Isotope analysis in the transmission electron microscope

Abstract

The Ångström-sized probe of the scanning transmission electron microscope can visualize and collect spectra from single atoms. This can unambiguously resolve the chemical structure of materials, but not their isotopic composition. Here we differentiate between two isotopes of the same element by quantifying how likely the energetic imaging electrons are to eject atoms. First, we measure the displacement probability in graphene grown from either ¹² C or ¹³ C and describe the process using a quantum mechanical model of lattice vibrations coupled with density functional theory simulations. We then test our spatial resolution in a mixed sample by ejecting individual atoms from nanoscale areas spanning an interface region that is far from atomically sharp, mapping the isotope concentration with a precision better than 20%. Although we use a scanning instrument, our method may be applicable to any atomic resolution transmission electron microscope and to other low-dimensional materials.

ARTICLE

Received 25 May 2016 | Accepted 26 Aug 2016 | Published 10 Oct 2016

DOI: 10.1038/ncomms13040 OPEN

Isotope analysis in the transmission electron microscope

Toma Susi1, Christoph Hofer1, Giacomo Argentero1, Gregor T. Leuthner1, Timothy J. Pennycook1, Clemens Mangler1, Jannik C. Meyer1 & Jani Kotakoski1

The ngstrm-sized probe of the scanning transmission electron microscope can visualize and collect spectra from single atoms. This can unambiguously resolve the chemical structure of materials, but not their isotopic composition. Here we differentiate between two isotopes of the same element by quantifying how likely the energetic imaging electrons are to eject atoms. First, we measure the displacement probability in graphene grown from either 12C or

13C and describe the process using a quantum mechanical model of lattice vibrations coupled with density functional theory simulations. We then test our spatial resolution in a mixed sample by ejecting individual atoms from nanoscale areas spanning an interface region that is far from atomically sharp, mapping the isotope concentration with a precision better than 20%. Although we use a scanning instrument, our method may be applicable to any atomic resolution transmission electron microscope and to other low-dimensional materials.

1 Faculty of Physics, University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, Austria. Correspondence and requests for materials should be addressed to T.S. (email: mailto:[email protected]

Web End [email protected] ) or to J.K. (email: mailto:[email protected]

Web End [email protected] ).

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number of atoms in the unit cell), oj(k) the eigenvalue of the jth mode at k, and nj(k) the number of phonons with frequency oj(k).

After computing the internal energy U F T

@F @T

Spectroscopy and microscopy are two fundamental pillars of materials science. By overcoming the diffraction limit of light, electron microscopy has emerged as a particularly

powerful tool for studying low-dimensional materials such as graphene1, in which each atom can be distinguished. Through advances in aberration-corrected scanning transmission electron microscopy2,3 (STEM) and electron energy loss spectroscopy4,5, the vision of a synchrotron in a microscope6 has now been realized. Spectroscopy of single atoms, including their spin state7, has together with Z-contrast imaging3 allowed the identity and bonding of individual atoms to be unambiguously determined4,810. However, discerning the isotopes of a particular element has not been possiblea technique that might be called mass spectrometer in a microscope.

Here we show how the quantum mechanical description of lattice vibrations lets us accurately model the stochastic ejection of single atoms11,12 from graphene consisting of either of the two stable carbon isotopes. Our technique rests on a crucial difference between electrons and photons when used as a microscopy probe: due to their nite mass, electrons can transfer signicant amounts of momentum. When a highly energetic electron is scattered by the electrostatic potential of an atomic nucleus, a maximal amount of kinetic energy (inversely proportional to the mass of the nucleus, p 1M) can be transferred when the electron backscatters. When this energy is comparable to the energy required to eject an atom from the material, dened as the displacement threshold energy Tdfor instance, when probing pristine11 or doped13 single-layer graphene with 60100 keV electronsatomic vibrations become important in activating otherwise energetically prohibited processes due to the motion of the nucleus in the direction of the electron beam. The intrinsic capability of STEM for imaging further allows us to map the isotope concentration in selected nanoscale areas of a mixed sample, demonstrating the spatial resolution of our technique. The ability to do mass analysis in the transmission electron microscope thus expands the possibilities for studying materials on the atomic scale.

ResultsQuantum description of vibrations. The velocities of atoms in a solid are distributed based on a temperature-dependent velocity distribution, dened by the vibrational modes of the material. Due to the geometry of a typical transmission electron microscopy (TEM) study of a two-dimensional material, the out-of-plane velocity vz, whose distribution is characterized by the mean square velocity v2z T

, is here of particular interest.

In an earlier study11 this was estimated using a Debye approximation for the out-of-plane phonon density of states14 (DOS) gz(o), where o is the phonon frequency. A better justied estimate can be achieved by calculating the kinetic energy of the atoms via the thermodynamic internal energy, evaluated using the full phonon DOS.

As a starting point, we calculate the partition function Z Tr{e H/(kT)}, where Tr denotes the trace operation and k is

the Boltzmann constant and T the absolute temperature. We evaluate this trace for the second-quantized Hamiltonian H describing harmonic lattice vibrations15:

Z

X1nj1 k1 0 :::

V from the

partition function via the Helmholtz free energy F kT lnZ,

we obtain the Planck distribution function describing the occupation of the phonon bands (Methods). We must then explicitly separate the energy into the in-plane Up and out-of-plane Uz components, and take into account that half the thermal energy equals the kinetic energy of the atoms. This gives the out-of-plane mean square velocity of a single atom in a two-atom unit cell as

v2z T

Uz= 2M

2M

Z

oz

0 gz o

1

1

2

exp o= kT

1

o do;

2

where M is the mass of the vibrating atom, oz is the highest out-of-plane mode frequency, and the correct normalization of the number of modes

R

do 2

is included in the DOS.

Phonon dispersion. To estimate the phonon DOS, we calculated through density functional theory (DFT; GPAW package16,17) the graphene phonon band structure18,19 via the dynamical matrix using the frozen phonon method (Methods; Supplementary Fig. 1). Taking the density of the components corresponding to the out-of-plane acoustic (ZA) and optical (ZO) phonon modes (Supplementary Data 1) and solving equation 2 numerically, we obtain a mean square velocity v2z 3:17 105 m2s 2 for a 12C

atom in normal graphene. This description can be extended to heavy graphene (consisting of 13C instead of a natural isotope mixture). A heavier atomic mass affects the velocity through two effects: the phonon band structure is scaled by the square root of the mass ratio (from the mass prefactor of the dynamical matrix), and the squared velocity is scaled by the mass ratio itself (equation 2). At room temperature, the rst correction reduces the velocity by 3% in fully 13C graphene compared with normal graphene, and the second one reduces it by an additional 10%, resulting in v2z;13 2:86 105 m2s 2.

Electron microscopy. In our experiments, we recorded time series at room temperature using the Nion UltraSTEM100 microscope, where each atom, or its loss, was visible in every frame. We chose small elds of view (B1 1 nm2) and short

dwell times (8 ms) to avoid missing the relling of vacancies (an example is shown in Fig. 1; likely this vacancy only appears to be unreconstructed due to the scanning probe). In addition to commercial monolayer graphene samples (Quantifoil R 2/4, Graphenea), we used samples of 13C graphene synthesized by chemical vapour deposition (CVD) on Cu foils using

13C-substituted CH4 as carbon precursor, subsequently transferred onto Quantifoil TEM grids. An additional sample consisted of grains of 12C and 13C graphene on the same grid, synthesized by switching the precursor during growth (Methods).

From each experimental dataset (full STEM data available20) within which a clear displacement was observed, we calculated the accumulated electron dose until the frame where the defect appeared (or a fraction of the frame if it appeared in the rst one). The distribution of doses corresponds to a Poisson process12 whose expected value was found by log-likelihood minimization (Methods; Supplementary Fig. 2), directly yielding the probability of creating a vacancy (the dose data and statistical analyses are included in Supplementary Data 2). Figure 2 displays the corresponding displacement cross sections measured at voltages between 80 and 100 kV for normal (1.109% 13C) and heavy graphene (B99% 13C), alongside values measured earlier11 using

oz0 gz o

X1nj3r kN 0exp 1 kT

Xkj oj k nj k 1 2

!

1

where : is the reduced Planck constant, k the phonon wave vector, j the phonon branch index running to 3r (r being the

Ykjexp 12 oj k = kT

1

exp oj k

= kT

;

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a

b

c

d

e

f

g

Figure 1 | Example of the STEM displacement measurements. The micrographs are medium angle annular dark eld detector images recorded at 95 kV. (a) A spot on the graphene membrane, containing clean monolayer graphene areas (dark) and overlying contamination (bright). Scale bar, 2 nm.(b) A closer view of the area marked by the red rectangle in (a), with the irradiated area of the following panels similarly denoted. Scale bar, 2 . (cg) Five consecutive STEM frames (B1 1 nm2, 512 512 pixels (px), 2.2 s per frame) recorded at a clean monolayer area of graphene. A single carbon

atom has been ejected in the fourth frame (f, white circle), but the vacancy is lled already in the next frame (g). The top row of (cg) contains the unprocessed images, the middle row has been treated by a Gaussian blur with a radius of 2 px, and the coloured bottom row has been ltered with a double Gaussian procedure3 (s1 5 px, s2 2 px, weight 0.16).

high-resolution TEM (HRTEM). For low-probability processes, the cross section is highly sensitive to both the atomic velocities and the displacement threshold energy. Since heavier atoms do not vibrate with as great a velocity, they receive less of a boost to the momentum transfer from an impinging electron. Thus, fewer ejections are observed for 13C graphene.

Comparing theory with experiment. The theoretical total cross sections sd(T, Ee) are plotted in Fig. 2 for each voltage (Methods;

Supplementary Table 1, Supplementary Data 2). The motion of the nuclei was included via a Gaussian distribution of atomic outof-plane velocities P(vz, T) characterized by the DFT-calculated v2z, otherwise similar to the approach of ref. 11. A common displacement threshold energy was tted to the data set by minimizing the variance-weighted mean square error (the 100 kV HRTEM point was omitted from the tting, since it was underestimated probably due to the undetected relling of vacancies, also seen in Fig. 1). The optimal Td value was found to be 21.14 eV, resulting in a good description of all the measured cross sections. Notably, this is 0.8 eV lower than the earlier value calculated by DFT, and 2.29 eV lower than the earlier t to HRTEM data11. Different exchange correlation functionals we

tested all overestimate the experimental value (by o1 eV), with the estimate TdA[21.25, 21.375] closest to experiment resulting from the C09 van der Waals functional21 (Methods).

Despite DFT overestimating the displacement threshold energy, we see from the good t to the normal and heavy graphene data sets that our theory accurately describes the contribution of vibrations. Further, the HRTEM data and the STEM data are equally well described by the theory despite having several orders of magnitude different irradiation dose rates. This can be understood in terms of the very short lifetimes of electronic and phononic excitations in a metallic system22 compared with the average time between impacts. Even a very high dose rate of 108 e 2s 1 corresponds to a single electron passing through a 1 nm2 area every 10 10 s, whereas valence band holes are lled23 in o10 15 s and core holes24 in o10 14 s, while plasmons are damped25 within B10 13 s and phonons26 in B10 12 s. Our results thus show that multiple excitations do not contribute to the knock-on damage in graphene, warranting another explanation (such as chemical etching11) for the evidence linking a highly focused HRTEM beam to defect creation27. Each impact is, effectively, an individual perturbation of the equilibrium state.

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12 12 12

12

0.5

0.4

Cross section (barn)

0.3

0.2

0.1

80 85 90 95 100

0.0

Acceleration voltage (kV)

Figure 2 | Displacement cross sections of 12C and 13C measured at different acceleration voltages. The STEM data is marked with squares, and earlier HRTEM data11 with circles. The error bars correspond to the 95% condence intervals of the Poisson means (STEM data) or to previously reported estimates of statistical variation (HRTEM data11).

The solid curves are derived from our theoretical model with an error-weighted least-squares best-t displacement threshold energy of21.14 eV. The shaded areas correspond to the same model using the lowest DFT threshold TdA[21.25, 21.375] eV. The inset is a closer view of the low cross section region.

Local mapping of isotope concentration. Finally, to test the spatial resolution of our method, we studied a sample consisting of joined grains of 12C and 13C graphene. Isotope labelling combined with Raman spectroscopy mapping is a powerful tool for studying CVD growth of graphene28, which is of considerable technological interest. Earlier studies have revealed the importance of carbon solubility into different catalyst substrates to control the growth process29. However, the spatial resolution of Raman spectroscopy is limited, making it impossible to obtain atomic-scale information of the transition region between grains of different isotopes.

The local isotope analysis is based on tting the mean of the locally measured electron doses with a linear combination of doses generated by Poisson processes corresponding to 12C and

13C graphene using the theoretical cross section values. Although each dose results from a stochastic process, the expected doses for

12C and 13C are sufciently different that measuring several displacements decreases the errors of their means well below the expected separation (Fig. 3c). To estimate the expected statistical variation for a certain number of measured doses, we generated a large number of sets of n Poisson doses, and calculated their means and standard errors as a function of the number of doses in each set. The calculated relative errors scale as 1/n and correspond to the precision of our measurement, which is better than 20% for as few as ve measured doses in the ideal case. Although our accuracy is difcult to gauge precisely, by comparing the errors of the cross sections measured for isotopically pure samples to the tted curve (Fig. 2), an estimate of roughly 5% can be inferred.

Working at 100 kV, we selected spots containing areas of clean graphene (43 in total) each only a few tens of nanometers in size (Fig. 1a), irradiating 415 (mean 7.8) elds of view 1 1 nm2 in

size until the rst displacement occurred (Fig. 1f). Comparing the mean of the measured doses to the generated data, we can estimate the isotope concentration responsible for such a dose. This assignment was corroborated by Raman mapping over the same area, allowing the two isotopes to be distinguished by their

differing Raman shift. A general trend from 12C-rich to 13C-rich regions is captured by both methods (Fig. 3b), but a signicant local variation in the measured doses is detectable (Fig. 3c). This variation indicates that the interfaces formed in a sequential CVD growth process may be far from atomically sharp30, instead spanning a region of hundreds of nanometers, within which the carbon isotopes from the two precursors are mixed together.

DiscussionIt is interesting to compare our method to established mass analysis techniques. In isotope ratio mass spectrometry precisions of 0.01% and accuracies of 1% have been reported31. However, these measurements are not spatially resolved. For spatially resolved techniques, one of the most widely used is time-of-ight secondary ion mass spectroscopy (ToF-SIMS). It has a lateral resolution typically of several micrometers, which can be reduced to around 100 nm by nely focusing the ion beam32. In the case of ToF-SIMS, separation of the 13C signal from 12C1H is problematic, resulting in a reported33 precision of 20% and an accuracy of B11%. The state-of-the-art performance in local mass analysis can be achieved with atom-probe tomography34 (APT), which can record images with sub-nanometer spatial resolution in all three dimensions. A recent APT study of the

13C/12C ratio in detonation nanodiamonds reported a precision of 5%, but biases in the detection of differently charged ions limited accuracy to B25% compared to the natural isotope abundances35.

A limitation of ToF-SIMS is its inability to discriminate between the analyte and contaminants and that it requires uniform isotope concentrations over the beam area for accurate results. APT requires the preparation of specialized needle-like sample geometries, a laborious reconstruction process to analyse its results36, and its detection efciency is rather limited37. In our case, we are only able to resolve relative mass differences between isotopes of the same element in the same chemical environment. While we do not need to resolve mass differences between different elements, since these differ in their scattering contrast, we do need to detect the ejection of single atoms, limiting the technique to atomically thin materials. However, our method captures the isotope information concurrently with atomic resolution imaging in a general-purpose electron microscope, without the need for additional detectors.

We have shown how the ngstrm-sized electron probe of a scanning transmission electron microscope can be used to estimate isotope concentrations via the displacement of single atoms. Although these results were achieved with graphene, our technique should work for any low-dimensional material, including hexagonal boron nitride and transition metal dichalcogenides such as MoS2. This could potentially extend to van der Waals heterostructures38 of a few layers or other thin crystalline materials, provided a difference in the displacement probability of an atomic species can be uniquely determined. Neither is the technique limited to STEM: a parallel illumination TEM with atomic resolution would also work, although scanning has the advantage of not averaging the image contrast over the eld of view. The areas we sampled were in total less than 340 nm2 in size, containing B6,600 carbon atoms of which 337 were ejected. Thus, while the nominal mass required for our complete analysis was already extremely small (131 zg), the displacement of only ve atoms is required to distinguish a concentration difference of less than twenty per cent. Future developments in instrumentation may allow the mass-dependent energy transfer to be directly measured from high-angle scattering39,40, further enhancing the capabilities of STEM for isotope analysis.

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a

b

1.0

2 3 4 5 6

8 9 10 11 12

13 16 18

24

30

12 C concentration

0.8

0.6

0.4

0.2

0.0

14 15 17

20 21 22 23

26 27 28 29

32 33 34 35

19

25

31 36

c

Electron dose (109 e )

12 16 30 33 34

17 20 22 23 24 26 27 28 29

10

5

1

0.5

Figure 3 | Local isotope analysis. (a) A STEM micrograph of a hole in the carbon support lm (1.3 mm in diameter), covered by a monolayer of graphene. In each of the overlaid spots, 415 elds of view were irradiated. The dimensions of the overlaid grid correspond to the pixels of a Raman map recorded over this area. (b) Isotope concentration map where the colours of the grid squares denote 12C concentration based on the tting of the Raman 2D band response (Methods; Supplementary Fig. 3). The overlaid spots correspond to (a), with colours denoting the concentration of 12C estimated from the mean of the measured doses. (c) Locally measured mean doses and their standard errors plotted on a log scale for each grid square. The horizontal coloured areas show the meanss.e. of doses simulated for the theoretical 12C and 13C cross sections. Note that a greater variation in the experimental doses is expected for areas containing a mix of both carbon isotopes.

Methods

Quantum model of vibrations. The out-of-plane mean square velocity v2

z T

can

be estimated by calculating the kinetic energy via the thermodynamic internal energy using the out-of-plane phonon DOS gz(o), where o is the phonon frequency. In the second quantization formalism, the Hamiltonian for harmonic lattice vibrations is ref. 15

H

XN k

and the internal energy of a single unit cell, therefore, becomes15

U F T

@F @T

V X

kj

1

2 oj k

coth oj k

= 2kT

3r

Z

1

2 coth o= 2kT

g o

o do;

X3rj1 oj k bykjbkj 1 2

6

where in the last step the sum is expressed as an average over the phonon DOS. Using the identity 12 coth x=2

12 1= exp x

1

yields the Planck distribution

function describing the occupation of the phonon bands, and explicitly dividing the energy into the in-plane Up and out-of-plane Uz components, we can rewrite this as

U Up Uz Z

o

0 gp o

; 3

where k is the phonon wave vector, j is the phonon branch index running to3r (r being the number of atoms in the unit cell), oj(k) the eigenvalue of the jth

mode at k, and by

kj and bkj are the phonon creation and annihilation operators, respectively.

Using the partition function Z Tr{e H/(kT)}, where Tr denotes the trace

operation and k is the Boltzmann constant and T the absolute temperature, and evaluating the trace using this Hamiltonian, we have

Z

X1n k 0:::

1

gz o

1

2

o do; 7

where the number of modes is included in the normalization of the DOSes, that is,

R

o0 gz o

do 2, corresponding to the out-of-plane acoustic (ZA) and optical

(ZO) modes (the in-plane DOS gp(o) being correspondingly normalized to 4), and od is the highest frequency of the highest phonon mode.

Since half of the thermal energy equals the average kinetic energy of the atoms, and the graphene unit cell has two atoms, the out-of-plane kinetic energy of a single atom is

Ek;z

1

2 Mv

2 z

exp o= kT

1

X1n k 0exp 1 kT

Xkj oj k nj k 1 2

!

Ykjexp 12 oj k = kT

1

exp oj k

= kT

;

4

1

2 Uz: 8

Thus, the out-of-plane mean square velocity of an atom becomes

v2

z T

Uz= 2M

1

2

where nj k bykjbkj is the number of phonons with frequency oj(k).

The Helmholtz free energy is thus

F kT ln Z kTXkjln 2 sinh oj k = 2kT

5

2M

Z

o

0 gz o

1

2

1

o do; 9

where oz is now the highest out-of-plane mode frequency. This can be solved numerically for a known gz(o).

exp o= kT

1

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For the in-plane vibrations, we would equivalently get

The statistical analyses were conducted using the Wolfram Mathematica software (version 10.5), and the Mathematica notebook is included as Supplementary Data 2. Outputs of the Poisson analyses for the main data sets of normal and heavy graphene as a function of voltage are additionally shown as Supplementary Fig. 2.

Displacement cross section. The energy transferred to an atomic nucleus from a fast electron as a function of the electron scattering angle y is ref. 47

E y

Emaxsin2

y 2

v2

p

v

2 x

v2y Up= 2M

2M

Z

o

0 gp o

1

2

1

o do: 10

Frozen phonon calculation. To estimate the phonon DOS, we calculated the graphene phonon band structure via the dynamical matrix, which was computed by displacing each of the two primitive cell atoms by a small displacement (0.06 ) and calculating the forces on all other atoms in a 7 7 supercell (frozen phonon

method; the cell size is large enough so that the forces on the atoms at its edges are negligible) using DFT as implemented in the grid-based projector-augmented wave code (GPAW) package17. Exchange and correlation were described by the local density approximation41, and a G-centered Monkhorst-Pack k-point mesh of42 42 1 was used to sample the Brillouin zone. A ne computational grid

spacing of 0.14 was used alongside strict convergence criteria for the structural relaxation (forces o10 5 eV 1 per atom) and the self-consistency cycle (change in eigenstates o10 13 eV2 per electron). The resulting phonon dispersion (Supplementary Fig. 1) describes well the quadratic dispersion of the ZA mode near G, and is in excellent agreement with earlier studies18,19. Supplementary

Data 1 contains the out-of-plane phonon DOS.

Graphene synthesis and transfer. In addition to commercial monolayer graphene (Graphenea QUANTIFOIL R 2/4), our graphene samples were synthesized by CVD in a furnace equipped with two separate gas inlets that allow for independent control over the two isotope precursors29 (that is, either B99%

12CH4 or B99% 13CH4 methane). The as-received 25 mm thick 99.999% pure Cu foil was annealed for B1 h at 960 C in a 1:20 hydrogen/argon mixture with a pressure of B10 mbar. The growth of graphene was achieved by owing50 cm3 min 1 of CH4 over the annealed substrate while keeping the Ar/H2 ow, temperature and pressure constant. For the isotopically mixed sample with separated domains, the annealing and growth temperature was increased to 1,045 C and the ow rate decreased to 2 cm3 min 1. After introducing 12CH4 for 2 min the carbon precursor ow was stopped for 10 s, and the other isotope precursor subsequently introduced into the chamber for another 2 min. This procedure was repeated with a ow time of 1 min. After the growth, the CH4 ow was interrupted and the heating turned off, while the Ar/H2 ow was kept unchanged until the substrate reached room temperature. The graphene was subsequently transferred onto a holey amorphous carbon lm supported by a TEM grid using a direct transfer method without using polymer42.

Scanning transmission electron microscopy. Electron microscopy experiments were conducted using a Nion UltraSTEM100 scanning transmission electron microscope, operated between 80 and 100 kV in near-ultrahigh vacuum(2 10 7 Pa). The instrument was aligned for each voltage so that atomic

resolution was achieved in all of the experiments. The beam current during the experiments varied between 8 and 80 pA depending on the voltage, corresponding to dose rates of B550 107 e 2s 1. The beam convergence semiangle was

30 mrad and the semi-angular range of the medium-angle annular-dark-eld detector was 60200 mrad.

Poisson analysis. Assuming the displacement data are stochastic, the waiting times (or, equivalently, the doses) should arise from a Poisson process with mean l. Thus the probability to nd k events in a given time interval follows the Poisson distribution

f k; l

Pr X k

lke l

exp o= kT

1

; 14

which is valid also for a moving target nucleus for electron energies 410 keV as noted by Meyer and co-workers11. For purely elastic collisions (where the total kinetic energy is conserved), the maximum transferred energy Emax corresponds to electron backscattering, that is, y p. However, when the impacted atom is

moving, Emax will also depend on its speed.

To calculate the cross section, we use the approximation of McKinley and Feshbach48 of the original series solution of Mott to the Dirac equation, which is very accurate for low-Z elements and sub-MeV beams. This gives the cross section as a function of the electron scattering angle as

s y

sR 1 b2 sin2 y=2

p

Ze2

c b sin y=2

1 sin y=2

; 15

where b v/c is the ratio of electron speed to the speed of light (0.446225 for

60 keV electrons) and sR is the classical Rutherford scattering cross section

sR

Ze2 4pE02m0c2

2

1 b2

b4 csc4 y=2

: 16

Using equation 14 this can be rewritten as a function of the transferred energy49 as

s E

Ze2 4pE02m0c2

21

b2

b4 1 b2

Emax

E

EEmax p

Ze2

c b

E Emax

r

E Emax

:

17

Distribution of atomic vibrations. The maximum energy (in eV) that an electron with mass me and energy Ee eU (corresponding to acceleration voltage U) can

transfer to a nucleus of mass M that is moving with velocity v is

Emax v; Ee

r t

2

2M

2 En p

2

p

2 Mvc

2Mc2 ;

18

where r 1c

Ee Ee 2mec

Ee En

E 2mec

p

2 Mv and t 1c

Ee Ee 2mec

2 En p

are

Ee En

Ee 2mec

the relativistic energies of the electron and the nucleus, and En Mv2/2 the initial

kinetic energy of the nucleus in the direction of the electron beam.

The probability distribution of velocities of the target atoms in the direction parallel to the electron beam follows the normal distribution with a standard deviation equal to the temperature-dependent mean square velocity v2

z T

,

P vz; T

q

1 exp v2z

2v2

z T

: 19

Total cross section with vibrations. The cross section is calculated by numerically integrating equation 17 multiplied by the Gaussian velocity distribution (equation 19) over all velocities v where the maximum transferred energy (equation 18) exceeds the displacement threshold energy Td:

s T; Ee

Z

E v;E

T

P v; T

2pv2

z T

k ! : 11 To estimate the Poisson expectation value for each sample and voltage, the cumulative doses of each data set were divided into bins of width w (using one-level recursive approximate Wand binning43), and the number of bins with 0, 1, 2... occurrences were counted. The goodness of the ts was estimated by calculating the Cash C-statistic44 (in the asymptotically-w2 formulation45) between a tted Poisson distribution and the data:

C

2 N

s Emax v; Ee

dv 20

Z

1 exp v2

2v2

z T

Emax v; Ee

E

21

b2

b4

v

Ze2 4pE02m0c2

0

2pv2

z T

XNi1ni ln niei ni ei

q

; 12

where N is the number of occurence bins, ni is the number of events in bin i, and ei is the expected number of events in bin i from a Poisson process with mean l.

An error estimate for the mean was calculated using the approximate condence interval proposed for Poisson processes with small means and small sample sizes by Khamkong46:

CI95%

l

Z22:5

2n Z2:5

21

where Emax(v, Ee) is given by equation 18, the term Y[Emax(v, Ee) Ed] is the

Heaviside step function, T is the temperature and Ee is the electron kinetic energy.

The upper limit for the numerical integration vmax 8

v2

z

!

" #

Y Emax v; Ee Ed

dv;

1 b2

EEmax v; Ee

p Ze2

c b

s

E Emax v; Ee

E Emax v; Ee

q

s ; 13

where l is the estimated mean and Z2.5 is the normal distribution single tail cumulative probability corresponding to a condence level of (100 a) 95%,

equal to 1.96.

was chosen so that the

l n

velocity distribution is fully sampled.

Displacement threshold simulation. For estimating the displacement threshold energy, we used DFT molecular dynamics as established in our previous studies12,13,50,51. The threshold was obtained by increasing the initial kinetic energy

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of a target atom until it escaped the structure during the molecular dynamics run. The calculations were performed using the grid-based projector-augmented wave code (GPAW), with the computational grid spacing set to 0.18 . The molecular dynamics calculations employed a double zeta linear combination of atomic orbitals basis52 for a 8 6 unit cell of 96 atoms, with a 5 5 1 Monkhorst-Pack

k-point grid53 used to sample the Brillouin zone. A timestep of 0.1 fs was used for the Velocity-Verlet dynamics54, and the velocities of the atoms initialized by a MaxwellBoltzmann distribution at 50 K, equilibrated for 20 timesteps before the simulated impact.

To describe exchange and correlation, we used the local density approximation41, and the Perdew-Burke-Ernzerhof (PBE)55, Perdew-Wang 1991 (PW91, ref. 41), RPBE56 and revPBE57 functionals, yielding displacement threshold energies of 23.13, 21.88, 21.87, 21.63 and 21.44 eV (these values are the means of the highest simulated kinetic energies that did not lead to an ejection and the lowest that did, respectively). Additionally, we tested the C09 (ref. 21) functional to see whether inclusion of the van der Waals interaction would affect the results. This does bring the calculated threshold energy down to [21.25, 21.375] eV, in better agreement with the experimental t. However, a more precise algorithm for the numerical integration of the equations of motion, more advanced theoretical models for the interaction, or time-dependent DFT may be required to improve the accuracy of the simulations further.

Varying mean square velocity with concentration. Since the phonon dispersion of isotopically mixed graphene gives a slightly different out-of-plane mean square velocity for the atomic vibrations, for calculating the cross section for each concentration, we assumed the velocity of mixed concentration areas to be linearly proportional to the concentration

vmix cv12 1 c

v13; 22

where c is the concentration of 12C and v12/13 are the atomic velocities for normal

and heavy graphene, respectively.

Raman spectroscopy. A Raman spectrometer (NT MDT Ntegra Spectra) equipped with a 532 nm excitation laser was used for Raman measurements.

A computer-controlled stage allowed recording a Raman spectrum map over the precise hole on which the electron microscopy measurements were conducted, which was clearly identiable from neighboring spot contamination and broken lm holes.

The frequencies o of the optical phonon modes vary with the atomic mass M as opM 1/2 due to the mass prefactor of the dynamical matrix. This makes the

Raman shifts of 13C graphene (12/13) 1/2 times smaller, allowing the mapping and localization of 12C and 13C domains28 with a spatial resolution limitedby the size of the laser spot (nominally B400 nm). The shifts of the G and 2D bands compared with a corresponding normal graphene sample are given

by o c

o12 1

12 c

12 1 c

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q

h i

, where o12 is the G (2D) line frequency of the normal sample, c013 0.01109 is the natural abundance of 13C, and c is the

unknown concentration of 12C in the measured spot.

Due to background signal arising from the carbon support lm of the TEM grid, we analyzed the shift of the 2D band, where two peaks were in most locations present in the spectrum. However, in many spectra these did not correspond to either fully 12C or 13C graphene58, indicating isotope mixing within the Raman coherence length. To assign a single value to the 12C concentration for the overlay of Fig. 3, we took into account both the shifts of the peaks (to estimate the nominal concentration for each signal) and their areas (to estimate their relative abundances) as follows:

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tted 2D spectra, arranged in the same 6 6 grid as the overlay, can be found as

Supplementary Fig. 3

Data availability. The full STEM time series data on which the determination of the 12C and 13C displacement cross sections (Fig. 2) are based are available on gshare with the identier http://dx.doi.org/10.6084/m9.figshare.c.3311946doi:10.6084/m9.figshare.c.3311946

Web End =http://dx.doi.org/10.6084/m9.gshare.c.3311946 (ref. 20). The STEM data of Fig. 3 are available upon request. All other data are contained within the article and its Supplementary Information les.

o12

o12

oB

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o12

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NATURE COMMUNICATIONS | 7:13040 | DOI: 10.1038/ncomms13040 | http://www.nature.com/naturecommunications

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Acknowledgements

T.S. acknowledges funding from the Austrian Science Fund (FWF) via project P 28322-N36, and the computational resources of the Vienna Scientic Cluster.

J.K. acknowledges funding from the Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) via project MA14-009. C.H., G.A., C.M. and J.C.M. acknowledge funding by the European Research Council Grant No. 336453-PICOMAT. T.J.P. was supported by the European Unions Horizon 2020 research andinnovation programme under the Marie Skodowska-Curie grant agreementNo. 655760-DIGIPHASE. We further thank Ondrej Krivanek for useful feedback.

Author contributions

T.S. performed theoretical and statistical analyses and DFT simulations, participated in STEM experiments and their analysis, and drafted the manuscript. C.H. performed sample synthesis, and participated in STEM experiments, their analysis, and the Raman analysis. G.A. participated in sample synthesis and the Raman analysis. G.T.L. participated in STEM experiments and their analysis. C.M. and T.J.P. prepared special alignments for the STEM instrument with J.K., who supervised the theoretical and statistical analyses and STEM experiments. J.K. and J.C.M conceived and supervised the study.

Additional information

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Competing nancial interests: T.S., J.C.M. and J.K. are named on a patent application relating to this method of isotope analysis (application number EP16183371). All other authors declare no competing nancial interests.

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How to cite this article: Susi, T. et al. Isotope analysis in the transmission electron microscope. Nat. Commun. 7, 13040 doi: 10.1038/ncomms13040 (2016).

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