 ¼ s2S S:
ð5Þ
2.2. Gaussian excitation In the case of Gaussian excitation (see Fig. 1b), the total signal from an ensemble of emitters is given by the sum of the signal from single emitters: N X
10 8
Comparing Eqs. (4) and (5), one can see that b has the meaning of the correction factor, which upgrades the Poisson distribution of the total signal from regular to compound. The signal from a single center S1 depends on the excitation power as well as on the quantum yield of luminescence and the sensitivity of detection. In contrast, the average number of optical centers does not depend on the quantum yield as long as it is the same for all centers. What really matters is whether the optical center emits with detectable intensity, regardless of its quantum yield.
S¼
14
2
2
S1 eri =w :
ð6Þ
i
Here S1 is the signal from the single emitter, when it is positioned in the center of the laser spot. The absorption coefficient 2 2 for the Gaussian excitation I ¼ I0 er =w can be written as
aðr, gÞ ¼
a0 1þ ger2 =w2
:
6 4 2 0 0
1
2
3
4
5
Fig. 1. Intensity distribution in the laser spot is shown for case of (a) flat-top laser beam that corresponds to the Poissonian statistics of the total signal and (b) Gaussian laser beam that corresponds to the compound Poissonian statistics of the signal. The correction factor between both cases is shown in (c) as a function of the dimensionless focal spot size k.
saturations: low g ¼0.01, medium g ¼1.0 and high g ¼100. As can be seen, the larger the saturation parameter, the closer the flat-top case. Two saturation limits can be recovered from Eq. (8): (i) g-N, flat-top case 2 with b ¼1 and (ii) g-0, non-saturated case with 2 b ¼ k2 coth k2 . For medium and low saturation levels the correction factor depends on the size of the useful area k; therefore we need a criterion for its determination.
ð7Þ
The dimensionless ratio g I0/Is is the saturation parameter. A special feature of Gaussian excitation is the absence of complete saturation of the total signal. The signal grows as ln(1 + g) due to the increase in size of the useful area. This is different from flat-top excitation, when the signal saturates as g/(1+ g). Calculating the average of the signal and the average of the squared signal, the expression for the correction factor can be presented in the following form: 2 2 2 ln ð1þ gÞ=ð1 þ gek Þ g 1ek =ð1 þ gÞ 1 þ gek 2 : ð8Þ b¼k 2 ln ð1þ gÞ=ð1 þ gek2 Þ The behavior of the correction factor as a function of dimensionless radius of the useful area k is shown in Fig. 1c for three levels of
2.3. Useful area size and conversion efficiency One possible way to choose k is based on the ratio of the signal from the optical centers to noise. The sample surface around the focal spot can be divided into the inside of the circle of radius R¼kw (useful area), and the outside. Since the intensity in the Gaussian beam quickly decreases with r, the signal from the outside of the circle is a small fraction of the total signal: 2
Sout ¼ S0 lnð1 þ gek Þ,
ð9aÞ
Stotal ¼ S0 lnð1 þ gÞ:
ð9bÞ
We have neglected Sout since it is equal to or less than the noise. It may even be totally lost if the spectra are smoothed to remove the noise. If SNR denotes the signal to noise ratio, the expression for
Y. Deshko et al. / Journal of Luminescence 131 (2011) 489–493
491
dimensionless radius of the useful area k is given by the formula ! k2 ¼ ln
g
ð1 þ gÞ1=SNR 1
:
ð10Þ
As mentioned above, the size of the useful area grows as the saturation level increases. To estimate the conversion efficiency, one has to calculate the average number of emitting centers inside the useful area Nemit, and the average number of implanted ions Nimpl determined by the implantation dose r0 and the size of the useful area. The expression for the quantum efficiency is then given by 2
q¼
bðS =s2S Þ Nemit ¼ : Nimpl pk2 w2 r0
ð11Þ
Experimentally, one has to establish the size of the focal spot w, the saturation parameter g and collect the signal at different points on the sample to determine the statistics of the total signal. In the discussed model, we assumed that all centers are equivalent with respect to the excitation conditions (i.e. polarization). This is consistent with our experiment (see Section 3.1). In the case of two or more non-equivalent groups of emitters (i.e. due to non-equivalent orientations), the total signal S is the sum of the signals from different groups of centers each one being described by the compound Poisson. In this case, there is no simple relationship between the statistics of the total signal and the average numbers of emitters. Nevertheless, if the population and the signal between centers in each group are linearly related and known, the model can be reduced to the all equivalent centers description.
3. Experimental The model presented above has been applied to Xe + ion implanted diamond. Our interest in the Xe center originated from the fact that this center is one of a few (Ni, Si, Cr, Xe) centers in diamond having sharp emission lines in the infrared spectral region. At low temperatures, the photoluminescence spectra features the single zero phonon lines (ZPL) at 811.7 nm and a weak phonon sideband. Room temperature luminescence consists of a zero phonon line at 813 nm and a weaker line at 794 nm (see Fig. 4a). Our previous studies proved that vacancies are involved in the formation of this defect [5], it contains a single Xe ion [6] and is a /1 1 1S oriented defect [7]. These results are in agreement with the previous calculations [8], which concluded that due to its large size, the Xe ion generates stresses and yields a stable configuration in the semi-divancy site V–Xe–V. 3.1. Sample and experimental setup The sample studied was a single crystal CVD film grown along the (0 0 1) crystallographic plane and implanted with Xe + with energy 180 keV at doses of 1010, 1011 and 5 1012 ion/cm2. Following Xe + implantation, annealing at 1400 1C was applied for 1 h. SRIM simulation [9] shows that the Xe centers form a thin (9.6 nm) layer close to the surface of the sample. Spectra were collected in a backscattering geometry (Fig. 2) using a Horiba-JY T64000 spectrometer equipped with a confocal microscope and the cover glass corrected objective 60, N.A. ¼0.7, a XYZ scanning stage, a half-wave plate to control polarization of the incident light at 514.5 nm and a LiN2 cooled CCD detector. In the cubic diamond lattice there are four equivalent /1 1 1S directions; therefore the crystal and laser light polarization were oriented in a way such that Xe centers along each equivalent direction were excited uniformly. To increase the signal/noise ratio, the sample was placed in an optical cryostat and cooled to T¼80 K.
Fig. 2. Experimental setup for micro-luminescence mapping.
3.2. Focal spot size and saturation parameter To measure the size of the focal spot w, we used the knife-edge method [10] based on Raman signal intensity variation during scanning across the border between graphitic and diamond phases of the sample. These two phases have different Raman spectra (see Fig. 3a): a graphite-like phase shows broad band centered at 1380 and 1600 cm 1, while diamond exhibits a sharp line at 1331 cm 1. A thin layer of graphite-like phase was introduced in the diamond sample by irradiation with a high dose of implantation. For a Gaussian beam, the intensity of the 1600 cm 1 band during linear scanning changes according to the expression Imax xX0 1erf , ð12Þ I1600 ðxÞ ¼ 2 w where x is the coordinate of the center of the laser beam and X0 the position of the border. One can find w by the fitting formula (12) to the experimental data (Fig. 3b). The measurements revealed a slight ellipticity of the focal spot, with the approximate values of wx ¼0.4770.04 mm and wy ¼0.3370.04 mm. The 20% error in the determination of the product wx and wy turned out to be one of the major sources of uncertainty in the conversion efficiency. Luminescence spectra of our sample are shown in Fig. 4a. The saturation parameter g was determined from the dependence of 811.7 nm line intensity on laser power Il. The measured signal together with its fit according to formula (9b) is shown in Fig. 4b. The saturation parameter was found to be g ¼0.6Il, it varied in the experiments from 0.47 up to 60.
3.3. Micro-luminescence maps, results and discussion Having determined w and g, we performed a series of mappings. A single mapping implies measurements of Xe center photoluminescence spectra at different points on the sample’s surface, arranged in a periodic 2D array. A regular map consists of 20 20 points, separated by 2–4 mm from each other. The representative map is shown in Fig. 5a and the respective histogram is shown in Fig. 5b. Each pixel is visualized with pseudo-color, corresponding to the ZPL intensity at 811.7 nm. After accumulation of the map, the statistics of the peak’s intensity was extracted and analyzed; the average of the signal, its variance and the signal to noise ratio were determined. The correction factor b was calculated and the average number of 2 emitting centers was estimated as N emit ¼ b sS 2 . The number of S
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4000 T = 80 K
3500
Intensity, c/s
Intensity, c/s
40 30
2000 1500
500
10
0 785 790 795 800 805 810 815 820 825 830 Wavelength, nm
0 1200
1300
1400
1500
1600
1700
1800
Wavenumber, 1/cm
600
30
500
PL signal, c/s
25 20 I1600
2500
1000
20
15 10 5
400 300 200 100
0
0 30
31
32
33
34
Fig. 3. (a) Raman spectrum of graphitic phase with bands at 1380 and 1600 cm 1 and of pure diamond with the sharp line at 1332 cm 1 and (b) variation of 1600 cm 1 band intensity during a linear scan across the border between two phases. Experimental data and fit to formula (12) are shown.
implanted ions is given by Nimpl ¼ pk2wxwyr0; thus one can estimate the conversion efficiency q¼Nemit/Nimpl. Results are shown in the table below for three accumulated maps measured under different conditions. The signal to noise ratio was of the order 102. Based on the data gathered from these and other mappings, we conclude that the conversion efficiency is close to 0.30 with 20% relative error.
q0 (1/cm2) S (c/s) r (c/s) c 1010 1010 1011
T = 300 K
3000
50
92.5 218 43.8
12 24.5 2.52
k 7.8 2.49 60 2.73 0.96 2.21
Nest 1.71 102 1.4 111 2.01 607
b
Nimpl 302 363 2380
q 0.34 0.31 0.26
Our analysis shows that the parameters w and k are the main sources of uncertainty in q. To increase accuracy, one has to use a method that is less sensitive to these parameters. One possible way to accomplish this is to ‘‘force’’ the size of the useful area. In a confocal microscope, this can be done by changing the size of the confocal aperture. Making it comparable to the size of the laser spot image in the aperture plane and working at high saturation levels is
0
25
50 Il, a. u.
75
100
Fig. 4. (a) Photoluminescence spectra of the Xe optical center in diamond (implantation dose of 5 1012 ion/cm2) at T¼300 K (top) and 80 K (bottom) and (b) intensity of the 811.7 nm luminescence line as a function of power of the exciting laser beam Il. The experimental data is fitted with Eq. (9b); Il ¼100 a.u. corresponds approximately to 4 mW.
likely to be a more accurate way to measure conversion efficiency. However, in this approach one has to take into account diffraction effects.
4. Summary We proposed a method for the determination of the probability of implanted ions to form optical centers. The method is based on statistical analysis of the measured signal from many ensembles of emitters. The method is applicable to any type of optical center and is independent of their quantum yield. The analysis revealed that the method is rather sensitive to errors in the size of the focal spot of the laser used to excite the centers, and requires an unambiguous criterion for choosing the useful (working) area. The approach was applied to Xe optical centers in a diamond crystal, and the conversion efficiency was estimated as 0.30 with 20% relative error. An alternative more accurate approach, based on the ability through confocal aperture to change the useful area of the sample, is in progress.
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Acknowledgments The authors thank Dr. A. Zaitsev for thermal annealing of the samples and for useful discussions. This work was supported in part by PSC/CUNY. References [1] F. Jelezko, J. Wrachtrup, Phys. Status Solidi A 203 (2006) 3207. [2] J. Meijer, B. Burchard, M. Domhan, C. Wittmann, T. Gaebel, I. Popa, F. Jelezko, J. Wrachtrup, J. Appl. Phys. 87 (2005) 261909. ¨ [3] W. Demtroder, Laser Spectroscopy, Springer, 1998 Section 3.6.1. [4] E.H. Lloyd, Probability, Handbook of Applicable Mathematics, John Wiley & Sons, 1980 Chapter 14. [5] V.A. Martinovich, A.V. Turukhin, A.M. Zaitsev, A.A. Gorokhovsky, J. Lumin. 102–103 (2003) 785. [6] A.A. Bergman, A.M. Zaitsev, Mengbing Huang, A.A. Gorokhovsky, J. Lumin. 129 (2009) 1524. [7] A.A. Bergman, A.M. Zaitsev, A.A. Gorokhovsky, J. Lumin. 125 (2007) 92. [8] A.B. Anderson, E.J. Grantscharova, Phys. Rev. B 54 (1996) 14341. [9] L.G. Jacobson, J.-K. Lee, B.L. Bennett, R.E. Muenchausen, M. Nastasi D.W. Cooke, J. Lumin. 124 (2007) 5–9. [10] A.H. Firester, M.E. Heller, P. Sheng, Appl. Opt. 16 (1977) 1971.
Fig. 5. (a) Map of luminescence signal of ZPL at 811.7 nm from a 40 40 mm2 (20 20 points) region at T¼ 80 K and (b) histogram of the peak luminescence signal for the map (a). Statistical analysis of the map gives the average intensity S¼ 218.1 c/s and standard deviation sS ¼24.5 c/s.