1. Theory

Definition: Reflection high-energy electron diffraction (RHEED) directs a 5 to 20 $keV$ electron beam at 1 to 3 degrees onto a growing surface and images the diffracted beams on a phosphor screen, revealing surface structure, reconstruction and roughness during growth (Krukowski and Zytkiewicz lecture).
Surface sensitivity: The glancing geometry limits penetration to a few atomic layers (background knowledge, Ichimiya and Cohen).
Oscillations: In layer-by-layer growth the specular intensity oscillates with a period of exactly one monolayer, giving an absolute growth-rate monitor (Larsen and Joyce).
Diffraction: Refraction by the inner potential shifts Bragg angles noticeably at glancing incidence (Mitura).

2. Working Principle

Gun and screen: Electron gun, phosphor screen and camera on the MBE chamber (Krukowski and Zytkiewicz lecture).
Patterns: Streaks for flat 2D surfaces, spots for 3D islands, and reconstruction orders along chosen azimuths (Krukowski and Zytkiewicz lecture).
Intensity tracking: Specular spot intensity recorded against time after opening the shutters (Larsen and Joyce).
Analysis: Period, phase and damping fitted to obtain growth rate and growth mode (growth-rate patent; Mitura).

3. Applications

Growth-rate calibration: GaAs, AlAs and alloy rates and compositions (Larsen and Joyce; background knowledge).
Oxide desorption and temperature calibration: Reconstruction fix points (Krukowski and Zytkiewicz lecture).
Growth mode control: Layer-by-layer, step-flow and islanding (Krukowski and Zytkiewicz lecture).
Quantum wells and lasers: Monolayer-precise layers for devices (Quantum-Well Laser page; background knowledge).

Electron Optics and Diffraction

Name	Formula	Related Formulae	Variables/Constants
Relativistic electron wavelength	$λ = \frac{h}{2 m _{0} e U ( 1 + \frac{e U}{2 m _{0} c ^{2}} )}$	$K^{2} = (1 + \frac{e U}{2 m _{0} c ^{2}}) \frac{2 m _{0} e U}{ℏ ^{2}}, K = \frac{2 π}{λ}$	$U : a cce l er a t in g v o l t a g e$ $m_{0} : e l ec t r o n r es t ma ss$ $K : w a v e v ec t or (E w a l d s p h er e r a d i u s); M i t u r a : E q . 1 w i t h K^{2} = 2650 \overset{˚}{A}^{- 1} s q u a r e d a t 10 keV (co n s t an t w r i tt e n w i t h h, w hi c h ma t c h es ℏ); K r u k o w s k i an d Z y t k i e w i cz l ec t u r e : 5 t o 20 keV, w a v e l e n g t h ab o u t 0.1 \overset{˚}{A}, g l an c in g an g l e 1 t o 3 d e g r ees$ $(co m p u t e d : 0.173, 0.122 an d 0.0859 \overset{˚}{A} a t 5, 10 an d 20 keV; t h e n o n - r e l a t i v i s t i c v a l u e i s 0.24 t o 0.97 p er ce n t l o n g er; K = 51.5 \overset{˚}{A}^{- 1})$
Bragg condition with refraction	$sin^{2} θ_{n} = \frac{n ^{2} π ^{2} + v ~ _{R e} d ^{2}}{K ^{2} d ^{2}}$	$nλ = 2 d sin θ (no refraction)$	$θ_{n} : g l an c in g an g l e o f t h e n - t h p e ak$ $d : l a y er s p a c in g$ $\tilde{v}_{R e} : r e a l m e an inn er p o t e n t ia l t er m; M i t u r a : d = 5.65 \overset{˚}{A}, \tilde{v}_{R e} ab o u t - 3.483 \overset{˚}{A}^{- 1} s q u a r e d, p r e d i c t e d 1.347, 4.498 an d 7.148 d e g r ees f or n = 4, 8 an d 12, o b ser v e d 4.49 an d 7.14 d e g r ees$ $(co m p u t e d : w i t h r e f r a c t i o n 1.347, 4.499 an d 7.149 d e g r ees; w i t h o u t i t 2.48, 4.96 an d 7.45 d e g r ees)$
Streak spacing	$Δ x = \frac{L λ}{a _{s}}$	$streaks: 2D surface; spots: 3D islands$	$Δ x : s t r e ak se p a r a t i o n o n t h e scr ee n$ $L : s am pl e - t o - scr ee n d i s t an ce$ $a_{s} : s u r f a ce r o w s p a c in g; s ma l l - an g l e r e l a t i o n (ba c k g r o u n d k n o w l e d g e, I c himi y a an d C o h e n); K r u k o w s k i an d Z y t k i e w i cz l ec t u r e : i d e a l 2 D s u r f a ce g i v es p a r a l l e l s t r e ak s, r o ug h s u r f a ce a d i f f er e n t p a tt er n, G a A s (2 x 4) s h o w in g 2 x a l o n g [110] an d 4 x a l o n g [- 110]$ $(a ss u m e d : L = 0.3 m, a_{s} = 5.65/ 2 \overset{˚}{A}; co m p u t e d : 9.2 mm a t 10 keV)$

Intensity Oscillations

Name	Formula	Related Formulae	Variables/Constants
Growth rate from oscillation period	$G = \frac{1 ML}{τ}, f = \frac{1}{τ}$	$v_{g} = \frac{d _{M L}}{τ}$	$G : g r o w t h r a t e in m o n o l a y er s p er seco n d$ $τ : osc i l l a t i o n p er i o d$ $d_{M L} : m o n o l a y er t hi c k n ess; K r u k o w s k i an d Z y t k i e w i cz l ec t u r e : g r o w t h r a t e = 1 M L / τ; L a r se n an d J oy ce : p er i o d e x a c tl y o n e m o n o l a y er, t y p i c a l l y 1 ML/s; g r o w t h - r a t e p a t e n t : f r e q u e n cy 1/ τ, 0.2 t o 0.5 ML/s w i t h e x am pl es a t 0.25 an d 0.4, a v er a g e d o v er s i x p er i o d s, er r or u n d er 3 p er ce n t, 0.282 nm p er G a A s m o n o l a y er$ $(co m p u t e d : 1, 0.25 an d 0.4 ML/s a r e 1.02, 0.25 an d 0.41 μ m/h)$
Oscillation phase and coverage	$I (Θ) minimum near Θ = \frac{1}{2}$	$ϕ = \frac{t _{32}}{T}, 1 \leq ϕ \leq 2$	$Θ : co v er a g e o f t h e g r o w in g l a y er$ $ϕ : p ha se o f t h e t hi r d minim u m r e l a t i v e t o t h e p er i o d; L a r se n an d J oy ce : ma x im u m r o ug h e nin g (minim u m s p ec u l a r in t e n s i t y) a t ha l f - l a y er co v er a g e; M i t u r a : p ha se d e f in e d a s t_{32} / T b e tw ee n 1 an d 2, d o u b l e minima f or θ b e l o w ab o u t 0.8 d e g r ees; g r o w t h - r a t e p a t e n t : f i r s t o b ser v ab l e p h o t oe mi ss i o n minim u m a t 1.55 pl u s or min u s 0.05 M L, ma x ima a t in t e g r a l an d minima a t ha l f - in t e g r a l m o n o l a y er s$ $(m y r e a d in g : t h e p ha se d e p e n d s o n an g l e an d o n h o w t h e s i g na l i s f or m e d, b u t t h e p er i o d d oes n o t, w hi c h i s w h y g r o w t h r a t e i s t ak e n f r o m t h e p er i o d)$
Damping and growth mode	$A (t) \approx A_{0} e^{- t / τ_{d}}$	$no oscillations in step-flow growth$	$A : osc i l l a t i o n am pl i t u d e$ $τ_{d} : d am p in g t im e; L a r se n an d J oy ce : osc i l l a t i o n s d am p e d w i t h g r a d u a l l y d ecr e a s in g am pl i t u d e; K r u k o w s k i an d Z y t k i e w i cz l ec t u r e : 2 D n u c l e a t i o n an d g r o u p - V - r i c h co n d i t i o n s n ee d e d, n o osc i l l a t i o n s f or s t e p f l o w, G a s h u tt er c l os u r e g i v es r eco v er y an d A l s h u tt er c l os u r e d am p in g b ec a u se o f l o w A l a d a t o m m o bi l i t y; e x p o n e n t ia l f or m (ba c k g r o u n d k n o w l e d g e, I c himi y a an d C o h e n)$ $(m y r e a d in g : d am p in g r e f l ec t s g r o w in g r o ug hn ess a s s u ccess i v e l a y er s n u c l e a t e b e f or e t h e p r e v i o u s o n e co m pl e t es)$

Composition, Reconstruction and Momentum

Name	Formula	Related Formulae	Variables/Constants
Composition from rate ratio	$x = \frac{G _{A l A s}}{G _{A l A s} + G _{G a A s}}$	$G_{A l G a A s} = G_{A l A s} + G_{G a A s}$	$x : A l A s f r a c t i o n in A l_{x} G a_{1 - x} A s$ $G : g r o u p - I I I - l imi t e d g r o w t h r a t es; r a t i o r e l a t i o n (ba c k g r o u n d k n o w l e d g e, I c himi y a an d C o h e n); g r o w t h - r a t e p a t e n t : co v er s G a A s o n l y$ $(a ss u m e d : 0.25 an d 0.75 ML/s; co m p u t e d : x = 0.25 a t 1.0 ML/s)$
Surface reconstructions	$GaAs(001): 4 \times 2, 4 \times 6, 1 \times 6, 2 \times 4, c (4 \times 4)$	$order of increasing As coverage$	$L a r se n an d J oy ce : r eco n s t r u c t i o n s in or d er o f in cr e a s in g A s p o p u l a t i o n, 2 x 4 t h e m os t s t ab l e; K r u k o w s k i an d Z y t k i e w i cz l ec t u r e : R H E E D a s a s u r f a ce t h er m o m e t er, G a A s o x i d e d esor pt i o n a t ab o u t 560^{\circ} C, S i (111) 7 x 7 t o 1 x 1 a t 830^{\circ} C$ $(m y r e a d in g : r eco n s t r u c t i o n t r an s i t i o n s g i v e t e m p er a t u r e f i x p o in t s t ha t c a l ib r a t e p y r o m e t er s an d t h er m oco u pl es in s i t u)$
Parallel momentum conservation	$k_{∥} = k_{∥, v a c} + g_{∥}$	$k_{⊥} = K sin θ$	$g_{∥} : s u r f a ce r ec i p r oc a l - l a tt i ce v ec t or$ $k_{⊥} : p er p e n d i c u l a r co m p o n e n t; L a r se n an d J oy ce : p a r a l l e l w a v e - v ec t or co n ser v a t i o n (f or p h o t oe mi ss i o n); M i t u r a : k = K s in θ w i t h r oc k in g c u r v es in 0.01 d e g r ee s t e p s; r e l a t i o n s (ba c k g r o u n d k n o w l e d g e, I c himi y a an d C o h e n)$ $(co m p u t e d : a t 2 d e g r ees an d 10 keV, k_{⊥} = 1.80 \overset{˚}{A}^{- 1})$

Important Questions

Ques: How does RHEED measure growth rate?

From the period of specular intensity oscillations. In layer-by-layer growth the surface is smooth when a monolayer is complete and roughest at half coverage, so the specular spot oscillates with a period of exactly one monolayer (Larsen and Joyce), giving a growth rate of one monolayer per period $τ$ (Krukowski and Zytkiewicz lecture). For GaAs, where one monolayer is 0.282 $nm$ (growth-rate patent), Larsen and Joyce’s typical 1 $ML/s$ is 1.02 $μ m/h$ (computed). Oscillations require 2D nucleation and group-V-rich conditions, and they vanish in step-flow growth, where adatoms attach to step edges without nucleating islands (lecture). Damping of the amplitude with time indicates gradual roughening. Opening one group-III shutter at a time gives each rate separately. Reference: P. K. Larsen and B. A. Joyce, ‘Surface Studies of MBE-Grown Semiconductor Films’, Europhysics News (1985), Philips Research Laboratories (fetch-tool summary); S. Krukowski and Z. R. Zytkiewicz, ‘Crystal Growth: Physics, Technology and Modeling’, Lecture 4 ‘Molecular beam epitaxy’, Institute of High Pressure Physics, Warsaw (March 2025) (fetch-tool summary); ‘Growth rate monitor for molecular beam epitaxy’, US patent 4,812,650 (continuation of 094,247, filed September 1987) (fetch-tool summary; title from the search listing, assignee not in the text).

Ques: Is a typical MBE growth rate 1 ML/s or 0.2 to 0.5 ML/s?

Larsen and Joyce give typically one monolayer per second, while the growth-rate patent used 0.2 to 0.5 $ML/s$ , with examples at 0.25 and 0.4. These are 1.02 against 0.25 to 0.41 $μ m/h$ of GaAs (computed). Both are realistic: 1 $ML/s$ suits thick buffer layers, while slower rates give better interface control for quantum wells and make oscillation periods of 2.5 to 4 s easier to resolve (my reading). The patent’s slower rates were chosen for its photoemission method, which averaged six periods for under 3 per cent error. For general growth, about 1 $ML/s$ is more likely the typical rate, and 0.25 to 0.5 $ML/s$ a typical calibration or interface rate (my reading). Reference: P. K. Larsen and B. A. Joyce, ‘Surface Studies of MBE-Grown Semiconductor Films’, Europhysics News (1985), Philips Research Laboratories (fetch-tool summary); ‘Growth rate monitor for molecular beam epitaxy’, US patent 4,812,650 (continuation of 094,247, filed September 1987) (fetch-tool summary; title from the search listing, assignee not in the text).

Ques: Why does the oscillation minimum not always fall at half a monolayer?

Because the measured phase depends on the diffraction conditions and the signal. Larsen and Joyce place maximum roughening, the specular minimum, at half-layer coverage. Mitura’s calculations at 10 $keV$ show a phase between 1 and 2 periods for the third minimum and double minima at glancing angles below about 0.8 degrees, because the intensity combines interference between layers with diffuse scattering. The growth-rate patent, using photoemission, saw its first observable minimum at 1.55 plus or minus 0.05 monolayers, though it states that maxima and minima correspond to integral and half-integral coverage. The period, not the phase, is therefore used for growth rate, and the half-monolayer minimum is more likely only at out-of-phase (anti-Bragg) conditions (my reading). Reference: P. K. Larsen and B. A. Joyce, ‘Surface Studies of MBE-Grown Semiconductor Films’, Europhysics News (1985), Philips Research Laboratories (fetch-tool summary); Z. Mitura, ‘Computations of RHEED intensities for growing surfaces of GaAs’, Computer Methods in Materials Science 15, 447 (2015) (fetch-tool summary; equations reconstructed from OCR); ‘Growth rate monitor for molecular beam epitaxy’, US patent 4,812,650 (continuation of 094,247, filed September 1987) (fetch-tool summary; title from the search listing, assignee not in the text).

Ques: What is the electron wavelength in RHEED?

About 0.1 $\overset{˚}{A}$ , as the Krukowski and Zytkiewicz lecture states for 5 to 20 $keV$ . With the relativistic expression the wavelength is 0.173, 0.122 and 0.0859 $\overset{˚}{A}$ at 5, 10 and 20 $keV$ (computed), and the non-relativistic value is 0.24 to 0.97 per cent too long. Mitura’s $K^{2}$ of 2650 $\overset{˚}{A}^{- 1}$ squared at 10 $keV$ gives $K$ = 51.5 $\overset{˚}{A}^{- 1}$ and 0.1221 $\overset{˚}{A}$ , matching the relativistic calculation (computed). The Ewald sphere radius, equal to $K$ , is far larger than surface reciprocal-lattice spacings, so it cuts the surface rods almost as a plane, producing streaks (background knowledge, Ichimiya and Cohen). For an assumed 0.3 m camera distance, GaAs streaks 4.0 $\overset{˚}{A}$ apart in real space are 9.2 mm apart on the screen (computed). Reference: S. Krukowski and Z. R. Zytkiewicz, ‘Crystal Growth: Physics, Technology and Modeling’, Lecture 4 ‘Molecular beam epitaxy’, Institute of High Pressure Physics, Warsaw (March 2025) (fetch-tool summary); Z. Mitura, ‘Computations of RHEED intensities for growing surfaces of GaAs’, Computer Methods in Materials Science 15, 447 (2015) (fetch-tool summary; equations reconstructed from OCR); A. Ichimiya and P. I. Cohen, Reflection High-Energy Electron Diffraction, Cambridge 2004 (background knowledge; not opened).

Ques: Why must refraction be included for RHEED Bragg peaks?

Because at glancing incidence the mean inner potential bends the beam noticeably. Mitura’s Bragg condition $sin^{2} θ_{n} = (n^{2} π^{2} + \tilde{v}_{R e} d^{2}) / (K^{2} d^{2})$ , with $\tilde{v}_{R e}$ of about -3.483 $\overset{˚}{A}^{- 1}$ squared and $d$ = 5.65 $\overset{˚}{A}$ , gives 1.347, 4.499 and 7.149 degrees for n = 4, 8 and 12 (computed), matching his predicted 1.347, 4.498 and 7.148 and his observed 4.49 and 7.14 degrees. Without refraction, $nλ = 2 d sin θ$ gives 2.48, 4.96 and 7.45 degrees (computed), errors of 1.1, 0.46 and 0.30 degrees. The low-order peak is most affected, which is why the n = 4 peak was strongly deformed (my reading). Rocking curves in 0.01 degree steps resolve these differences. Reference: Z. Mitura, ‘Computations of RHEED intensities for growing surfaces of GaAs’, Computer Methods in Materials Science 15, 447 (2015) (fetch-tool summary; equations reconstructed from OCR).

Ques: How does RHEED help calibrate substrate temperature?

Through reconstruction transitions at known temperatures. GaAs(001) shows reconstructions of 4x2, 4x6, 1x6, 2x4 and c(4x4) in order of increasing arsenic coverage, with 2x4 the most stable (Larsen and Joyce), and these depend on temperature and arsenic-to-gallium ratio, making RHEED a surface thermometer (Krukowski and Zytkiewicz lecture). The lecture gives native oxide desorption from GaAs at about 560 $^{\circ} C$ and the Si(111) 7x7 to 1x1 transition at 830 $^{\circ} C$ . These fix points calibrate pyrometers and thermocouples in situ (my reading). The growth-rate patent measured 580 to 590 $^{\circ} C$ by optical pyrometer during GaAs growth with an arsenic-to-gallium beam pressure ratio of 15 to 40, typically 17. Reference: P. K. Larsen and B. A. Joyce, ‘Surface Studies of MBE-Grown Semiconductor Films’, Europhysics News (1985), Philips Research Laboratories (fetch-tool summary); S. Krukowski and Z. R. Zytkiewicz, ‘Crystal Growth: Physics, Technology and Modeling’, Lecture 4 ‘Molecular beam epitaxy’, Institute of High Pressure Physics, Warsaw (March 2025) (fetch-tool summary); ‘Growth rate monitor for molecular beam epitaxy’, US patent 4,812,650 (continuation of 094,247, filed September 1987) (fetch-tool summary; title from the search listing, assignee not in the text).

References

P. K. Larsen and B. A. Joyce, “Surface Studies of MBE-Grown Semiconductor Films,” Europhysics News 16(4), 12 (1985). Available: https://www.europhysicsnews.org/articles/epn/pdf/1985/04/epn19851604p12.pdf
S. Krukowski and Z. R. Zytkiewicz, “Crystal Growth: Physics, Technology and Modeling,” Lecture 4, “Molecular beam epitaxy,” Institute of High Pressure Physics PAS, March 2025. Available: https://korfu.unipress.waw.pl/~stach/cg-2024-25/w16.pdf
“Growth rate monitor for molecular beam epitaxy,” US patent 4,812,650 (continuation of US application 094,247, filed September 8, 1987). Available: https://image-ppubs.uspto.gov/dirsearch-public/print/downloadPdf/4812650
Z. Mitura, “Computations of RHEED intensities for growing surfaces of GaAs,” Computer Methods in Materials Science 15(4), 447 (2015). Available: https://cmms.agh.edu.pl/public_repo/2015_4_0550.pdf
A. Ichimiya and P. I. Cohen, Reflection High-Energy Electron Diffraction, Cambridge University Press, 2004 (cited from background knowledge).

Quartz 5

Explorer

Reflection High-Energy Electron Diffraction (RHEED)