Protocol for drain current measurements

Transcription

1 Protocol for drain current measurements 1 Basics: the naive model of drain current A beam of intensity I (number of photons per unit area) and cross section impinges with a grazing angle Φ on a sample. Neglecting all the effects of reflection/refraction/multiple interference, and being Λ the electron free collision path, the excited volume contributing to the drain current can be written as V sin, (.1) which gives, within a linear approximation (i.e. di Idz ), for the drain current Y I IV, (.) sin where is the optical absorption coefficient and is the function converting the absorbed photon into the electrons contributing to drain current. In the case that the reflectivity R of the sample is not negligible, the drain current is given by Y I 1 R V. (.3) The reflectivity of the sample depends on the degree of linear polarization 1 P of the incoming light, and is given by 1P P1P R R R, (.4) where the reflectivities and P are related to the real and imaginary part of the refraction index n n ik by and R sin n cos sin n cos P n cos n sin R n cos n sin (.5). (.6) Being that the extinction coefficient k is related to the absorption coefficient by 4 k, (.7) and the refraction index (real part) n is related to the extinction coefficient k by the KK relations, the determination of the absorption coefficient by a drain current measurement becomes more complicated. As can be seen from Figure 1.1, at the carbon K edge, at around 84 ev, the reflectivity of an ideal Ag mirror

2 (with negligible roughness) is below 1% and can be neglected for angles greater 1 degrees. For angles below 1 degrees, the measurement of reflectivity of the sample should be done. Figure 1.1: alculated reflectivity of an ideal Ag mirror at photon energy = 84 ev as a function of the grazing angle. Let us suppose the case of an experiment whose goal is the measure of the optical absorption coefficient α a of an overlayer, with electron free collision path Λ, deposited onto a semi-infinite substrate of optical absorption coefficient α s, with electron free collision path Λ s. Note: the substrate absorption coefficient is supposed to be flat, i. e. ( ) cost. Otherwise or a numerical correction (f.i. by Henke table, tabulated cross section) or a third calibration measurement is needed to correct for the shape of the substrate absorption coefficient. The measurement proceeds in two steps. First step: measurement of substrate drain current, Y O, before depositing the overlayer (the incoming flux is simultaneously monitored for normalization/comparison purposes by measuring the drain current, W O, of a tungsten mesh. ee also below). In summary the result of the measurement can be written

3 Y, ti, t 1 R ; I, t I ( ) f O t (.8) sin with a mesh drain current W (, t ), where t is the instant at which the O measurement is taken. Y, t Let us introduce also the quantity Y, t O of use in the data O W, t O reduction (note that Y (, t) being the ratio of two currents, is an a-dimensional O quantity). Moreover by knowing at one energy point (e.g. from Henke tables) and measuring independently (e.g. by a calibrated photodiode I(, t) ) the values of the quantity in the Y (, t) I (, t) ( ) O sin sin can be determined. econd step: it consists in the measurement of the drain current Y A, in presence of the overlayer of thickness d (the incoming flux is simultaneously monitored for normalization/comparison purposes by measuring the drain current, W A, of a tungsten mesh). Two cases must be considered a) overlayer thickness d The drain current is given, neglecting reflectance, by Y, t I, t d I, t ( d), (.9) sin a a sin s s s with a mesh drain current W,t, where the optical absorption in the overlayer can be neglected disregarding the absorption of I in the adsorbate layer. ividing by the drain current measured in the step 1 (absence of overlayer) and utilizing the mesh currents, W O, t and W, t, for relative normalization one obtains I (, t ) I (, t ) d ( d) a a s s s Y (, t ) W (, t ) sin W (, t ) sin Y (, t ) I (, t) O ( ) W (, t) sin O (.1) I (, t ) I (, t ) d a a ( d s ) s s W (, t ) W (, t ) I (, t) ( ) W (, t) O

4 I (, t) I ( ) f ( t) Introducing the relations, where I ( ) is the photon I (, t ) I ( ) f ( t) distribution due to source plus transport optics and f(t) is the function taking into account the time dependence of such distribution (e.g. due to time decay, fluctuations etc.), the above expression becomes I ( ) f( t ) I ( ) f( t ) d ( d) a a s s s Y (, t ) I ( ) f( t ) I ( ) f( t) W W Y (, t ) I ( ) f( t) O ( ) I ( ) f( t) W 1 1 d ( d ) a a s s s d ( ) ( d) ( ) W W a a s s s 1 ( ) ( ) W d d a d d a a ( ) ( ) (.11) d This expression shows that, through the known coefficient ( ), the ratio of the two measured drain currents is proportional to the absorption coefficient of the overlayer plus a constant term, also in principle known. In the case that reflectivity of the substrate is not negligible, the reflectivity of the sample changes with the addition of adsorbate. Then, the ratio between the drain current in step and 1 has to take into account the reflectivity measured only on the substrate R and on substrate + molecule R (is shown in Fig. 3 the difference between them). Y (, t ) d 1R d 1R a (.1) a Y (, t ) ( ) 1R 1R O The reflectivity of the substrate + adsorbate depends on the substrate and overlayer optical constants. The reflectivity of the system can be calculated by 1P P 1P the equation R R R, where the reflectivity and P are given by the transmittance and reflectance Fresnel coefficients related to the interface vacuum-adsorbate and adsorbate-substrate. sp, sp, sp, r a tva rva expid na sin sp, P, R r Va. (.13) sp, sp, 1rVa ra expid na sin

5 The variation of calculated reflectivity of the sample as the thickness is shown in Figure 1. and Figure 1.3 in the case of an Ag sample + a H 14 layer with increasing thickness. Figure 1.: alculated reflectivity of on an ideal Ag mirror + a H 14 layer with different thickness as a function of the grazing angle at photon energy = 84 ev. The calculation was done at the carbon K edge at 84 ev and taking as parameter P=. The figures show that at grazing angles, where the substrate reflectivity is not negligible, the variation of reflectivity changes significantly (also with interference effects).

6 Figure 1.3: The same as Fig. but for low grazing incidence angles. b) overlayer thickness d At these coverages, the contribution of the substrate can be considered d negligible, the constant term vanishes and the drain current ratio is proportional to the overlayer absorption coefficient I ( ) f( t ) 1 a a a a Y (, t ) I ( ) f( t ) W W a a. (.14) Y (, t ) I ( ) f( t) 1 ( ) ( ) O ( ) I ( ) f( t) W W Taking into account reflectivity, this ratio becomes Y (, t ) 1R a a. (.15) Y (, t ) ( ) 1 R Experimental: error evaluation O Let s evaluate the error present in the determination of the ratio between the drain current from the sample I and the current from the monitor (mesh) W.

7 The error of this measure is given by the usual error propagation, where W are shown in Figure 1.4 and Figure 1.5: I and f f f I W I W. (.16) Figure 1.4: Noise in drain current measurements.

8 Figure 1.5: Noise in mesh current measurements. Usually the signal of the monitor is lower then that of the sample and we can assume I W (.17) and the ratio is given by I W f W W. (.18) Then, the error in the ratio between drain and monitor is given by 1 1W f I W W 4 W. (.19) I W W 4 W If the same instrument (or two equal instruments) is used for the measure of I and of W, then we can assume that the errors in the two measures are the same: inst 1 f 1 1, (.) W 4 W

9 being σ inst the error of the instrument. Equations (.16) and (.) show that the error is minimum when the difference in magnitude of the two signals is, and it is f inst 5 W 4 (.1)

10 ata analysis: smoothing with splines and energy shift The data reduction shows little energy shifts due or to the grating position or to source movements. Then, the spectra analysis requires before a smoothing and then, often, a manual shift that permits to overlap them correctly. Energy shifts generally can be related to the uncertainty in the reading of the position of the grating axis. In the case of the carbon K edge, at 84 ev, for example, the encoder resolution corresponds to about 6 mev. Another possible cause of energy shift between spectra could be the movement of the light source The formula that relates the variation Δ to the energy shift ΔE is E cos E. (.) cnqp1 EXP The algorithm for the smoothing with splines makes the minimization of the quantity among all the functions x n g x dx, (.3) x g x for which n g xi yi i, (.4) i where gx i is the value of the smoothing spline curve at x i, y i is the is the standard deviation, and is the smoothing measured value at x i, i factor. The number of point n for the smoothed signal can be chosen arbitrary. The σ s are essentially the instrumental errors of the ammeters. The best estimates we have for them are: 13 1 A for Keithley 1 for drain current 14 1 A for Keithley for mesh current The smoothing factor is 1.. One starts interpolating the data using these values producing a curve with about points. Then the smoothing can be refined changing and/or the number of points for the output array. The results of this smoothing procedure on the signal from the mesh is shown in Figure.1 and in Figure...

11 Figure.1: smoothing of the mesh current data for scan #41. Figure.: zoom of the Figure.1. In order to take into account the possible photon energy shift between the two scans, the signals from the meshes, smoothed with the described procedure, have been compared.

12 Figure.3: comparison of the mesh signals. The minima of the curves, at energies around 85.3 ev, have been fitted with two Gaussians (Figure.4): the difference of their positions give the energy shift of the photon, since we can consider that the mesh is passivated and is stable in time. Figure.4: gaussian fit of the minima of the mesh signals.

13 The values for the centers of the two Gaussians are: Emin ev. Emin ev The difference is ΔE =.6 ev. An estimation of the same order of magnitude is obtained starting the analysis from the signal of the BPM. Figure.5: variation of the value, as consequence of a movement of the beam at the entrance of the beamline, between the two scans. In the interval about 85 ev (from 84 and 86 ev), the mean values of are.713 and.7. Therefore Δ =.443. and then results ΔE = -.357, close to the value obtained after the spline procedure.

14 3 Angular dependence of the intensity of an emitting dipole In the reference frame of the laboratory the electric field of the radiation has the following form: itkr EL EH e 1. (.5) i e In the reference system of the sample, the components of the electric field can be obtained using rotations around X axis (+ψ ) and around the Y axis (-θ M ) (in this order): cosm sinm 1 itkr E EHe 1 cos sin 1 i sinm cos M sin cos e i sinm e cos sin, (.6) itkr i Ee H cos e sin i cosm e cos sin where M and are both positive quantities. Note that the term k r is invariant under rotations since both vectors rotate in the same way. Let us suppose we want to measure a physical quantity of the sample represented by a vector p, which can be, for example, the direction of a bonding. What we really measure is some signal proportional to f E p i I. (.7) E Therefore we must evaluate E p EpEp *. p can be expressed, in the reference system of the sample, in polar coordinates as: sin cos p p sin sin cos itkr Ep pehe asinm sincos cosm cosb sin sin, (.8) where i a e cos sin. (.9) i b cos e sin It results * itkr * * Ep pehe a sinm sincos cosm cosb sin sin (.3) The product of (.8) and (.3) gives

15 * * E p p EH aa sinm sincos cosm cos bb sin sin (.31) Let s calculate separately the terms * * sinm sincos cosmcossin sinab ba * * aa, bb, and * * ab (note that ba * ab * * i i aa cos sin cos sin e e cos sin cos sin cos i i bb cos sin cos sin e e * cos sin cos sin cos * i i ). (.3) (.33) ab cos sin 1 e cos e sin (.34) ba cos sin 1 e cos e sin (.35) * i i ombining (.34) and (.35) we obtain * * ab ba cos sin 1 cos cos sin Finally we have: Ep p E H. (.36) cos sin cos sincossinm sincos cosm cos cos sin cos sincossin sin cos sin 1cos cos sin sinm sincos cosm cossin sin Let us examine some limit cases: Pure circular polarization In this case 1 and. The expression (.37) reduces to: Ep p E sin sincos cos cos sin sin, (.38) H M M i.e. the signal is independent from the position of the chamber. Pure linear polarization For linear polarization is. The (.37) becomes Ep p E sin sin sincos cos cos cos sin sin H M M cos sin sinm sincos cosm cos sin sin Ep p E sin sin sin cos cos cos cos sin sin H M M (.39) (.37)

16 Horizontal chamber It is. Then Ep p E H sin M sin cos cos M cos sin sin sin sincos sin sincos cos cos M M (.4) Vertical chamber It is. Then Ep p E H sin M sin cos cos M cos sin sin sin sincos sin sincos cos cos M M (.41) In the following figure is shown the aspect of four different values of ε. E p as function of for Figure 3.1: calculated dipole intensity as a function of for 9. c If during the measurements I use only the central part of the beam, on the sample arrives an equal amount of light where and light with

17 , indicating with the changing in the phase induced by the optics of the beamline. Therefore the signal is composed by I I I ince it turns out that cos sin, cos sin I I I I (.4) Figure 3.: calculated intensity as a function of for 11. c Figure 3.3: calculated intensity as a function of for 7 c

18 Figure 3.4: um of the intensities calculated for the two cases 7 and 11.