Fermionic Lensing in Smooth Graphene P-N Junctions
FFermionic Lensing in Smooth Graphene P-N Junctions
Võ Tiến Phong and Jian Feng Kong ∗ Massachusetts Institute of Technology, Department of Physics, Cambridge, Massachusetts 02139 (Dated: October 1, 2016)Focusing of electron waves in graphene p-n junctions is a striking manifestation of fermionicnegative refraction. We analyze lensing in smooth p-n junctions and find that it differs in severalinteresting ways from that in the previously studied sharp p-n junctions. Most importantly, whilethe overall negative-refraction behavior remains unchanged, the image at the focal point under-goes additional broadening due to Klein tunneling in the junction. We develop a theory of imagebroadening and estimate the effect for practically interesting system parameter values.
Focusing of ballistic electrons has been extensively ex-plored using electrostatic and magnetostatic lensing intwo-dimensional electron gas (2DEG) systems [1–4]. Un-fortunately, this technique faces some practical challengesfrom low operating temperatures, high fields, and so-phisticated equipment. Using semiconductor p-n junc-tions (PNJs) as electron lens provides some solutionsbut has other problems: the large band gap and widedepletion region of conventional semiconductors are too“opaque” for effective electron focusing [5]. Due to itsunique properties, graphene has been proposed to over-come these shortcomings as an electronic lens [6]. Onesuch property is Klein tunneling, where a graphene PNJis transparent to ballistic electrons, even for a smoothjunction [7–10]. Recent experimental efforts have suc-cessfully demonstrated the use of graphene as an elec-tronic lens [11, 12].Theoretical efforts have been focused on sharpgraphene PNJs [6, 10], however, in realistic experimentalset-up, the PNJ interface width is always finite. For in-stance, PNJs experimentally fabricated using suspendedgraphene have interface width on the order of 200 nm.With boron nitride (BN) encapsulation, a finer interfacewidth can be achieved ∼
20 nm [11]. An understanding ofsmooth PNJs is thus necessary to fully realize fermionicfocusing in practical applications.In this paper, we demonstrate that a smooth PNJ ex-hibits negative refraction and lensing similar to a sharpPNJ, as illustrated schematically in Fig. 1. However, theeffect of junction smoothness is to blur the image at thefocal point. In particular, for an incident wavepacket ofsize a , its image is broadened to a width a given by a = a + πv F (cid:126) eE , (1)where v F ≈ m/s, and E is the electric field strengthnear the interface ( eE > E , U ( x ) = − eEx, near the junction. We assumethat U ( x ) is asymptotically constant at large x . Let us (a) (b) tunneling type- n type- p type- n type- p ncollimatio Figure 1. Comparison of focusing in sharp and smoothgraphene PNJs. (a)-(b) show focusing in a PNJ, where thewavevectors in the conduction band k c and the wavevectorsin the valence band k v are on the n -type and p -type side re-spectively. In this case, both a sharp PNJ (a) and a smoothPNJ (b) produce focusing at the focal point. However, in thesmooth junction, waves coming from the left undergo tunnel-ing near the junction, whereby the transmission is suppressedexponentially by a factor given in Eq. 2. This is illustrated in(b) by the blurring out of wave components incident at largeangles. start with a wave packet with size a incident from the leftside of the junction ψ ( y ) ∝ e − y / a . In Fourier space, thecomponents are ψ ( k y ) ∝ e − k y a / . Each harmonic aftertunneling through the junction acquires a transmissionfactor [13, 14] t ( k y ) = exp (cid:32) − πv F (cid:126) k y eE (cid:33) . (2)This formula, which is exact for a linear potential, re-mains valid for a wide range of E and k y when the po-tential is smooth. The harmonics on the other side of thejunction then become ψ ( k y ) ∝ t ( k y ) e − k y a / = e − k y a / , where a given by Eq. 1. Fourier transforming to posi-tion space, we see that a is indeed the size of the image.This argument shows that the image is broadened aftertunneling by a factor inversely proportional to the fieldstrength E. We will derive this result carefully, and il-lustrate that electron focusing in graphene is a robustphenomenon.In graphene’s band structure, at the corners of itsBrillouin zone, the conduction band and valence bandare degenerate [15, 16], as shown in Fig. 2a. In un-doped graphene, the Fermi level crosses these degeneracy a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t points, and is shared by both bands. Fermions in this en-ergy regime behave as relativistic Dirac particles obeyingthe Hamiltonian [17], H = v F σ · p , where v F ≈ m/s, σ = ( σ x , σ y ) are the Pauli matrices in pseudospin (sub-lattice) space, and p = ( p x , p y ) are the momentum oper-ators. The energy dispersion for electrons with quasi-momentum (cid:126) k = (cid:126) ( k x , k y ) in the conduction band is ε c = v F (cid:126) | k | ; likewise, the dispersion for holes in the va-lence band is ε v = − v F (cid:126) | k | . Carrier densities in graphenecan be precisely tuned through electrical gating [18] ordoping [19, 20].In the presence of an electrostatic potential energy U ( x ) produced by electrical gating, the degeneracy pointsare shifted as shown in Fig. 2a. The region where U ( x ) < E , with E being the chemical potential, is of n -type with an electron density ρ c ∼ k c , where k c is theFermi radius of the conduction band far away from thejunction. Likewise, the p -type region is where U ( x ) > E , and the hole density is ρ v ∼ k v . Suppose we have a sharpPNJ. An electron wave on the n -type side with wave vec-tor k = k c (cos θ i , sin θ i ) has corresponding wave vector k = − k v (cos θ t , sin θ t ) on the other side of the junction.Due to symmetry in the y -direction, the component of k parallel to the junction is conserved, leading to thefermionic analogue of Snell’s law for refractionsin θ i sin θ t = − k v k c = n. (3)As shown in Eq. 3, the refractive index n is negative,indicating that graphene can be used as a Veselago lens[21] to focus electron flows, as shown in Fig. 1.In optics, a Veselago lens is superior to a conven-tional lens because it can achieve a resolution below thediffraction limit [22]. This is possible because evanescentFourier modes are amplified by a Veselago lens so thatwhen an image is reconstructed on the other side of thelens, all of the Fourier modes of the original field con-tribute, not just those of propagating waves. A Veselagolens is thus a so-called “superlens” or perfect lens, be-cause sub-wavelength imaging is possible, and the sizeof the image, a, is equal to the size of the source, a , i.e. a = a and a (cid:28) λ , where λ is the wavelength ofthe imaging light. A similar story can be told for sharpgraphene PNJs, except that fermionic waves are beingfocused instead of electromagnetic waves.If the graphene PNJ is smooth, then the image reso-lution is reduced: a > a . This is due to tunneling offermions near the junction, and we have already heuris-tically established this fact above. Since our argumentrelies on Eq. 2, we will now show a semi-classical deriva-tion of it in the paraxial approximation. We considerthe case where the junction is smooth, with width w larger than the Fermi wavelength λ F . The opposite limit, λ F (cid:29) w, corresponds to the sharp junction limit. Since λ F ∼ ρ − / , where the fermion density ρ can be easilytuned using gating [23], λ F can always be tuned to be x )( xU c ε y k x k v ε y k x k c k v k type- n type- p w (a) x x w ForbiddenII RegionAllowedI Region
AllowedIII Region (b) g v g v Figure 2. (a) Dirac points are shifted by an electrostatic po-tential energy U ( x ) produced by gating. The chemical poten-tial is E = 0 . (b) This illustrates the allowed regions of Diracfermions when p ( x ) > , with x and x as the classicalturning points. smaller than w experimentally. In this case, we will showthat wave vectors which are not perpendicular to thejunction undergo quantum tunneling near the junction.However, a diverging wave-front on one side still con-verges on the other side, resulting in focusing of fermionsshown in Fig. 1b. The quality of the image at the focalpoint depends on the sharpness of the junction, | ∂ x U ( x ) | . Charge carriers in graphene are described by the Diracequation [ v F σ · p + U ( x )] ψ ( x, y ) = E ψ ( x, y ) , (4)where without loss of generality, we can set E = 0 . Anal-ogous to methods in Fourier optics [24], it is possibleto obtain approximate solutions that illustrate focusingproperties. To obtain semi-classical solutions, we firstmultiply Eq. 4 by [ v F σ · p − U ( x )] to write (cid:20) v F p − U ( x ) − iv F (cid:126) dUdx (cid:21) ψ ( x, y ) = 0 . (5)Clearly, this transformation is not unitary. We have ar-rived at an effective Schrodinger equation that is non-Hermitian. However, as shown in [25], this transforma-tion is still useful in studying chiral tunneling in single-layer graphene. Since the Hamiltonian has translationalinvariance in the y − direction, we can Fourier transform ψ in the y − direction ψ ( x, y ) = (cid:90) dk y π e ik y y ψ ( x, k y ) . (6)Substituting Eq. 6 into Eq. 5, we find the fermionic ana-logue of the Helmholtz equation [24] in optics d ψ ( x, k y ) dx = − (cid:126) (cid:18) p ( x ) + iσ x (cid:126) v F dUdx (cid:19) ψ ( x, k y ) , (7)where p ( x ) = [ U ( x ) /v F ] − (cid:126) k y , the classical momen-tum in the x -direction. We now diagonalize Eq. 7 bywriting ψ ( x, k y ) = (cid:18) (cid:19) η ( x, k y ) + (cid:18) − (cid:19) η ( x, k y ) . (8)The matrix equation in Eq. 7 now becomes scalar equa-tions for η and η d η , ( x, k y ) dx = − (cid:126) (cid:18) p ( x ) ± i (cid:126) v F dUdx (cid:19) η , ( x, k y ) . (9)To ensure self-consistency, we substitute Eq. 8 into Eq. 4to obtain the relationship between η and η η ( x, k y ) = 1 k y (cid:18) ddx + i (cid:126) k y U ( x ) (cid:19) η ( x, k y ) . (10)Eq. 9 can be solved approximately using the Jeffreys-Wentzel–Kramers–Brillouin (JWKB) method with caretaken when selecting appropriate propagating modes forhole and electron regions. We use the ansatz η ( x, k y ) =exp[ iS ( x, k y ) / (cid:126) ] and expand the phase in a power seriesof (cid:126) , S ( x, k y ) = (cid:80) ∞ j =0 (cid:126) j S j ( x, k y ) . In the allowed regionswhere v F (cid:126) | k y | < | U ( x ) | , we have traveling waves of elec-trons (if U ( x ) <
0) and holes (if U ( x ) > v F (cid:126) | k y | > | U ( x ) | , no plane-wave solutions are allowed,and we instead have a decaying wavefunction. The elec-tron wavefunction on the left side of the PNJ with bound-ary condition at x = x is η ∼ (cid:112) p ( x ) exp (cid:18) i (cid:126) (cid:90) x x φ ( ξ ) dξ + i (cid:126) (cid:90) xx φ ( ξ ) dξ (cid:19) , (11)where φ ( x ) = p ( x ) − (cid:126) v F i ∂ x Up ( x ) . (12)The corresponding forward-propagating hole wavefunc-tion on the right side of the PNJ is η ∼ t (cid:112) p ( x ) exp (cid:18) i (cid:126) (cid:90) x x φ ( ξ ) dξ − i (cid:126) (cid:90) xx φ ( ξ ) dξ (cid:19) , (13)where t ∼ exp (cid:18) i v F (cid:90) x x ∂ x U | p ( ξ ) | dξ (cid:19) exp (cid:18) − (cid:126) (cid:90) x x | p ( ξ ) | dξ (cid:19) (14) is the transmission amplitude. The spinor wavefunctioncan be found using Eqs. 8, 10, 11, and 13.To study lensing from a PNJ, we choose a model poten-tial U ( x ) such that U → U ∞ for large x and U → U −∞ for large −| x | ; for intermediate | x | ∼ w, we assumethat U ( x ) can be well-approximated by a linear poten-tial U ( x ) = eEx, where eE is a constant. For instance, U ( x ) = U tanh( x/w ) . In the paraxial limit where k y issmall, we obtain precisely Eq. 2 as desired. Suppose westart with a localized Gaussian wavefunction at ( x , ψ in ( x, y ) = (cid:90) dk y π ψ e ( x, k y ) e − a k y / e ik y y , (15)where a is the width of the wave-packet in coordinatespace, and ψ e ( x, k y ) is the electron momentum eigenstatefor each k y on the left side of the PNJ. The transmittedwave packet on the right side of the PNJ is simply ψ tr ( x, y ) = (cid:90) dk y π ψ h ( x, k y ) e − a k y / e ik y y , (16)where ψ h ( x, k y ) is the hole momentum eigenstate for each k y such that ψ h ( x, k y ) ∝ exp (cid:32) i (cid:126) (cid:90) − vF (cid:126) | ky | eE x φ ( ξ ) dξ − i (cid:126) (cid:90) x vF (cid:126) | ky | eE φ ( ξ ) dξ (cid:33) , (17)and a is as defined in Eq. 1. For a symmetric PNJ inwhich U ∞ = − U −∞ , φ ( x ) is approximately an even func-tion. In this case, the momentum states interfere con-structively at x = − x , at which point the phase cancelsout in Eq. 17. This is the point of focusing of fermionson the right side of the PNJ, as shown in Fig. 3a. Thiscorresponds to the case with refractive index n = − . For the case of an asymmetric PNJ, n = U −∞ /U ∞ , and focusing occurs near x = nx . In this case, a cuspis formed near the focal point, as shown in [6] and illus-trated in Fig. 3b. In both cases of symmetric and asym-metric junctions, the wave packet at the focal point is aGaussian of momentum eigenstates with width a givenby Eq. 1. Thus, the size of the image at the focal point islarger than the original size of the object. The effect ofa smooth PNJ is to broaden the image, resulting in im-perfect focusing. Eq. 1 suggests that the minimal imagesize is experimentally limited to the ability to fabricatea sharp PNJ. Suppose we have a graphene PNJ with anelectrostatic potential of about 20 V, and interface widthabout 20 nm. Consequently, eE = 300 pN, and the min-imal image size is approximately 1 nm.It is interesting to ask whether the sub-wavelengthresolution regime (superlensing) can be realized for asmooth potential. To achieve superlensing in this setting,Eq. 1 must remain true even when a (cid:28) λ F , the Fermiwavelength. For such to be true, a necessary conditionis πv F (cid:126) eE (cid:28) λ F , which is satisfied by increasing the field (a) (b) Figure 3. Classical trajectories of fermions. We see that inboth a symmetric PNJ (a) and an asymmetric PNJ (b), adiverging wave packet on the left side converges to a focus onthe right side. The shaded regions indicate tunneling. strength without changing the Fermi wavelength. To seethat Eq. 1 is true for a (cid:28) λ F , let us consider a symmetricPNJ such that U −∞ = − U ∞ with k F = | U ∞ | /v F (cid:126) and λ F = 2 π/k F . We follow the same approach as done in[22]. We begin with a localized wavepacket on the leftside far away from the junction. For x (cid:28) , we can writethe wavefunction as a superposition of its Fourier modesas ψ ( x, y ) = (cid:88) k x , | k y |≤ k F ψ ( k x , k y ) e ik x x + ik y y + (cid:88) κ, | k y | >k F ψ ( k x , k y ) e − κx + ik y y , (18)where k x = (cid:113) k F − k y and κ = (cid:113) k y − k F . The first partof Eq. 18 contains the propagating Fourier modes, whilethe second part consists of the evanescent waves. Aswe continue this wavefunction to the right side of thejunction, the wavevectors in the x -direction change sign,i.e. k x (cid:55)→ − k x and κ (cid:55)→ − κ , because we are now in thevalence band of graphene. The transmitted componentof the wavefunction on the right side far away from thejunction with x (cid:29) ψ ( x, y ) = (cid:88) k x , | k y |≤ k F t ˜ ψ ( k x , k y ) e − ik x x + ik y y + (cid:88) k x , | k y | >k F t ˜ ψ ( k x , k y ) e κx + ik y y , (19)where the tilde is used to emphasize that the Fourier com-ponents on the right side are not necessarily the same ason the left side; and are to be determined by boundaryconditions. We see from Eq. 19 that the evanescent com-ponents are being amplified on the right side of the PNJ.Thus, at the focal point of the image, all the Fourier com-ponents contribute to the image reconstruction. This ar-gument suggests that Eq. 1 is true even when the size ofthe image is smaller than λ F , and that lensing below thediffraction limit in a smooth symmetric graphene PNJ isa possibility. We now consider the question of sub-wavelength reso-lution in an asymmetric PNJ. In this case, U ∞ (cid:54) = − U −∞ ,and we need to replace k F by k c = | U −∞ | /v F (cid:126) on the leftside, and k F by k v = | U ∞ | /v F (cid:126) on the right side. Here,we do not get focusing at a single point as shown earlier,but instead get a cusp. However, it is still true that theevanescent waves are also being amplified in this caseby an identical argument to the symmetric case. Sub-wavelength resolution is thus possible also in an asym-metric PNJ.Our calculations suggest that fermionic focusing is arobust feature in smooth graphene PNJs that can poten-tially operate below the diffraction limit. 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