ULTRAFAST DYNAMICS

Four-dimensional imaging of carrierinterface dynamics in p-n junctionsEbrahim Najafi,1 Timothy D. Scarborough,1 Jau Tang,1,2 Ahmed Zewail1*

The dynamics of charge transfer at interfaces are fundamental to the understanding ofmany processes, including light conversion to chemical energy. Here, we report imaging ofcharge carrier excitation, transport, and recombination in a silicon p-n junction, wherethe interface is well defined on the nanoscale. The recorded images elucidate thespatiotemporal behavior of carrier density after optical excitation. We show that carrierseparation in the p-n junction extends far beyond the depletion layer, contrary to theexpected results from the widely accepted drift-diffusion model, and that localization ofcarrier density across the junction takes place for up to tens of nanoseconds, dependingon the laser fluence. The observations reveal a ballistic-type motion, and we provide amodel that accounts for the spatiotemporal density localization across the junction.

Multiple spectroscopic techniques haveenhanced the understanding of chargecarrier dynamics in semiconductors (1–3).However, as the size approaches the crit-ical limit of the nanoscale (4), it becomes

necessary to investigate the behavior of carriersat a high spatiotemporal resolution to elucidatethe extent of spatial and density localizations atthese scales. In semiconductors, carrier excita-tion, transport, and recombination occur ontime scales that span a wide range, from a fewfemtoseconds to hundreds of microseconds (5),and these processes are dramatically altered innanostructures due to their low dimensionalityand the large surface-to-volume ratio (6, 7). Al-though a large body of literature exists oncarrier dynamics in bulk semiconductors, studiesof surfaces and interfaces demand the resolutionof the dynamics in both space and timewith highenough sensitivity. Recently developed scanningelectron-probe microscopies, which combine thespatial resolution of electronmicroscopy and thetemporal resolution of laser spectroscopy, makesuch studies possible (8).Here, we report direct imaging of carrier in-

terfacial dynamics in the silicon p-n junction byscanning ultrafast electron microscopy (SUEM).We image the spatiotemporal evolution of carrierdensity after excitation and follow the behaviorof transport and recombination of carriers. It isshown, with the spatial resolution of the electronprobe, that optically induced long-range carriertransport can span tens of micrometers. Snap-shots of the carrier density on the picosecondtime scale indicate the localization of excesscarriers and the associated electric field acrossthe junction. The displaced carriers remain lo-calized in both space and time (up to tens ofnanoseconds) until they cross the junction and

recombine. With high excitation fluence, carrierlocalizationdistorts the effective junctionpotentialand acts as a locally time-dependent field, pro-viding an energetically favorable pathway to re-combination. These observations were accountedfor by developing amodel that describes both theballistic-type carrier transport and the decay ofenergy to the lattice.The p-n junction diodes were purchased as

wafers fromEL-CAT Inc. and usedwithout furthermodifications. The wafers consisted of phosphorus-doped n-type silicon (1.4 × 1014 cm−3) epitaxiallygrown on boron-doped p-type silicon (9.4 ×1018 cm−3) with (111) crystal orientation; this re-sults in an intrinsic potential (Vintrinsic) of 0.79 V (fig.S1). No external field was applied to the junction.The samples were cleaved and immediately trans-ferred into the SUEM chamber, where the pres-sure was maintained at 1.2 × 10−6 torr. Surfaceswith the root mean square (RMS) roughness ofless than 100 nm [orders of magnitude smallerthan the estimated diffusion length in silicon (9)]

were selected for this experiment to minimizeinterruptions of carrier diffusion. As shown be-low, the ballistic coherence length of carriers istens of micrometers, a much larger value thanthe roughness scale, and thus it is relatively ir-relevant on the picosecond time scale. Images ofa typical surface studied statically—i.e., withouttime resolution—are provided in Fig. 1 and fig. S2.In SUEM, femtosecond infrared pulses (1030nm,

300 fs) are split to generate green (515 nm)and ultraviolet (UV) (257 nm) pulses. The greenis used to pump the specimen samples, whereasthe UV is directed toward the microscope’s pho-tocathode to generate short pulses of electronsfrom the field emission tip; details of the designand operation were given previously (10). Thetime delay between the optical pump and elec-tron probe pulses was controlled, covering therange from −680 ps to þ3:32 ns with the snap-shot resolution of ~2 ps. The pulsed electrons,accelerated to 30.0 kV by electrostatic lenses,produce secondary electrons (SEs) from the top1 to 5 nm (11) of the sample’s surface, which arethen collected by the SE detector. To generate animage, the electron probewas scanned across thearea of interest, which included the image regionfor carrier transport, typically using a resolutionof 200 nm, which is sufficient for a surface scanover tens of micrometers. It has been shown thatthis spatial resolution can be made better than10 nm (10). The experimental setup is schemat-ically depicted in Fig. 1.Under our experimental conditions, the tran-

sient surface charge density in the local vacuumis estimated to be less than ~107 cm−2 due to thefluences used (maximum 1.28 mJ/cm2), indicat-ing that surface field transients do not stronglyinterfere with the electron probe (12). Within agiven material, the number of emitted SEs de-pends on the topography and composition (13),but, in all SUEM experiments presented here, weobtain the change in carrier density by sub-tracting the image recorded at negative time

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1Physical Biology Center for Ultrafast Science andTechnology, Arthur Amos Noyes Laboratory of ChemicalPhysics, California Institute of Technology, Pasadena, CA91125, USA. 2Institute of Photonics, National Chiao TungUniversity, 300 Hsinchu, Taiwan.*Corresponding author. E-mail: zewail@caltech.edu

Fig. 1. Schematic ofSUEM and imaging ofthe junction.The 515-nmoptical pump initiatescarrier excitation to theconduction band. Transportof charges to the junctionthen takes place in thediode while the scanningelectron probe measuresthe induced changes atdifferent time delays. Theinset is an SUEM imageof the diode recorded at–680 ps; the well-knownbut unintuitive contrastbetween n-type and p-typesilicon in the diode is due tothe energetics of the localbending of the vacuum level(15–18) and simply reflectsthe (T) surface charge involvement in enhancing or suppressing SEs in the p (n) regions.

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(reference image recorded at −680 ps) from theimage obtained at a given positive time, i.e.,Iðrx;y; tþÞ − Iðrx;y; t−Þ, where rx,y is the spatialpixels scanned and tT are the frame times aftertime zero (t+) and before time zero (t–). Thisresults in the so-called “contrast image” in whichthe bright and dark contrasts correspond, respec-tively, to increases in local electron and holedensities (fig. S3), and the difference image becomesinsensitive to collection and other possible fluctua-tions. Such images provide the snapshots thatare used to construct movies S1 to S3.After optical excitation in the specimen, elec-

trons are promoted to the conduction band, andin this case the junction separates electron-hole(e-h) pairs, with e and h drifting in opposite di-rections, a process usually described by the drift-diffusion model (14); this locally increases thedensity of majority carriers in both n-type andp-type. In the absence of additional fields, thecharge separation occurs solely in the vicinityof the junction, where excited carriers are in-fluenced by the static junction potential. Theinset in Fig. 1 shows the raw SUEM image ofthe diode recorded at a negative time delay

(−680 ps), long before the arrival of the ini-tiating pump pulse. The observed difference inbrightness between p-type and n-type regionsin the image agrees with previous observations(15, 16) and is attributed to the difference ineffective electron affinities between the tworegions (17, 18). The pump pulse was guided tosymmetrically illuminate n-type and p-type, ashighlighted by the dotted ellipse in Fig. 1. Adeviation of up to several micrometers in beamplacement may occur, although it can be ac-counted for, as discussed below.To understand the influence of the junction

on the dynamics, we first examined n-type andp-type silicon individually without the junctionand with doping levels and crystal orientationssimilar to their counterparts in the diode. Figure 2,A and B, shows contrast images recorded forn-type and p-type silicon, respectively, at varioustime delays. The images clearly display a brightcontrast in both samples after the optical excita-tion; the contrast is rather weak in n-type, with aspatial spread comparable to the laser profile onthe surface, but is considerably stronger in p-type,with a larger spatial extent. This is attributed to

the larger absorption cross section inheavily dopedp-type silicon (19). In both samples, the brightcontrast gradually decays at long times. Such be-havior in lightly doped n-type and heavily dopedp-type is independent of laser fluence. If thepresence of the junction potential does not alterthe dynamics, we would expect to observe thesame behaviors in the n-type and p-type regionsof the diode as those shown in Fig. 2, A and B.The temporal behavior of the junction is sum-

marized in Fig. 2C for 1.28 mJ/cm2 fluence; fullsets of contrast images can be viewed in moviesS1 to S3. The dynamics can qualitatively be de-scribed as follows. Immediately after the excita-tion (the frame at +6.7 ps), both layers are bright,mirroring their individual behaviors in the ab-sence of the junction. After 36.7 ps, charges aretransported toward the junction, resulting in ex-cess electron and hole density in n-type and p-type,respectively; the depletion layer at the junctionremains dark due to surface patch fields thathinder SE detection (17, 18). After transport acrossthe junction (+80 ps), the density of excess car-riers reaches its maximum. The spatial extent ofcharge separation is estimated to span tens of mi-crometers within the first ~80 ps. The observed“bending” at the junction is due to the asymmetryin the pump beam placement, as discussed in thesupplementary materials. Such drastic differencein contrast behavior, with and without the junc-tion, provides the underpinning for obtainingthe density evolution.At longer times, the diode relaxes toward equi-

librium as carriers move back across the junctionto recombine. Analysis of images for all regions(fig. S11) indicates that the relaxation towardequilibrium occurs more quickly at higher flu-ences. In fact, at 0.16 mJ/cm2 fluence, the decaysessentially plateau within the experimental timescale of 3.32 ns. The transients suggest that therecombination dynamics are influenced by thenumber of displaced carriers; specifically, asthe system is removed further from equilibriumat larger fluences, recombination is increasinglyfacilitated. Moreover, the high spatial resolutionof SUEM provides a range of apparent dynam-ics throughout the illuminated region due to thespatial profile of the laser pulse; carriers near thecenter of the laser profile, where the local fluenceis highest, recombine more quickly than regionsfarther away.The rapid transport of carriers over tens of

micrometers in only ~80 ps is not expected fromthe conventional drift-diffusion model. Accord-ing to such a model, carrier drift occurs onlywithin the depletion layer, where the intrinsicelectric field is nonzero. The longitudinal diffu-sion coefficient of electrons in silicon as a func-tion of electric field strength (at 300 K) has beenmeasured (9) and ranges from 10 to 30 cm2/s. Itfollows that such values for a 50-mmdistance willyield dynamics on the microsecond time scale,contrary to our observation. Furthermore, theinvolvement of lattice phonons at a velocity of103 m/s cannot account for the observed pico-second time scale behavior. For these reasons, wehave developed the following model, illustrated

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Fig. 2. Dynamics with and without the junction. Silicon (A) n-type and (B) p-type samples without ajunctionwere studied as individual wafers. Both n-type (1.4 × 1014 cm−3 P-doped) and p-type (9.4× 1018 cm−3

B-doped) show a bright signal (increased electron density) under laser illumination, in contrast to thebehavior when the junction is present. (C) A series of contrast images in which the bright and dark contrastscorrespond to the local electron and hole densities, respectively, relative to the signal at negative time.Theseframes were chosen to display the unperturbed signal (–470 ps), excitation (6.7 ps), carrier separation aftertransport (36.7, 80 ps), and relaxation toward equilibrium (653.3 ps, 3.32 ns).

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in Fig. 3, A and D, which accounts for the ob-served results.The excitation of the junction with 2.41 eV

photons results in carriers with excess energyequivalent to a temperature of ~104 K. In themodel, the important point is that these carriersmove ballistically at a relatively high speed, onthe order of 106 m/s, and on the picosecond timescale the lattice phonons cannot fully quenchsuchmotion. In the supplementarymaterials, weconsider the influence of the lattice, which drainsthe energy (and reduces the speed) as a functionof time. It is shown that, even for electron-latticeinteraction times on the order of 10 ps, the ob-served carrier behavior is robust. The initial ex-

cited carrier population follows the laser profile.With the junction placed at x ¼ 0 in the surface(x-y) plane, the initial e-h pair density is deter-mined for p-type (rpe;h) and n-type (r

ne;h) and takes

on a Gaussian profile (see the supplementarymaterials). The velocities of motion in the x-yplane can similarly be expressed for a given tem-perature and for the subpicosecond equilibratedcarriers.As shown in the supplementary materials, the

initial density profile at t = 0 moves (x – vt) withvelocity v. By integrating over all velocities of theproduct of density and the Maxwell-Boltzmannvelocity distribution, and considering the carrierdensity asymmetry at the interface, we can ex-

press the spatiotemporal evolution of the densityin the x-y plane. For p-type, we have

rpe ðx; y; tÞ ¼Np

2pðs2 þ v2e t2Þ exp −

x2 þ y2

2ðs2 þ v2e t2Þ

� �� 1 − erf

s

vetffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðs2 þ v2e t

2Þp x

!" #ð1Þ

rphðx; y; tÞ ¼Np

pðs2 þ v2ht2Þ exp −

x2 þ y2

2ðs2 þ v2ht2Þ

� �� hð−xÞ ð2Þ

whereNp is the total number of photoexcited e-hpairs; s is the RMS half-width of the laser focus;

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Fig. 3. Theoretical modeling of carrier separa-tion and transport. Shown are the behaviors ofisotropically expanding electrons (red) and holes(blue) after the excitation in (A) p-type and (D)n-type; the arrows represent the initial (x) velocitydirections. In both layers, the minority carriers areable to cross the junction, whereas the majoritycarriers are reflected by the junction.This results innet charge separation represented in the figure bythe shaded regions. Calculated also are the (B)electron and (C) hole densities originated in p-typeand the (E) electron and (F) hole densities orig-ated in n-type. All density calculations reflect thebehavior after +20 ps of propagation time. Thesecalculations show the extent of carrier expansionin space and at different times. The scale of thenormalized density shown (0 to 2 × 10−4) whenmultiplied by Np, which is 109 e-h pairs, gives theactual density.

Fig. 4. Comparison between experiment and theory.Shown are comparisons between experimental chargedensity and electric potential and those predicted by thetheoretical model following transport. (A) The net theo-retical charge density at +20 ps (the asymmetry of ex-citation is included).The presence of long-range transport,up to tens of micrometers, is evident. The scale of thenormalized density shown (–0.16 × 10−5 to 4.1 × 10−5)when multiplied by Np, which is 109 e-h pairs, gives theactual density. (B) The landscape calculated directly fromthe Coulomb interactions between separated carriers.The dynamic potential (due to net charge localization)reaches more than 300 meV, which is of the same orderof magnitude as the junction potential (0.79 eV). Thisinfluences charge localization and carrier recombination.(C) Experimental contrast image of the junction that mir-rors charge densities in n-type and p-type after chargeseparation. (D) The dynamic potential map calculateddirectly from the experimental data after charge sepa-ration by considering each pixel of the net electrons andholes corresponding to bright and dark contrast and cal-culating the Coulomb potential. The apparent tilt in thefigure is due to the inclined angle of the incident laserbeam, which is about 15° with respect to the junction.

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ve and vh are the average thermal velocities ofelectrons and holes, respectively; and h(–x) is theHeaviside step function. Similar analysis for n-typeis presented in the supplementary materials.Figure 3 displays the density distributions in

space as a snapshot at t = +20 ps; adding up thedistributions for carriers originating in p-type(Fig. 3, B and C) and n-type (Fig. 3, E and F)affords the net carrier density. The net electron(red) and hole (blue) densities at t = +20 ps (fromEqs. 1 and 2) in Fig. 3, A and D, are shown asdotted curves. Taking p-type as an example, elec-trons with initial velocity toward the junctionare transported to n-type, whereas holes movingtoward the junction face an impassable barrier.This gating behavior is illustrated in Fig. 3A(charges initially fromp-type) and Fig. 3D (chargesinitially from n-type). The net charge separationleads to localization of electrons on one side andholes on the other, shown as shaded regions.This creates a long-range electric field, the po-tential of which is given in Fig. 4.The sum of these four images, weighted to

reflect the increased absorption in p-type, ispresented in Fig. 4A. The figure depicts the long-range separation of carriers in agreement withthe experimental observations. Separation dueto the gating effect creates a net carrier distribu-tion localized across the junction. The resultantCoulomb interactions distort the electric poten-tial landscape as illustrated in Fig. 4B. For com-parison, we present an experimental contrastimage (Fig. 4C), which mirrors the net carrierdistribution, and a potential map (Fig. 4D) calcu-lated directly from the experimental data. Com-parison between simulation and experiment showsagreement on the spatial extent of the distortions;this agreement validates the ballistic expansionof carriers as the underlying mechanism. Theresulting electric potential, which opposes theintrinsic built-in junction potential, acts as spa-tiotemporally isolated “forward bias.” At high flu-ences, this distortion induces a larger restoringforce, which explains the fluence-dependent re-combination rates observed in fig. S11.Four-dimensional space-time imaging of car-

rier dynamics at interfaces, using SUEM, has thepotential of unraveling in other complex struc-tures the mechanism of charge transport, withultrafast time resolution and nanometer-scalescanning capability. The unexpected ballisticcarrier velocity (toward the junction) and thegated localization of charges (at the junction)are not predicted by the widely accepted drift-diffusion model, and these features may be char-acteristics of othermaterials when examinedwiththe spatiotemporal resolutions of SUEM. At in-terfaces, the created transient density and elec-tric field are dependent on the optical excitation,and it should be possible to control the proper-ties of nano-to-micrometer–scale heterojunctionsusing the temporal, spatial, and pulse-shape char-acteristics of applied light fields.

REFERENCES AND NOTES

1. K.-T. Tsen, Ed., Ultrafast Dynamical Processes in Semiconductors(Springer-Verlag, New York, 2004).

2. J. Shah, Ultrafast Spectroscopy of Semiconductors andSemiconductor Nanostructures (Springer-Verlag, New York,1999).

3. D. H. Auston, Top. Appl. Phys. 60, 183–233 (1993).4. P. Rodgers, Ed., Nanoscience and Technology: A Collection of

Reviews from Nature Journals (World Scientific, London, 2010).5. M. Saraniti, U. Ravaioli, Nonequilibrium Carrier Dynamics in

Semiconductors: Proceedings of the 14th IinternationalConference, July 25-29, 2005, Chicago, USA (SpringerProceedings in Physics, Berlin, 2006).

6. P. Hommelhoff, M. F. Kling, M. I. Stockman, Ann. Phys. 525,A13–A14 (2013).

7. R. P. Prasankumar, P. C. Upadhya, A. J. Taylor, Phys. StatusSolidi B. 246, 1973–1995 (2009).

8. A. H. Zewail, Science 328, 187–193 (2010).9. R. Brunetti et al., J. Appl. Phys. 52, 6713–6722 (1981).10. O. F. Mohammed, D. S. Yang, S. K. Pal, A. H. Zewail, J. Am.

Chem. Soc. 133, 7708–7711 (2011).11. Y. Lin, D. C. Joy, Surf. Interface Anal. 37, 895–900 (2005).12. S. Schäfer, W. X. Liang, A. H. Zewail, Chem. Phys. Lett. 493,

11–18 (2010).13. J. Goldstein et al., Scanning Electron Microscopy and X-ray

Microanalysis (Springer, New York, 2007).14. B. Sapoval, C. Hermann, Physics of Semiconductors

(Springer-Verlag, New York, 1995).15. K. W. A. Chee, C. Rodenburg, C. J. Humphreys, J. Phys. Conf. Ser.

126, 012033 (2008).

16. K. W. A. Chee, C. Rodenburg, C. J. Humphreys, Microscopy ofSemiconducting Materials 2007, A. G. Cullis, P. A. Midgley, Eds.(Springer-Verlag, Berlin, 2008).

17. S. L. Elliott, R. F. Broom, C. J. Humphreys, J. Appl. Phys. 91,9116–9122 (2002).

18. C. P. Sealy, M. R. Castell, P. R. Wilshaw, J. Electron Microsc. 49,311–321 (2000).

19. G. E. Jellison Jr., F. A. Modine, C. W. White, R. F. Wood,R. T. Young, Phys. Rev. Lett. 46, 1414–1417 (1981).

ACKNOWLEDGMENTS

This work was supported by NSF grant DMR-0964886 and AirForce Office of Scientific Research grant FA9550-11-1-0055 in thePhysical Biology Center for Ultrafast Science and Technology atCalifornia Institute of Technology, which is supported by theGordon and Betty Moore Foundation.

SUPPLEMENTARY MATERIALS

www.sciencemag.org/content/347/6218/164/suppl/DC1Materials and MethodsSupplementary TextFigs. S1 to S11Movies S1 to S3References (20–24)

8 October 2014; accepted 2 December 201410.1126/science.aaa0217

QUANTUM GASES

Critical dynamics of spontaneoussymmetry breaking in ahomogeneous Bose gasNir Navon,*† Alexander L. Gaunt,† Robert P. Smith, Zoran Hadzibabic

Kibble-Zurek theory models the dynamics of spontaneous symmetry breaking, which playsan important role in a wide variety of physical contexts, ranging from cosmology tosuperconductors. We explored these dynamics in a homogeneous system by thermallyquenching an atomic gas with short-range interactions through the Bose-Einstein phasetransition. Using homodyne matter-wave interferometry to measure first-order correlationfunctions, we verified the central quantitative prediction of the Kibble-Zurek theory, namelythe homogeneous-system power-law scaling of the coherence length with the quench rate.Moreover, we directly confirmed its underlying hypothesis, the freezing of the correlation lengthnear the transition. Our measurements agree with a beyond-mean-field theory and supportthe expectation that the dynamical critical exponent for this universality class is z = 3/2.

Continuous symmetry-breaking phase tran-sitions are ubiquitous, from the cooling ofthe early universe to the l transition of su-perfluid helium. Near a second-order tran-sition, critical long-range fluctuations have

a diverging correlation length x, and details ofthe short-range physics are largely unimportant.Consequently, all systems can be classified intoa small number of universality classes, accordingto their generic features such as symmetries, di-mensionality, and range of interactions (1). Closeto the critical point, many physical quantitiesexhibit power-law behaviors governed by criticalexponents characteristic of a universality class.Specifically, for a classical phase transition,

x e jðT−TcÞ=Tcj−n, where Tc is the critical tem-perature and n is the (static) correlation-lengthcritical exponent. Importantly, the correspondingrelaxation time t, needed to establish a divergingx, also diverges: t e xz , where z is the dynamicalcritical exponent (2). An elegant framework forunderstanding the implications of this criticalslowing down for the dynamics of symmetrybreaking is provided by the Kibble-Zurek (KZ)theory (3, 4).Qualitatively, as T is reduced toward Tc at a

finite rate, beyond some point in time the cor-relation length can no longer adiabatically followits diverging equilibrium value. Consequently, attime t ¼ tc, the transition occurs without x everhaving reached the size of thewhole system. Thisresults in the formation of finite-sized domainsthat display independent choices of the symmetry-breaking order parameter (Fig. 1A). [At the domain

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Cavendish Laboratory, University of Cambridge, J. J. ThomsonAvenue, Cambridge CB3 0HE, UK.*Corresponding author. E-mail: nn270@cam.ac.uk †Theseauthors contributed equally to this work.

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