Crystal truncation rods from miscut surfaces
Trevor A Petach, Apurva Mehta, Michael F Toney, David Goldhaber-Gordon
CCrystal truncation rods from miscut surfaces
Trevor A Petach and David Goldhaber-Gordon
Department of Physics, Stanford University, Palo Alto, CA 94305, USA
Apurva Mehta and Michael F Toney
SSRL, SLAC National Accelerator Laboratory, Menlo Park, CA 94205, USA (Dated: June 5, 2017)Crystal truncation rods are used to study surface and interface structure. Since real surfacesare always somewhat miscut from a low index plane, it is important to study the effect of miscuton crystal truncation rods. We develop a model that describes the truncation rod scattering frommiscut surfaces that have steps and terraces. We show that non-uniform terrace widths and jaggedstep edges are both forms of roughness that decrease the intensity of the rods. Non-uniform terracewidths also result in a broad peak that overlaps the rods. We use our model to characterize theterrace width distribution and step edge jaggedness on three SrTiO (001) samples, showing excellentagreement between the model and the data, confirmed by atomic force micrographs of the surfacemorphology. We expect our description of terrace roughness will apply to many surfaces, even thosewithout obvious terracing. I. INTRODUCTION
Surface x-ray diffraction is a critical tool for under-standing surface structure on an atomic scale. One use-ful surface diffraction technique is analysis of crystaltruncation rods, which are streaks of scattering extendingaway from the Bragg peaks parallel to the surface normal.Crystal truncation rods have been used to solve surfacereconstructions, locate adatoms, study self-assembledmonolayers, and understand buried interfaces, solid-liquid interfaces, and solid-gas interfaces. Thus, theyare a critical tool for understanding surface structure de-termination.There are two approaches to simulating the truncationrod intensity. The first “continuum” approach presumesthat a crystal can be described as an infinite lattice mul-tiplied by a shape function, which is unity in the bulkand zero outside the crystal. The Fourier transform ofthe shape function determines the shape of the trunca-tion rod. The second “atomistic” approach is to add upthe scattering from every atom in the crystal, with anappropriate phase factor that depends on the position ofthe atom. The square of the magnitude of the sum isproportional to the truncation rod intensity.Roughness can be included in both models. In thecontinuum approach, roughness is captured by a broad-ening of the shape function. In the atomistic approach,roughness is modeled as a series of partially occupiedlayers near the surface. In the best-known formulation,called β -roughness, the occupancy of each layer abovethe bulk is a constant fraction, β , of the layer below. In both approaches, roughness reduces the intensity ofthe truncation rod, with the largest effect at the anti-Bragg points. Other models have also been developed.For example, co-existence of two-dimensional and three-dimensional growth modes in a thin film results in a morecomplex roughness factor. These approaches generallywork well when the crystal surface is parallel to a low-index plane. However, no real surface is parallel to a low-indexplane. Even if the surface is locally parallel to the plane,steps, often one unit cell tall, divide the surface into ter-races whose lateral spacing depends on the miscut angle.Provided the coherence length exceeds the terrace width,a separate “sub-rod” will extend from each Bragg point,as shown in Fig. 1(a). Measuring the truncation rodsfrom such a surface with a point detector is challeng-ing because each sub-rod must be measured separately,and small misalignments or imperfections in the diffrac-tometer require frequent alignment scans to find the sub-rods. Thus, experimentalists frequently use samples withsmall miscuts ( < . ◦ ) and align the miscut with thelow-resolution direction of the beam or diffractometer toavoid the need to track the individual sub-rods. Re-cently, area detectors have made the task of measuringtruncation rods from miscut samples much easier sincethe detector usually intercepts several sub-rods simulta-neously and small misalignments have minimal impacton the data collection. In light of this much easier datacollection, it is necessary to develop a theory of trunca-tion rods from miscut samples so that a wider variety ofsamples can be studied.In this paper, we develop a model for the crystal trun-cation rods from miscut surfaces with terraces and stepsand show that it agrees well with data collected frommiscut SrTiO (001) surfaces, whose morphology is sep-arately characterized by atomic force microscopy. II. MODEL
It is well known that scattering from a surface en-codes information about the height-height correlationfunction.
Several models have been developed to de-scribe diffraction from vicinal, stepped surfaces that in-clude a variety of terrace width distributions and stepedge roughnesses.
Experimentally, the details of thestep distribution can be elucidated using these models. a r X i v : . [ c ond - m a t . m t r l - s c i ] J un For example, step edge repulsion and phase separationwere observed in miscut silicon, and anisotropy in aroughening transition was observed in Ag (110). These models for step distributions generally don’t de-scribe the intensity along the entire truncation rod, butrather only the in-plane shape. However, it has beenshown in a simple model with two terrace widths that un-equal terrace widths reduce the truncation rod intensity,especially at the anti-Bragg points. A numerical calcu-lation of truncation rod intensities from an atomic forcemicroscope image of a terraced, miscut surface shows asimilar effect. Building on these principles, we develop an atomisticmodel of a miscut surface, presuming a cubic crystal com-prised of bulk unit cells (with structure factor F b ) cov-ered by a single layer of a different surface unit cell (withstructure factor F s ). We presume that all step edgesare one bulk unit cell tall and that the average terracewidth is M unit cells, so the miscut angle is arctan(1 /M ).However, we do not presume that the terraces edges arestraight, or that each terrace has the same width. Asshown in Fig. 1(b), the position of the step edge at theend of the m th terrace, n unit cells along the step, is(( m + 1) M + D m,n ) a , where D m,n is the deviation ofthe position from the ideal value and a is the lattice con-stant. For an ideal surface with straight-edged, uniformlyspaced terraces, all D m,n = 0.To begin, we add the structure factors from a singlerow of unit cells, outlined in black in Fig. 1(b), to find F m,n = x m,n /a − (cid:88) j = −∞ F b e iq x aj + x m +1 ,n /a − (cid:88) j = x m,n /a F s e iq x aj × e iq y an e − iq z am . (1)If the beam were perfectly coherent, the scattered am-plitudes from the entire illuminated surface would addcoherently. To account for partial coherence, we add scat-tered amplitudes from a local region or “patch,” weight-ing amplitudes farther from the center of the patch lessthan those near the center: F patch = ∞ (cid:88) n = −∞ ∞ (cid:88) m = −∞ F m,n e − m M a /ξ x e − n a /ξ y , (2)where ξ x and ξ y are the coherence lengths in the x and y directions. Presuming that any correlation in the devi-ations D m,n decays on a shorter scale than either coher-ence length, the scattered intensity is proportional to I = Aa π ξ x ξ y (cid:104) F patch F ∗ patch (cid:105) , (3) xyz y m,n = naz m,n = − ma x m +1 ,n y m +1 ,n z m +1 ,n F b F s average width Ma (a)(b) (10 l ) (10 l ) (101)(102)(100) l − Lx m,n = ( mM + D m,n ) a xz hl detector FIG. 1. Schematic of the truncation rods and the samplesurface. (a) Miscut results in splitting of the truncation rod,with separate sub-rods extending away from each Bragg pointin a direction perpendicular to the surface. L is an integerlabeling the out of plane index of the Bragg points. We labeleach sub-rod with a subscript indicating the Bragg point fromwhich it emanates. It is possible to intercept all sub-rodsfrom a given rod using an area detector. (b) Terraces are onebulk unit cell tall with an average width Ma . The deviationfrom the zero-roughness position of the step edge on the m thterrace, n unit cells along the step is D m,n a . F s and F b arethe the structure factors of the surface unit cell and bulk unitcell, respectively. where A is the illuminated area, and the brackets denotethe spatial average over the whole sample. Expandingthis expression, we find I = Aa π ξ x ξ y (cid:42) (cid:88) m,n,m (cid:48) ,n (cid:48) F m,n F ∗ m (cid:48) ,n (cid:48) × e − ( m + m (cid:48) ) M a /ξ x e − ( n + n (cid:48) ) a /ξ y (cid:43) . (4)Defining˜ F m,n ≡ ( F s − F b ) e iq x aD m,n − F s e iq x a ( M + D m +1 ,n ) − e iq x a (5)and θ ≡ q x aM − q z a, (6)the expression for intensity becomes I = Aa π ξ x ξ y (cid:88) m,n,m (cid:48) ,n (cid:48) (cid:68) ˜ F m,n ˜ F ∗ m (cid:48) ,n (cid:48) (cid:69) e iθ ( m − m (cid:48) ) × e iq y a ( n − n (cid:48) ) e − ( m + m (cid:48) ) M a /ξ x e − ( n + n (cid:48) ) a /ξ y , (7)where (cid:68) ˜ F m,n ˜ F ∗ m (cid:48) ,n (cid:48) (cid:69) = | F s − F b | (cid:68) e iq x a ( D m,n − D m (cid:48) ,n (cid:48) ) (cid:69) − ( F s − F b ) F ∗ s e − iq x aM (cid:68) e iq x a ( D m,n − D m (cid:48) +1 ,n (cid:48) ) (cid:69) − ( F s − F b ) ∗ F s e iq x aM (cid:68) e iq x a ( D m +1 ,n − D m (cid:48) ,n (cid:48) ) (cid:69) + | F s | (cid:68) e iq x a ( D m +1 ,n − D m (cid:48) +1 ,n ) (cid:69) . (8)In order to simplify the calculation, we assume that thedeviations D m,n have zero mean and a Gaussian distribu-tion. Then, we can use the Baker-Hausdorff Theorem to calculate the spatial average, (cid:68) e iq x a ( D m,n − D m (cid:48) ,n (cid:48) ) (cid:69) = e − ( q x a ) (cid:104) ( D m,n − D m (cid:48) ,n (cid:48) ) (cid:105) / . (9)To proceed, we must calculate the average differ-ence between the step positions at different locations, (cid:104) ( D m,n − D m (cid:48) ,n (cid:48) ) (cid:105) . For most surfaces, this quantity is acomplicated function of m − m (cid:48) and n − n (cid:48) that dependson the details of the step distribution on that particularsurface. In order to proceed, we use a simple step distri-bution. As shown in Fig. 2, we presume that there areonly two non-idealities in the step edges. First, any sin-gle step edge is jagged, with standard deviation from theaverage position σ s (“s” for “step”) and no correlation inthe jaggedness along the step. Second, the terrace widthchanges from terrace to terrace, with standard deviationfrom the average width σ w (“w” for “width”) and no cor-relation between widths on subsequent terraces. These (a)(b) σ s > σ w > FIG. 2. Roughness on a terraced surface can arise from (a)step edge jaggedness and/or (b) width variation from terraceto terrace. two types of roughness result in the average correlationfunction (cid:104) ( D m,n − D m (cid:48) ,n (cid:48) ) (cid:105) n = n (cid:48) , m = m (cid:48) σ a n (cid:54) = n (cid:48) , m = m (cid:48) σ + σ a m (cid:54) = m (cid:48) (10)Many surfaces will have much more complicated formsof this correlation function. However, this simple formcaptures the main features of many terraced surfaces.Evaluating the sum in Eq. (7) using Eq. (9) and Eq. (10),we find three distinct components of the total intensity I = I p + I w + I s , (11)where I p , I w , and I s are all functions of q x , q y , and q z , F b and F s , and the two roughness parameters σ s and σ w .These functions are shown in Fig. 3 for fixed q z . Eq. (11)is a significant result. It states that the scattering froma miscut surface can be divided into three distinct partsarising from 1) the splitting of the truncation rod dueto the miscut, 2) a broad peak due to variable terracewidths, and 3) a diffuse background from jagged stepedges. We now discuss these parts in more detail. A. I p – Sharp Peaks from Sub-Rods Most of the scattering in our model comes in a seriesof sharp peaks, given by q x q y I p I w I s I ptot I wmax I p (a)(b) (c)(d) I p + I w + I s q x q y tot ,L Rel. Intensity
FIG. 3. The three components of the scattered intensity froma miscut surface at fixed q z . (a) I p – sharp peaks given byEq. (12). I tot ,L p is the intensity in a single peak integrated over q x and q y . I totp is the sum of the integrated intensity in all ofthe peaks. (b) I w – a broad peak that arises from variationsin terrace width from terrace to terrace given by Eq. (22). Wedefine I maxw as the sum of the column containing the highestintensity pixel. (c) I s – a diffuse background that arises fromjagged step edges given by Eq. (26). The intensity is usuallytoo low to observe experimentally. (d) The total scatteredintensity. I p = Aa π ξ x ξ y H ( θ, ξ x /M a ) H ( q y a, ξ y /a ) × (cid:12)(cid:12) ( F s − F b ) − F s e iq x aM (cid:12)(cid:12) e − q x ( σ + σ ) ( q x a/ , (12)(with “p” for “peaks”), where H ( x, b ) = ∞ (cid:88) j = −∞ πb e − b ( x − πj ) . (13)Eq. (12) is plotted in Fig. 3(a) for fixed q z . The function H ( x, b ) is a periodic series of peaks spaced by 2 π in x .Thus, the product of the two H functions in Eq. (12)restricts scattering to a series of rods in reciprocal space.With no miscut, there would be a single rod for each inte-ger value of h and k , and the rods would be labeled (00 l ),(10 l ), and so on, in the usual representation. As shown inFig. 1(a), miscut splits these rods into separate sub-rodseach emanating from a single Bragg point, which we callthe “primary” Bragg point for that sub-rod. Thus, welabel each sub-rod by the usual notation plus a subscriptnoting the l value of the primary Bragg point, so the(10 l ) rod now splits into several sub-rods, labeled (10 l ) ,(10 l ) , and so on. An area detector often intercepts sev-eral of these sub-rods simultaneously, displaying a seriesof sharp peaks. These peaks will be spaced by 2 π/M a in q x . The larger the miscut, the larger the spacing betweenpeaks. We have assumed that the coherence length is signifi-cantly longer than the average terrace width, so that thepeaks from the sub-rods are well defined. If the coher-ence length is shorter than or comparable to the averageterrace width, then the width of each sub-rod will bebroad enough that the individual sub-rods will be indis-tinguishable, and the scattering will appear like a singlerod connecting Bragg peaks in the out-of-plane direction.In that case, the analysis is more complicated.The second line of Eq. (12) modulates the intensityof the sub-rods in two ways. First, there is the stan-dard interference between the surface and the bulk, inwhich the bulk dominates near the primary Bragg point(and whenever l is an integer), whereas they contributeequally whenever l is a half-integer. Second, there is aroughness factor which reduces the intensity of the sub-rod away from the primary Bragg point. Larger terracewidth variation, σ w , and larger step edge jaggedness, σ s ,both result in a faster reduction of intensity moving awayfrom the primary Bragg point.Experimentally, in a truncation rod measurement, I p will be observed as a series of peaks on an area detector,one from each sub-rod. One useful way to treat such datais to subtract a background from each peak and then addthe total intensity in all of the peaks. To calculate thetotal integrated intensity observed in this case, we needto integrate over q x and q y . We note that (cid:90) ∞−∞ πb e − b x dx = π √ πb. (14)Even though F s and F b are functions of q x and q y ,we treat them as constants during the integration sinceonly a few peaks contribute to the integral and F s and F b vary only slightly from peak to peak (for M >> H ( θ, ξ x /M a ) is peaked at θ = 2 πL , where L isan integer, we approximate it as a series of δ -functionsand make the substitution q x = ( q z a − πL ) /M a , whichallows us to write the result as a sum: I totp = AM a (cid:12)(cid:12) ( F s − F b ) − F s e iq z a (cid:12)(cid:12) × (cid:88) L e − ( q z a − πL ) ( σ tot /Ma ) (cid:0) ( q z a − πL ) / M (cid:1) , (15)where σ tot ≡ (cid:112) σ + σ (16)is the total “terrace roughness.” For M >>
1, only termsnear L = q z a/ π contribute to the sum, so the argumentof sin x in the denominator is small. Expanding and re-arranging, we find I totp = Aa (cid:12)(cid:12)(cid:12)(cid:12) F s + F b e − iq z a − e − iq z a (cid:12)(cid:12)(cid:12)(cid:12) × (cid:88) L ( q z a/ q z a − πL ) e − ( q z a − πL ) ˜ σ , (17)where ˜ σ tot ≡ σ tot /M a is the total roughness as a fractionof the terrace length, and ˜ σ w and ˜ σ s are similarly defined.Each term in the sum in Eq. (17) is the integrated inten-sity in the sharp peak from the L th sub-rod, which wedenote I tot ,L p .In order to isolate the effect of roughness, we define I ≡ Aa (cid:12)(cid:12)(cid:12)(cid:12) F s + F b e − iq z a − e − iq z a (cid:12)(cid:12)(cid:12)(cid:12) (18)and a roughness factor c p ≡ (cid:88) L ( q z a/ q z a − πL ) e − ( q z a − πL ) ˜ σ (19) ≈ − (cid:18) √ π ˜ σ tot + O (˜ σ ) (cid:19) sin (cid:16) q z a (cid:17) , (20)where the first order approximation in Eq. (20) is validfor ˜ σ tot < ∼ . The ex-pression for I totp can then be written I totp = c p I , (21)where I is the intensity had there been no miscut, and c p is a factor that depends only on q z and σ tot and not on F b or F s . It is unity when the total roughness is zero and lessthan one otherwise. (We discuss the roughness factor indetail in Sec. IV.) Thus, the total integrated intensity inall of the peaks is proportional to the intensity had therebeen no miscut and a roughness factor which dependsonly on q z and σ tot . B. I w – Broad Peak If the terraces have non-uniform widths, regardless ofwhether the step edges are straight or jagged, then abroad peak develops underneath the sharp peaks, de-scribed by I w = Aa (cid:112) π/ π ξ y M H ( q y a, ξ y /a ) (cid:12)(cid:12) ( F s − F b ) − F s e iq z a (cid:12)(cid:12) × e − q x σ − e − q x ( σ + σ ) ( q x a/ . (22)Eq. (22) is plotted in Fig. 3(b) for fixed q z . The widthin q y is inversely proportional to the coherence length. However, the width in q x is much wider than the inverseof the coherence length and depends on the terrace widthnonuniformity. When the width variation σ w is small, thepeak is weak and broad in q x . As σ w increases, the peakbecomes narrower and stronger in such a way that theintegrated intensity increases. Step edge jaggedness ( σ s )reduces the peak width in q x but does not change themaximum intensity. In any crystal with uniform terracewidths (but where the step edges may or may not bejagged), σ w is zero and I w is zero.Experimentally, the scattering from a sample withnonuniform terrace widths is a series of sharp peaks fromthe sub-rods with a broad peak underneath. One ap-proach to analyzing such data is to integrate the totalintensity in all of these peaks. To calculate the totalintegrated intensity observed in this case, we need to in-tegrate I w over q x and q y and add the result to I totp .Again treating F s and F b as constants since they varyonly slightly over the extent of the broad peak in q x and q y , we find that the integrated intensity in the broad peakis I totw = Aa (cid:12)(cid:12)(cid:12)(cid:12) F s + F b e − iq z a − e − iq z a (cid:12)(cid:12)(cid:12)(cid:12) × sin ( q z a/
2) 4 √ π (˜ σ tot − ˜ σ s ) . (23)Defining c w ≡ sin ( q z a/
2) 4 √ π (˜ σ tot − ˜ σ s ) , (24)the expression for I totw can be written I totw = c w I , (25)where c w is a factor that depends on σ w , σ s , and q z , andnot on F b and F s . Thus, the integrated intensity in thebroad peak is always proportional to the scattering hadthere been no miscut. Depending on how the detectorimages from an experiment are analyzed, this scatteringmay or may not need to be included during modeling.As we discuss in Sec. IV, including it changes the shapeof the rod in q z . C. I s – Diffuse Background When the step edges are jagged, there is a diffuse back-ground. It does not depend on σ w and has the functionalform I s = A π M (cid:12)(cid:12) ( F s − F b ) − F s e iq z a (cid:12)(cid:12) − e − q x σ ( q x a/ . (26)Eq. (26) is shown in Fig. 3(c) for constant q z . Thisbackground is broad in all directions and cannot be easilymeasured experimentally. It would be subtracted in mostreasonable background subtraction procedures. III. EXPERIMENTAL RESULTS
To test our model, we prepared three SrTiO (001) sur-faces with different terrace morphologies that correspondto different values of σ w and σ s . We call the samples A,B, and C. We etched the samples in 1:6 buffered oxideetch for 2 minutes to achieve TiO termination. Wethen annealed the samples differently: A at 1025 C in1:10 O :Ar for 1 hr, B at 950 C in 1:10 O :Ar for 1 hr, Cno anneal. These three annealing conditions resulted inthree different surface morphologies, as shown by atomicforce microscopy (AFM) in Fig. 4(a)-(c).We measured the specular crystal truncation rod fromthe three samples at beamline 7-2 at SSRL in four cir-cle mode with a double crystal Si (111) monochroma-tor and a Rh-coated mirror to focus the beam to a spotapproximately 100 x 500 µ m FWHM. The energy was15.5 keV. Scattered photons were collected on a Pilatus100k area detector located approximately 1 m from thediffractometer center. Lorentz and illuminated area cor-rections were applied to all data. The SupplementalMaterial contains a complete discussion of the correc-tions.The scattering from the different surfaces agrees quali-tatively with our theory. For all three samples, there areseveral sharp peaks and a single broad peak, as shownin Fig. 4(d)-(f). For Sample A, with the smoothest stepedges and the most uniform widths, the sharp peaks falloff most slowly away from the center, and the broad peakis only faintly visible. The larger width variations in Sam-ple B result in a stronger, narrower broad peak, and afaster falloff in intensity of the sharp peaks. For SampleC, the broad peak is similar to sample B, suggesting thatthe terrace width variance is similar, but the sharp peaksfall off more quickly, consistent with the more jagged stepedges. The extra diffuse background from the jagged stepedges, I s , is too weak to be visible. The scattering pat-tern is rotated for Sample C because the miscut directionis rotated relative to the crystal axes (see SupplementalMaterial for further discussion of miscut rotation).To make a quantitative estimate of σ w and σ s , we com-pare both the integrated intensity of individual sharppeaks and the height of the broad peak to the total in-tensity in all the sharp peaks.Each sharp peak arises from the intersection of onesub-rod with the detector. Each sub-rod emanates fromits primary Bragg point as an elliptic cylinder with itsaxis slightly tilted relative to the [00 l ] direction, where l = q z a/ π . For the specular rod, the primary Braggpoints are located at (00 L ), where L is an integer. Welabel the sub-rods using these integers. We plot the inte-grated intensities of the sharp peaks from sub-rods L = 0 to L = 5 in Fig. 4(g)-(i). The integrated intensity in eachsub-rod reaches a maximum at its primary Bragg point.The detector occupies a region of the Ewald sphere,and thus in general intersects each sub-rod at a slightlydifferent value of l , as shown in Fig. 1(a). However, forsmall miscut ( M a >> thickness of surface unit cell), theintensity in the sub-rods varies slowly with l , and weapproximate the intersection as occurring at the samevalue of l for each sub-rod.With this approximation, we find using Eq. (15) and(17) that, at a given l , the ratio of the integrated inten-sity of the sharp peak from the L th sub-rod to the totalintensity of all sharp peaks is I tot ,L p I totp = sin ( πx ) π x e − x π ˜ σ (cid:30) (cid:18) − √ π ˜ σ tot sin ( πx ) (cid:19) , (27)where x ≡ l − L is the distance along the sub-rod, inthe q z direction, to the primary Bragg point. This ratiodepends only on the total roughness. Thus, by fitting theobserved ratios to this expression, shown in Fig. 5(d)-(f),we can extract ˜ σ tot . Even though the intensities of thesub-rods vary by many orders of magnitude, the ratioscollapse onto a single curve, and the fit is excellent.The easiest way to find σ w would be to compare theintegrated intensity in all of the sharp peaks to the inte-grated intensity in the broad peak. However, since thebroad peak is often wide and weak, it is hard to accu-rately measure the integrated intensity. Thus, we focuson the maximum intensity. As shown in dashed box inFig. 3(b), we define I maxw to be the sum of the intensi-ties of the pixels in the column that contains the broadpeak maximum. Using Eq. (17) and (23), we calculatethe ratio I maxw I totp = sin ( πl ) 2˜ σ π ∆ (cid:30)(cid:18) − √ π ˜ σ tot sin ( πl ) (cid:19) , (28)where ∆ is the width of a detector pixel in reciprocalspace. This ratio depends on ˜ σ w and ˜ σ tot . We plotthe observed ratio as red squares in Fig. 5(a)-(c). Us-ing Eq. (28) and our best fit value for ˜ σ tot , we do a leastsquares fit to extract ˜ σ w .To complete the test of our model, we compare theroughness parameters extracted from truncation rod fit-ting to the roughness parameters found directly from theAFM images in Table I. To find the roughness param-eters from the AFM images, we use a correlation length ξ y of 100 nm, inferred from the width of the truncationrods in q y , and presume that the ξ x exceeds the widthof the image. We calculate the average step position andthe step edge jaggedness in horizontal 100 nm strips, andthen average over all strips to find ˜ σ w and ˜ σ s . As weshow in Table I, the agreement between the two methods
400 nm 400 nm 400 nm l (r.l.u.) l (r.l.u.) l (r.l.u.) Sample A Sample B Sample C (a) (b) (c)(d) (e) (f)(g) (h) (i) -0.025 0.0250 q x (nm -1 ) -0.025 0.0250 q x (nm -1 ) -0.025 0.0250 q x (nm -1 )0.2 nm -1 -1 -1 (00 l ) (00 l ) (00 l ) (00 l ) (00 l ) (00 l ) ( c oun t s ) ( c oun t s ) ( c oun t s ) I p t o t , L FIG. 4. Specular truncation rod from miscut SrTiO (001). Atomic force micrographs of samples annealed at (a) 1025 C, (b)950 C, and (c) no anneal, show different terrace morphologies. Each step is one unit cell (3.905 ˚A) high. (d), (e), (f) Detectorimages where the center pixel is l = 1.6, with column sums below. The scale bar on the image indicates the change in the lengthof q across the image, whereas the x -axis labels on the plot indicate the change in q x only. (g), (h), (i) The integrated intensityof the sub-rods L = 0 to L = 5, plotted as a function of l . Each sub-rod is a different colored symbol. The SupplementalMaterial contains a complete discussion of the correction factors applied to the data to obtain I tot ,L p .SampleA B CBest fit to diffractionpattern ˜ σ w σ s ξ y = 100 nm ˜ σ w σ s ± .
01 for all parameters. is excellent. However, ˜ σ s from the AFM images system-atically exceeds ˜ σ s from the truncation rod fits, whilethe opposite is true for ˜ σ w . We hypothesize in the Sup-plemental Material how correlations in the step edgejaggedness, which are not captured in our model, maybe responsible for this discrepancy. Overall, the goodagreement between the parameters extracted from thetwo methods indicates that our model successfully de-scribes the terrace roughness of the three surfaces. Sample A l (r.l.u.) σ w = 0 . l (r.l.u.) σ w = 0 . -2 l (r.l.u.) σ w = 0 . -4 -2 σ tot = 0 . σ tot = 0 / I p l − L (r.l.u.) σ tot = 0 . σ tot = 0 l − L (r.l.u.) σ tot = 0 . σ tot = 0 l − L (r.l.u.) Sample B Sample C ˜ ˜˜ ˜ ˜ ˜˜˜˜ t o t I ptot I wmax I wmax I ptot / (a) (b) (c)(d) (e) (f) I p t o t , L FIG. 5. Extracting roughness parameters. (a), (b), (c) The total intensity in the sharp peaks ( I totp , black triangles) and theheight of the broad peak ( I maxw , blue crosses). The black lines are the best fit to the ratio of these intensities using Eq. (28),where ˜ σ w is the only fitting parameter. (d), (e), (f) The ratio of integrated intensity in a single sharp peak (plotted in Fig. 4(g)-(i)) to the total intensity in the sharp peaks, as a function distance from the primary Bragg peak. The black lines are thebest fit to the ratio of these intensities using Eq. (27), where ˜ σ tot is the only fitting parameter. IV. DISCUSSION
We have found that the truncation rods from miscutsurfaces have three components: 1) a series of evenlyspaced sharp peaks arising from the splitting of the trun-cation rod into sub-rods, 2) a single broad peak arisingfrom terrace width variation, and 3) a diffuse backgroundarising from step edge jaggedness. One of the most no-table aspects of our model is the separation between solv-ing the surface structure and evaluating the roughness.Indeed, we were able to characterize the terrace widthand step edge roughness on three different samples with-out any knowledge of the surface or bulk structure fac-tors, F s and F b .However, to do so, we had to examine in detail the in-tensities of the sub-rods and the broad peak. In a typicalmeasurement, with lower resolution or smaller miscut,it might not be possible to resolve these peaks. As dis-cussed in Sec. III, a typical analysis is likely to either1) add integrated intensities from the sharp peaks andsubtract I w and I s as background or 2) add integratedintensities from the sharp peaks and the broad peak, and subtract only I s as background. In the first case, the rodintensity is I times a roughness factor ( c p ) that dependsonly on σ tot and l . This factor is shown in Fig. 6(a). Itis periodic in l , reaching unity at the Bragg points and aminimum at the anti-Bragg points. In the second case,the rod intensity is I times a factor ( c p + c w ) that de-pends on σ w , σ s , and l . This factor is shown in Fig. 6(b).It is also periodic in l , reaching a minimum at the anti-Bragg points, where the minimum value depends on theratio of σ w to σ s in addition to σ tot . For two sampleswith the same σ tot , the sample with larger σ w will havethe shallower minimum.The effect of terrace roughness is similar to otherroughnesses because it decreases the intensity away fromthe Bragg peaks. As shown in Fig. 6(a), the l depen-dence is different than for β -roughness. The effect of ter-race roughness is concentrated near the anti-Bragg point,whereas β -roughness results in a broader reduction in in-tensity. We note that multiple types of roughness maybe present on a single sample. For example, it is possiblefor a surface to have β -roughness or partial occupancyacross the entire surface in addition to having terrace c p β = 0.064 σ tot = 0.4 σ tot = 0.2 σ tot = 0.1 β = 0.149 β = 0.420˜˜˜ l (r.l.u.) σ tot = 0.4 σ w = 0 σ s = 0.4 σ w = 0.35 σ s = 0.2 σ w = 0.38 σ s = 0.1 c p + c w ˜˜˜˜˜˜ ˜(a)(b) FIG. 6. Roughness factors. (a) c p , the ratio of the totalintensity in the sharp peaks for a rough sample relative to anideally miscut sample for various values of the total terraceroughness. The black dashed lines are β -roughness factors forcomparison, calculated assuming F s = F b , chosen to matchthe c p factor at l = 0 .
5. (b) The total intensity in the sharppeaks and the broad peak relative to the total intensity froman ideally miscut sample. The total roughness is the samefor all curves, but it is split differently between terrace widthvariation and step edge jaggedness. roughness. In that case, β -roughness would only impact I , and terrace roughness would only impact c p and c w .In conclusion, we have developed a new model for crys-tal truncation rods from miscut surfaces and applied it toa series of SrTiO samples, where we characterized theterrace roughness without needing to solve the surfacestructure. Our model gives a simple multiplicative factorto account for this roughness in the crystal truncationrods. Our approach is broadly applicable to analyzingtruncation rods from miscut samples and solving theirsurface structure. ACKNOWLEDGMENTS
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