1. Specular reflectivity and off-specular scattering
Tools for roughness investigation
Hugues Guerault 15/12/2000

2. Outline  Introduction  Flat surface/interface - Dynamical theory
Layer thickness and electronic density determination
 Rough surface/interface - Kinematical theory
Roughness and diffuseness (Non-)periodic roughness
Differential cross-section Correlation lengths
 Investigation geometries
Specular reflectivity (specular scan)
Off-specular scattering (longitudinal, transverse and detector scans)
 Conclusions

3. Introduction
 Increasing ability to structure solids in 1, 2 or 3D at nanoscopic scale
Mesoscopic layered superstructures
(multilayers, superlattices, layered gratings, quantum wires –and dots)
 Perfection depends on
Perfection of the superstructure (grating shape, periodicity, layer thickness)
Interface quality (roughness, interdiffusion)
Crystalline properties (strain, defects, mosaicity,…)
 Roughness affects the physical behavior of interfaces
Optical : reduces the specular reflectivity – creates diffuse scattering
Magnetic : changes the interface magnetization
Electronic : disturbs the band structure in semiconductor devices (resistivity)

4. Dynamical Theory Electric field in layer p
Substrate N
p+1 p 2 1
0 Air (n=1) ZS Zp hn - + kn
Through the layer p (Tp : Translation Matrix)
At p,p+1 interface (Rp,p+1 : Refraction Matrix ; pp, mp Fresnel coef.)
Transfer Matrix [M]ij M=R01T1R12……………TN-1RNN-1TNRNS
Reflection coefficient r=M12/M22 Absolute Reflectivity R=r.r*
Transmission coefficient t=1/M22

5. N=1 , N=2 Single Layer Bilayer 2 oscillation frequencies are evidenced
R=r.r* max. each time
As Then
(Kiessig fringes)
Cu thickness
CuO2 thickness
For <c  Total external reflection
For p=0, =c leading to el via
Bilayer 2 oscillation frequencies are evidenced

6. Kinematical Theory
Rough interfaces Dynamical theory not appropriated anymore
Born approximations No multiple reflections
No refraction
R function of d/dz
Substrate
100 A
300 A
TF-1
Disturbance of el
at interface

7. Interface disturbance
What kind of disturbance ?
Rough interface Diffuse interface 2
Diffuseness / Graded interface
Graded composition (electronic density)
from j to i layer with l steps

8. Differential cross-section
kin
ksc Q
q//(x,y) qz
Um+1(x,y)
Um (x,y)
Um (x’,y’)
hmideal
zm+1 zm
Q qz
If Then
Differential cross-section (detected intensity) depends on p(W=Um(x,y)) (Height distribution at interfaces)

9. Periodic Roughness Flat substrate Discrete Height Distribution D
Pure specular
Discrete Height Distribution
Kiessig fringes function of D U1 U2 D

10. Non-periodic Roughness
Random Height Distribution
Gaussian height distribution
Height-Height correlation function
Two contributions
Specular contribution observed in the specular direction
Diffuse contribution observed when Q(x,y)0 +

11. Correlation Lengths Height-Height correlation function
where h : roughness exponent
 : lateral correlation length
Increasing and decreasing roughness in periodic multilayers
 : vertical correlation length
No Increasing Partial Identical
replication roughness replication replication

12. Specular reflectivity

13. Off-specular (diffuse) scattering
inaccessible q-area
rocking curve
2q=1.5
detector scan
w=0.9
Transverse scan (Rocking curve) at 2=2º
Si layer (64 nm) on Si substrate
s=7 A , h=0.2 , various 
Large lateral correlation  at interface 
Specular peak
Yoneda wings : each time ai or af = ac
Yoneda Wings

14. Longitudinal and detector scans
Si (30 nm) // Ge (50 nm) // SiO2 (1.5 nm)
Schlomka et al. PRB 51(4) 1995
Offset (longitudinal) scan Detector scan
Curve depends on ai (penetration depth)
ai < ac No penetration
Increasing ai  different modulations
Specular contribution of the diffuse scattering
Same oscillations than reflectivity curve

15. Conclusions Grazing Incidence X-Ray Reflection
 Surface/interface investigation at atomic scale
 Non destructive technique
Vertical periodicity & in-plane morphology
 Layer thickness, electronic density profile (composition profile)
 Surface and interface roughness
 In-plane and between plane correlations
 No information on the crystalline structure
Application 01/2001: Collaboration IKS / VSM / IMEC
 Roughness characterization of Co1-xNixSi2 layers (MBE)
 Roughness influence on the resistivity