Analog Sauter-Schwinger effect in semiconductors for spacetime-dependent fields
AAnalog Sauter–Schwinger effect in semiconductors for spacetime-dependent fields
Malte F. Linder, Axel Lorke, and Ralf Schützhold ∗ Fakultät für Physik, Universität Duisburg-Essen, Lotharstr. 1, 47057 Duisburg, Germany (Dated: January 16, 2018)The Sauter–Schwinger effect predicts the creation of electron–positron pairs out of the quantumvacuum via tunneling induced by a strong electric field. Unfortunately, as the required field strengthis extremely large, this fundamental prediction of quantum field theory has not been verified experi-mentally yet. Here, we study under which conditions and approximations the interband tunneling insuitable semiconductors could be effectively governed by the same (Dirac) Hamiltonian, especiallyfor electric fields which depend on space and time. This quantitative analogy would allow us to testsome of the predictions (such as the dynamically assisted Sauter–Schwinger effect) in this area bymeans of these laboratory analogs.
PACS numbers: 12.20.-m, 77.22.Jp, 11.15.-q
I. INTRODUCTION
The are several fundamental predictions of quantumfield theory which have so far resisted a direct experimen-tal verification. One of the most prominent examples isthe Sauter–Schwinger effect [1–4] predicting the creationof electron–positron pairs out of the quantum vacuum viatunneling. For a constant electric field E , the associatedpair-creation probability behaves as P e + e − ∝ e − πE QEDcrit /E (1)and is thus exponentially suppressed for fields E well be-low the critical field E QEDcrit = m c (cid:126) q ≈ . × Vm (2)(often denoted by E S in the literature), where m is theelectron mass and q > the elementary charge. The factthat expression (1) does not admit a Taylor expansion in q already indicates that this is a nonperturbative effect,which renders calculations intrinsically difficult. Never-theless, apart from the constant-field case above, it ispossible to derive the pair-creation probability for sev-eral scenarios with varying fields. For example, a tempo-ral Sauter pulse of the form E ( t ) = E / cosh ( ωt ) doesalso facilitate an exact solution of the Dirac equation(see, e.g., Ref. [5]). In this situation, the absolute valueof the exponent (1) is reduced and thus the probabil-ity enhanced. Conversely, for a spatial Sauter profile E ( x ) = E / cosh ( kx ) , the absolute value of the expo-nent increases, leading to a suppression of the pair-crea-tion probability.As another interesting case, the superposition of aconstant (or slowly varying) strong field with a weakertime-dependent field can result in an enhancement of theprobability: the dynamically assisted Sauter–Schwingereffect [6]. The dependence of this effect on the shape ∗ [email protected] of the weaker time-dependent field and the momentumof the created electrons and positrons has been studiedin Refs. [7–10], for example. If the strong field is notconstant but spatially varying (such as a spatial Sauterprofile), there is an interesting interplay or competitionbetween the spatial dependence of the stronger field andthe temporal dependence of the weaker field; see Ref. [11].Most unfortunately, because the critical fieldstrength (2) is so large, these nonperturbative phe-nomena have not been observed yet, and thus it was notpossible to test the various predictions mentioned aboveexperimentally. This motivates the quest for other,experimentally more accessible, laboratory systemswhich display analogous effects, ideally governed by thesame Hamiltonian (under appropriate approximations)and thus the same equations of motion. To use thefamous quote by R. Feynman: “ The same equationshave the same solutions. ” Due to their high degreeof experimental control, one such option are ultracoldatoms in optical lattices; see also Refs. [12–14]. Otherpossible options include graphene [15–17] and trappedions [18]. In the following, we study interband tunnelingin semiconductors as another promising example. Notethat the qualitative analogy between Landau–Zenertunneling in semiconductors [19–21] and the Sauter–Schwinger effect in the case of a constant electric fieldhas already been discussed in, e.g., Refs. [16, 22–26].Here, the goal is to derive a quantitative analogy (inthe spirit of Feynman) and to specify the underlyingapproximations and assumptions, with special emphasison fields depending on time (see also Ref. [26]) andspace (as motivated above). The use of these analogiesis twofold: On the one hand, they allow us to test theabove predictions by means of laboratory analogs, whichare easier to access experimentally, and, on the otherhand, they help us to understand the physics of theselaboratory systems better. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n II. TIME-DEPENDENT CASE E = E ( t ) Let us start with the simpler case of a homogeneousand purely time-dependent external electric field in 1+1spacetime dimensions. We choose to describe the exter-nal electric field in temporal gauge E ( t ) = ˙ A ( t ) with theone-component vector potential A ( t ) . This potential cou-ples to the electron momentum operator via the covariantderivative ∂ x + i qA ( t ) ( c = ε = (cid:126) = 1 in the following,unless otherwise stated).The many-body Dirac Hamiltonian can be written as ˆ H D ( t ) = ∞ (cid:90) −∞ ˆΨ † { [ − i ∂ x + qA ( t )] σ x + mσ z } ˆΨ d x . (3)This form is obtained by expressing the Dirac matricesin terms of Pauli matrices via γ = σ z and γ = i σ y .The field operator consequently has two components, ˆΨ( t, x ) = (cid:0) ˆΨ + ( t, x ) , ˆΨ − ( t, x ) (cid:1) , which corresponds to theabsence of spin in 1+1 dimensions.We transform this Hamiltonian to momentum spaceby inserting the spatial Fourier transform ˆΨ( t, x ) = 1 √ π ∞ (cid:90) −∞ ˆ (cid:101) Ψ( t, k )e i kx d k (4)of the field operator. The result reads ˆ H D ( t ) = ∞ (cid:90) −∞ ˆ (cid:101) Ψ † ( t, k ) (cid:18) m k + qA ( t ) k + qA ( t ) − m (cid:19) ˆ (cid:101) Ψ( t, k ) d k .(5)The next step is to derive the crystal-momentum rep-resentation of the Hamiltonian for electrons in a semi-conductor which is exposed to the same external electricfield. This semiconductor Hamiltonian can then be com-pared to the Dirac Hamiltonian (5). A. Two-band semiconductor model
A direct, quantitative analogy between Dirac’s theoryand electrons in a semiconductor can only exist if the con-sidered semiconductor electrons can only occupy two ad-jacent energy bands: the higher (lower) band then corre-sponds to the positive (negative) relativistic continuum.In the ground state (no external field and zero temper-ature), the lower band must be completely filled withelectrons (analog of the Dirac sea), while the upper bandmust be empty. This is precisely the case if we restrict thesemiconductor model to the valence band and the con-duction band only. Our starting point is the well-knownKane model [27], but we only include the light-hole va-lence band in our theory and neglect the heavy-hole band(since lighter particles are more likely to be excited viathe pair-creation mechanism we are interested in) and thesplit-off valence band, which is energetically lowered due to spin–orbit interaction; see, e.g., Refs. [28–31], whichalso employ and describe this model.Let us start with the basic Hamiltonian. Since thepossible electron group velocities within the valence andconduction bands of typical semiconductors are far belowthe vacuum speed of light, we may describe the semi-conductor electrons with the nonrelativistic Schrödingerequation. The Bloch electrons, which we are interestedin, are subject to the lattice-periodic potential V ( x ) ofthe ion cores. We denote the lattice constant by (cid:96) , so thepotential satisfies V ( x + (cid:96) ) = V ( x ) . For simplicity, weneglect electron–electron interactions; see Sec. VI below.The Hamiltonian of the Bloch electrons in the externalfield E ( t ) = ˙ A ( t ) thus reads as ˆ H full s ( t ) = ∞ (cid:90) −∞ ˆ ψ † (cid:26) [ − i ∂ x + qA ( t )] m + V ( x ) (cid:27) ˆ ψ d x , (6)where ˆ ψ ( t, x ) is the scalar electron field operator.Note that the quadratic A term in this Hamiltoniancan be absorbed via a suitable gauge transformation (seeAppendix A), so we may consider the simplified Hamil-tonian ˆ H full s ( t ) = ∞ (cid:90) −∞ ˆ ψ † (cid:20) − ∂ x m + V ( x ) + qA ( t ) m ( − i ∂ x ) (cid:21) ˆ ψ d x (7)instead.For the derivation of the two-band model, we restrictthe Hamiltonian ˆ H full s to valence- and conduction-bandelectrons only. This should be a good approximation foranalogs of the Sauter–Schwinger effect since an excitationof a valence-band electron into the conduction band isassociated with a lower energy difference than any otherpossible transition in the (initial) ground state. Largerenergy differences lead to exponential suppression in thecontext of nonperturbative pair creation, so the two-bandmodel should reproduce the leading-order pair-creationprobability in the initial ground state correctly.We apply the two-band approximation by assumingthat only the valence and conduction Bloch bands con-tribute to the field operator: ˆ ψ ( t, x ) ≈ π/(cid:96) (cid:90) − π/(cid:96) ˆ a − ( t, K ) f − ( K, x ) + ˆ a + ( t, K ) f + ( K, x ) d K .(8)In this equation, the functions f n ( K, x ) = (cid:104) x | n, K (cid:105) arethe position-space representations of the Bloch states | n, K (cid:105) in the unperturbed semiconductor crystal [ A ( t ) =0 ]. The band index − ( + ) denotes the valence (conduc-tion) band. There is one independent Bloch state perband for each quasimomentum K in the first Brillouinzone, which is the range ( − π/(cid:96), π/(cid:96) ] . Hence, our fieldoperator (8) is per assumption a linear combination of allBloch states in the valence and the conduction bands ateach instant of time. The time-dependent “coefficients,”which are in fact operators, ˆ a ± ( t, K ) , are instantaneousannihilation operators for electrons in the correspond-ing Bloch states |± , K (cid:105) . For this statement to hold, theBloch states must be normalized, so that they obey theorthonormality relation (cid:104) n, K | n (cid:48) , K (cid:48) (cid:105) = ∞ (cid:90) −∞ f ∗ n ( K, x ) f n (cid:48) ( K (cid:48) , x ) d x = δ nn (cid:48) δ ( K (cid:48) − K ) . (9)We use the convention f n ( K, x ) = e i Kx u n ( K, x ) (10)throughout this paper, so our lattice-periodic Bloch fac-tors u n ( K, x ) are orthonormalized (at a fixed K ) accord-ing to the unit-cell Bloch-factor scalar product (cid:104) n, K | n (cid:48) , K (cid:105) u = 2 π(cid:96) (cid:96) (cid:90) u ∗ n ( K, x ) u n (cid:48) ( K, x ) d x = δ nn (cid:48) .(11)Inserting the approximation (8) into the full Hamilto-nian (7) yields the two-band semiconductor Hamiltonian ˆ H s , which neglects the dynamics of all other Bloch bands.In the calculation of ˆ H s , we use the fact that Bloch wavessatisfy the energy eigenvalue equation (cid:20) − ∂ x m + V ( x ) (cid:21) f n ( K, x ) = E n ( K ) f n ( K, x ) . (12)Furthermore, the Bloch-wave momentum matrix ele-ments (cid:104) n, K | − i ∂ x | n (cid:48) , K (cid:48) (cid:105) (also known as optical matrixelements) appear in the new Hamiltonian. It is wellknown that these matrix elements vanish unless K = K (cid:48) (see, e.g., Ref. [32]; a proof of this important theorem isgiven in Appendix B). There are thus three independentmomentum matrix elements in the two-band model foreach K : the interband element κ is given implicitly by (cid:104)− , K | − i ∂ x | + , K (cid:48) (cid:105) = κ ( K ) δ ( K (cid:48) − K ) [cf. Eq. (B5)] andcan be written κ ( K ) = (cid:104)− , K | − i ∂ x | + , K (cid:105) u (13)with the product defined in Eq. (11). This quantity iscomplex in general; however, we define the global phasesof the Bloch bands in a way such that the value κ = κ (0) is real and positive: κ > . The two intraband elementsare related to the group velocities v ± ( K ) = d E ± ( K ) / d K via (cid:104)± , K | − i ∂ x |± , K (cid:48) (cid:105) = mv ± ( K ) δ ( K (cid:48) − K ) ; (14)see, e.g., Refs. [31, 33]. The resulting two-band Hamiltonian in crystal-mo-mentum space reads ˆ H s ( t )= π/(cid:96) (cid:90) − π/(cid:96) ˆ a † ( t, K ) (cid:18) E + + qAv + qAm κ ∗ qAm κ E − + qAv − (cid:19) ˆ a ( t, K ) d K (15)(we have omitted to write explicitly the dependencies ofthe quantities in the matrix here) with ˆ a ( t, K ) = (cid:18) ˆ a + ( t, K )ˆ a − ( t, K ) (cid:19) . (16)Note that this Hamiltonian as well as the Dirac Hamil-tonian (5) have the form ˆ H ( t ) = (cid:82) ˆ H ( t, k ) d k , whichmeans that each k mode evolves independently, and k (or K in the semiconductor case) is thus a conserved quan-tity as expected in a purely time-dependent potential. B. Diagonalization of the Hamiltonians
In order to bring both Hamiltonians, ˆ H D and ˆ H s , intothe same form, so that we can compare them, we diago-nalize the × matrices in the Hamiltonians. To this end,we transform (“rotate”) the momentum-space field oper-ators ˆ (cid:101) Ψ( t, k ) (Dirac case) and the Bloch-electron opera-tors ˆ a ( t, K ) (semiconductor) to operators correspondingto the instantaneous energy eigenstates, respectively.In the Dirac case, the transformed field operators readas ˆΥ( t, k ) = 1 (cid:112) d ( t, k ) (cid:18) d ( t, k ) − d ( t, k ) 1 (cid:19) ˆ (cid:101) Ψ( t, k ) (17)with the abbreviations d ( t, k ) = k + qA ( t ) m + Ω( t, k ) (18)and Ω( t, k ) = (cid:112) m + [ k + qA ( t )] . (19)Note that Eq. (17) describes a unitary relation, which isalso a Bogoliubov transformation, so the two componentsof ˆΥ obey the canonical anticommutation relations. Interms of these field operators, the Dirac Hamiltonian (5)assumes the diagonal form ˆ H D ( t ) = ∞ (cid:90) −∞ ˆΥ † ( t, k ) (cid:18) Ω( t, k ) 00 − Ω( t, k ) (cid:19) ˆΥ( t, k ) d k .(20)Before we diagonalize the matrix in the semiconductorHamiltonian (15), we want to make its diagonal elementssymmetric like in the Dirac case, in which the originaldiagonal elements in Eq. (5) are ± m . In order to do this,we rewrite the Hamiltonian as ˆ H s ( t )= π/(cid:96) (cid:90) − π/(cid:96) ˆ a † ( t, K ) ∆ E + qA ∆ v qAm κ ∗ qAm κ − ∆ E + qA ∆ v ˆ a ( t, K ) d K + π/(cid:96) (cid:90) − π/(cid:96) E + ( K ) + E − ( K ) + qA ( t )[ v + ( K ) + v − ( K )]2 × (cid:104) ˆ a † + ( t, K )ˆ a + ( t, K ) + ˆ a †− ( t, K )ˆ a − ( t, K ) (cid:124) (cid:123)(cid:122) (cid:125) =1 for all t and K (cid:105) d K .(21)In this equation, we have introduced the ( K -dependent)band-energy difference ∆ E ( K ) = E + ( K ) − E − ( K ) andthe group-velocity difference ∆ v ( K ) = v + ( K ) − v − ( K ) .Since K is a conserved quantity, ˆ a † ( t, K )ˆ a ( t, K ) mustalways be because there is exactly one electron per K value in our two-band model, and the electron for agiven K must be either in the conduction band or inthe valence band at each point in time. The second K integral in the Hamiltonian (21) therefore yields a time-dependent constant, which can be eliminated by a gaugetransformation on the scalar potential again, as describedin Appendix A.The Bogoliubov transformation which diagonalizes theredefined semiconductor Hamiltonian [first K integral inEq. (21)] has the same form as in the Dirac case [complexversion of Eq. (17)], ˆ b ( t, K ) = 1 (cid:112) | d ( t, K ) | (cid:18) d ∗ ( t, K ) − d ( t, K ) 1 (cid:19) ˆ a ( t, K ) ,(22)but with different auxiliary functions d ( t, K ) = qA ( t ) κ ( K ) /m [∆ E ( K ) + qA ( t )∆ v ( K )] / Ω ( t, K ) (23)and Ω ( t, K )= (cid:115)(cid:20) ∆ E ( K ) + qA ( t )∆ v ( K )2 (cid:21) + (cid:20) qA ( t ) | κ ( K ) | m (cid:21) ,(24)so the resulting Hamiltonian reads as ˆ H s ( t ) = π/(cid:96) (cid:90) − π/(cid:96) ˆ b † ( t, K ) (cid:18) Ω ( t, K ) 00 − Ω ( t, K ) (cid:19) ˆ b ( t, K ) d K .(25)Now that we have derived the diagonal forms of bothHamiltonians, their physical differences including scalesand dependence on conserved (quasi)momentum are en-coded in the instantaneous energy eigenvalues Ω and Ω . C. Analogy between the modes k = 0 and K = 0 Let us start to point out the quantitative analogy be-tween the two Hamiltonians, ˆ H D (20) and ˆ H s (25), at the(quasi)momentum-space points k = K = 0 . We assumefor the moment that there is no electric field ( A = 0 ).The energy bands in the Dirac case are the two squareroots of the relativistic energy–momentum relation; seeFig. 1(a). The mode k = 0 thus coincides with the mini-mal mass gap m in the absence of an external field.The exact shapes of the valence band and the conduc-tion band in the semiconductor, E ± ( K ) , are not fixedbut depend on the periodic potential V ( x ) ; however, wemake the following assumptions about the semiconductorband structure, which shall be satisfied in the remainderof this paper: • no band crossing [ ∆ E ( K ) > for each K ] and • a direct band gap at the center K = 0 of the Bril-louin zone; that is, E g = ∆ E (0) is the minimal valueof ∆ E ( K ) .An example for such a band structure is plotted inFig. 1(b).Now, we reintroduce the electric field and compare theinstantaneous energy eigenvalues in Eqs. (19) and (24) at k = K = 0 with each other. In the Dirac case, we get Ω( t,
0) = (cid:112) ( mc ) + [ cqA ( t )] (26)with the speed of light written explicitly in this equa-tion. In the semiconductor case, we first note that bothgroup velocities [ E ± ( K ) derivatives] vanish at K = 0 and thus also ∆ v (0) = 0 . Comparing the resulting Ω ( t,
0) = (cid:112) ( E g / + [ qA ( t ) κ /m ] with Eq. (26), weimmediately see that the quantity c (cid:63) = κ /m plays therole of an effective speed of light in the semiconductor.We also want to define a suitable effective mass m (cid:63) such that m (cid:63) c (cid:63) (the analog of the rest energy mc inDirac theory) produces the term E g / in Ω ( t, above.Hence, we set m (cid:63) = m E g κ , (27)so we may write c (cid:63) = (cid:114) E g m (cid:63) (28)and Ω ( t,
0) = (cid:112) ( m (cid:63) c (cid:63) ) + [ c (cid:63) qA ( t )] . (29)Comparing Eqs. (26) and (29) shows that the Hamilto-nians of both systems are equivalent in the large-wave-length limit k = K = 0 . The semiconductor just exhibitsdifferent scales, which are given by the material constants E g and κ . The same effective constants have also beenfound in Refs. [28, 29]. - m m k ℰ k + m - k + m m - m (a) Relativistic dispersion relation. - π / ℓ π / ℓ K ℰ ℰ + ( K ) ℰ - ( K ) ℰ g (b) Reduced zone scheme of a two-band semiconductor. FIG. 1: Electron dispersion relations in the two systemsunder consideration, without an external electric field( A = 0 ). (a) Dirac case: the two branches of the rel-ativistic energy–momentum relation. (b) Semiconductorcase: example for an electronic two-band structure inthe first Brillouin zone (reduced zone scheme). We as-sume throughout this paper that the semiconductor hasa direct band gap at the center of the Brillouin zone,measuring E g = ∆ E (0) .Note that we refer to the quantity (27) as “effectivemass” because it allows us to write Ω ( t, in a way for-mally equivalent to Ω( t, above. Nevertheless, as wewill see in the next subsection, m (cid:63) is indeed related tothe parabolic energy-band curvatures in the semiconduc-tor, which is the usual notion of effective masses in thisarea of physics.Another point to notice here is that we could also de-fine an effective elementary charge via cq (cid:63) = qκ /m in-stead of the effective speed of light (28) to make the anal-ogy between the modes k = K = 0 work (in which casethe effective electron mass in the semiconductor mustbe defined by the equation m (cid:63) c = E g / ). Or, we canshift the factor in q (cid:63) into an effective vector potentialdefined by q (cid:63) A ( t ) = qA (cid:63) ( t ) . The concept of an effective external potential in analogs of the Sauter–Schwinger ef-fect is known from ultracold atoms in optical lattices;see Refs. [12, 13]. However, we will see in the next sub-section that defining an effective speed of light as above(and thus leaving the external potential and elementarycharge unchanged) is required to extend the analogy inthe semiconductor to more modes than just k = K = 0 . D. Analogy for long-wavelength modes
We now want to extend the analogy to all small(quasi)momenta, which means | k | (cid:28) m in the Diraccase and | K | (cid:28) π/(cid:96) in the semiconductor case. In thisrange, the dispersion curves in Fig. 1 are approximatelyparabolic in both cases. This is also the range withthe smallest energy difference between the bands, andwe consequently expect the corresponding modes to gen-erate the dominant contributions to the total pair-cre-ation yield via the Sauter–Schwinger effect. Note thatif the vector potential vanishes in the in and out states[i.e., A ( t → ±∞ ) = 0 ], then the conserved quantities k and K correspond to the initial and the final kinetic(quasi)momentum of the considered mode, respectively,so the long-wavelength modes can be identified with elec-tron states close to the minimal band gaps in Fig. 1 inthis case. Given a particular external field E ( t ) , we canalways satisfy this condition by letting A ( t ) start at zerofor t → −∞ , and, if A ( t ) (cid:54) = 0 after the field has beenswitched off (or has become very tiny), letting A ( t ) ap-proach zero again very slowly (adiabatically, such thatthis process does not cause band transitions).Concerning the analogy, let us start with the Dirac caseagain. The Taylor expansion of Ω around k = 0 reads Ω( t, k ) = Ω( t,
0) + c qA ( t )Ω( t, k + O ( k ) (30)with c written explicitly. In the second-order term, thesmall quantity ( k/m ) is suppressed by the prefactor / { q A ( t ) /m ] / } ≤ / . Hence, we will onlyconsider the first orders of k or K in Ω and Ω when com-paring the Hamiltonians for small (quasi)momenta, andwe ignore all higher-order terms.In the semiconductor case, we get Ω ( t, K ) = Ω ( t,
0) + KΩ ( t, × (cid:40) ∆ E (1) (0) + qA ( t )∆ v (1) (0)4 E g + (cid:20) qA ( t ) m (cid:21) κ Re κ (1) (0) (cid:41) + O ( K ) .(31)Superscripts of the form “ ( n ) ” denote the n -th derivativewith respect to K .In order to evaluate these derivatives at K = 0 , wefirst note that the parabolic parts of the energy bandsaround the minimal gap are usually written as (we ar-bitrarily locate the band gap symmetrically around thezero energy level) E + ( K ) = E g K m (cid:63),e + O ( K ) , E − ( K ) = − E g − K m (cid:63),h + O ( K ) , (32)where m (cid:63),e and m (cid:63),h denote the (positive) effectivemasses of conduction-band electrons and valence-bandholes in the crystal. These quantities can be calculatedanalytically from the band structure by expanding theband energies in powers of K up to the second order us-ing k · p perturbation theory (see, e.g., Ref. [34]). Whiledoing so, we apply the two-band approximation again,which means that we neglect contributions to m (cid:63),e and m (cid:63),h from other bands than the valence band and theconduction band. Within this model, we get the well-known relations (cf. Ref. [34]) m (cid:63),e = 1 m + 2 κ m E g , m (cid:63),h = − m + 2 κ m E g (33)according to k · p perturbation theory.By adding these two equations, we find that the effec-tive mass m (cid:63) (27) defined in the previous subsection isgiven by the harmonic mean of the effective charge-car-rier masses: m (cid:63) = 2 m − (cid:63),e + m − (cid:63),h , (34)which equals twice the reduced mass. This relation (34)between our effective mass m (cid:63) , which is related to κ (off-diagonal momentum matrix element) via Eq. (27), andthe parabolic curvatures of the energy bands is essentialfor extending the analogy at K = 0 to a neighborhoodof this point with the same effective physical constants, m (cid:63) and c (cid:63) , as before. Note that we are not required toassume m (cid:63),e = m (cid:63),h here in the time-dependent case inorder to draw the analogy.If we had defined an effective elementary charge q (cid:63) in-stead of c (cid:63) (as mentioned in the previous subsection),Eq. (34) would not be valid since we would have a dif-ferent m (cid:63) [given by E g / (2 c ) ] then. For this reason, theanalogy would not work for nonzero (quasi)momenta.Returning to the K derivatives in Eq. (31), we may uti-lize Eq. (34) to write the energy-band difference in thesemiconductor as ∆ E ( K ) = E g + K /m (cid:63) + O ( K ) . As ex-pected at an extremum, we get ∆ E (1) (0) = 0 . The groupvelocity difference ∆ v is given by the first K derivativeof the energy difference, so we get ∆ v (1) (0) = ∆ E (2) (0) =2 /m (cid:63) . The quadratic A term vanishes since κ (1) (0) = 0 (see Appendix C for the calculation). All in all, we arrive at Ω ( t, K ) = Ω ( t,
0) + E g qA ( t ) / (2 m (cid:63) ) Ω ( t, K + O ( K ) Eq. (28) = Ω ( t,
0) + c (cid:63) qA ( t ) Ω ( t, K + O ( K ) . (35)Comparing this result with Eq. (30) confirms the anal-ogy between the Dirac case and the semiconductor caseup to the first order in the conserved (quasi)momentumaround k = K = 0 . E. Analogy in the entire Brillouin zone
The analogy between the Hamiltonians can be ex-tended to the whole Brillouin zone, which means thateach K mode in ˆ H s can be mapped to a k mode in ˆ H D with a suitable effective speed of light and electronmass. In the previous two subsections, we have derivedthat these effective quantities are constant ( K indepen-dent) for long-wavelength modes and that k and K havean interchangeable meaning for these modes. However,this coincidence between k and K is not universal sincewe can always confine the crystal momentum of a Blochelectron in the two-band semiconductor to the first Bril-louin zone (a consequence of restricting ourselves to twobands), while each canonical wave vector in the Diraccase represents a unique mode. The distinction becomesimportant when we go beyond long-wavelength modes,which we will do in this subsection.The question we aim to answer is as follows: Givena semiconductor band structure—i.e., the functions ∆ E ( K ) , ∆ v ( K ) = d∆ E ( K ) / d K , and κ ( K ) are fixed—and a mode K ∈ ( − π/(cid:96), π/(cid:96) ] , can we then find effectiveconstants m (cid:63) ( K ) and c (cid:63) ( K ) and a wave vector k = k ( K ) such that the eigenvalue Ω[ t, k ( K )] in the Dirac case[Eq. (19)] with these K -dependent effective quantitiesequals the semiconductor analog Ω ( t, K ) in Eq. (24)? Wetherefore want to solve the equation m (cid:63) ( K ) c (cid:63) ( K ) + c (cid:63) ( K )[ k ( K ) + qA ( t )] = (cid:20) ∆ E ( K ) + qA ( t )∆ v ( K )2 (cid:21) + (cid:20) qA ( t ) | κ ( K ) | m (cid:21) (36)for an arbitrary potential A ( t ) , so we compare the coef-ficients with respect to the powers of A . This procedureyields three equations, which uniquely fix the three un-known quantities c (cid:63) ( K ) = (cid:114) ∆ v ( K )4 + | κ ( K ) | m , (37) m (cid:63) ( K ) = ∆ E ( K ) | κ ( K ) | mc (cid:63) ( K ) , (38) k ( K ) = ∆ E ( K )∆ v ( K )4 c (cid:63) ( K ) . (39)These K -dependent effective quantities are of coursecompatible with the results from the previous two sub-sections: for K = 0 , we get c (cid:63) (0) = κ /m , m (cid:63) (0) = E g m / (2 κ ) , and k (0) = 0 —exactly what we found inSec. II C. Furthermore, we find that the first K deriva-tives at K = 0 are c (1) (cid:63) (0) = m (1) (cid:63) (0) = 0 and k (1) (0) = 1 ;that is, for all long-wavelength modes, the effective quan-tities are constant, and the crystal momentum K in thesemiconductor has the same meaning as the momentum k in Dirac theory, which is basically the result of Sec. II D.In the remainder of this paper, we will focus on modeswith small conserved (quasi)momenta again. For brevity,we write c (cid:63) and m (cid:63) without a parameter again to denotethe respective value at K = 0 .Note that, in the gauge used here, K is conserved ex-actly for purely time-dependent fields A ( t ) . This is some-what different from other gauges where K becomes effec-tively time dependent K → K + qA ( t ) , and thus theanalogy between pair creation and Landau–Zener tun-neling during the temporal passage through an avoidedlevel crossing (at the gap K = 0 ) becomes even more ap-parent. In our representation (where K is conserved), wemay directly translate the momentum spectra from QEDcalculations (e.g., for the dynamically assisted Sauter–Schwinger effect [7, 8]) to the semiconductor scenario viaEqs. (37), (38), and (39). The only difference is that therange of K is reduced to the Brillouin zone in the semi-conductor case, and the density of states is given per K interval (instead of k for real QED), which introduces anadditional factor of d k/ d K .However, when comparing to experimental results, an-other important difference must be taken into account:the conserved wave numbers K (and k ) correspond tothe canonical momenta, which are generally differentfrom the mechanical momenta. The latter are not con-served, of course, because the electric field acceleratesthe charged particles after they have been created. Thisacceleration then depends on the shape of the dispersionrelation, such that here the analogy to QED eventuallybreaks down. Ergo, the analogy applies to the creation of particle–hole pairs (for a given K ), but not necessarilyto their trajectory after they have been created. F. Analog Sauter–Schwinger effect and dynamicalassistance in gallium arsenide
The fact that the Hamiltonians ˆ H D and ˆ H s do coincidefor long-wavelength modes (except for scales) allows us toinfer that we may directly transfer all findings regardingnonperturbative (tunneling) pair creation from quantumelectrodynamics to the semiconductor model (at least toleading order).
1. Constant electric field
Let us start with a constant electric field E stat with A ( t ) = E stat t as the simplest example. In the Dirac case,this corresponds to the ordinary Sauter–Schwinger effectwith the associated critical electric field strength E QEDcrit [see Eq. (2)]. In a semiconductor, interband tunnelingdue to a constant external field is typically described viathe Landau–Zener model [19–21]—but due to the analogywith quantum electrodynamics (QED), we may also usethe QED terminology and consequently define the analogcritical field strength E crit = c (cid:63) m (cid:63) q = √ m (cid:63) E / g q . (40)This expression, here simply derived from the analogywith QED, can be found in many papers which study thebehavior of semiconductors/insulators in strong electricfields; see, e.g., Ref. [35].As an example for a semiconductor with a direct bandgap at the Brillouin-zone center (as assumed in Sec. II C),we consider gallium arsenide (GaAs) here. The band gapof GaAs measures about E GaAs g = 1 . , and the effec-tive masses m GaAs (cid:63),e = 0 . m and m GaAs (cid:63),h = 0 . m (lightholes; see Ref. [36]) yield the value m GaAs (cid:63) ≈ . m ac-cording to Eq. (34). The resulting critical field strengthis thus E GaAscrit ≈ . × V / cm —a typical value forthis type of semiconductor according to, e.g., Ref. [37].This value is roughly one order of magnitude larger thanthe dielectric-breakdown field strength of GaAs given inRef. [36]. This relation seems reasonable since interbandtunneling starts below E GaAscrit of course, but it is sup-pressed exponentially by the factor exp( − πE GaAscrit /E stat ) .For E stat ≈ E GaAscrit / , this factor measures − . We donot consider the (nonexponential) prefactor in the pair-creation rate here, but one can easily imagine that theexponential term suppresses any realistic prefactor formuch smaller values of E stat .
2. Assisting temporal Sauter pulse
As a next example, we add a temporal Sauter pulse E Sauter / cosh ( ωt ) to the constant background field E stat and assume that the pulse amplitude is much smallerthan the static field; E Sauter /E stat (cid:28) . The effect of theweak pulse on nonperturbative pair creation has beenstudied in Ref. [6]. According to that paper, the pulseis negligible if its characteristic frequency scale, ω , issmaller than a certain critical value ω crit , which dependson the background field strength but (interestingly) noton the pulse amplitude. This critical frequency scale isreached when the combined Keldysh parameter γ ω = mωqE stat = E QEDcrit E stat ωm (41)takes on the value π/ . Above this threshold, the so-called dynamically assisted Sauter–Schwinger effect setsin, which means that the pulse exponentially enhancesthe pure Sauter–Schwinger pair-creation rate induced by E stat .Let us assume in our example that the background fieldis one order of magnitude below the critical field strength.In the Dirac case, that means E stat = E QEDcrit / , and weget a critical frequency scale in the hard X-ray spectrum: ω crit = 80 keV . In our semiconductor example ( E stat = E GaAscrit / ), the result ω crit = 0 .
12 eV lies in the infraredpart of the spectrum.
3. Assisting harmonic oscillation
The last example profile consists of the constant back-ground field E stat again plus a harmonic oscillation E wave cos( ωt ) . Similar to the Sauter pulse, such a wavecan increase the nonperturbative pair-creation rate ex-ponentially as studied in Ref. [10]. However, the criticalvalue of the Keldysh parameter (41) for dynamical assis-tance depends on the ratio E wave /E stat for this profile—or, if ω and E stat are fixed, we can inverse this relation todetermine a critical laser amplitude E critwave as a functionof E stat and ω .In the worldline instanton picture, the effect of theadditional oscillation is that it lowers the instanton ac-tion A , which in turn increases the pair-creation ratesince the rate is proportional to exp( −A ) . (We ignorethe nonexponential prefactor in the pair-creation ratehere; however, it has been shown in Ref. [38] that thebehavior of the exponent A plays the crucial role in thedynamical assistance mechanism.) Let us (arbitrarily)define the threshold of dynamical assistance as a config-uration according to which the pair-creation rate withthe oscillation [ ∝ exp( −A ω ) ] is 50% larger than the rate[ ∝ exp( −A ) ] in the constant background field E stat alone. We may derive from Eqs. (52) and (57) in Ref. [10]that this condition gives e −A ω e −A = exp (cid:34) π E QEDcrit E stat I ( γ ω ) γ ω E wave E stat (cid:35) ! = 1 . , (42)where I ( x ) denotes a modified Bessel function of thefirst kind. Assuming that only the oscillation amplitudeis variable, we solve this equation for E wave to find thecritical amplitude E critwave = ln 1 . π E stat E QEDcrit γ ω I ( γ ω ) E stat . (43)Let us now transfer this QED result (43) to the semi-conductor analog and do some estimations regarding theexperimental realization of assisted tunneling pair cre-ation in GaAs. We assume a rather pure sample ofGaAs placed in a background field E stat = E GaAscrit / again. The harmonic oscillation is generated by a CO Dirac theory Two-band semiconductor electron mass m ↔ m (cid:63) , effective mass [Eqs. (27) and (34)]GaAs: m (cid:63) ≈ . m speed of light c ↔ c (cid:63) = (cid:112) E g / (2 m (cid:63) ) , effective speedGaAs: c (cid:63) ≈ . c mass gap mc ↔ m (cid:63) c (cid:63) = E g , band gap ≈ ↔ GaAs: E g ≈ . Sauter–Schwinger effect: E QEDcrit = m c /q ↔ E crit = √ m (cid:63) E / g / (4 q ) ≈ V / m ↔ GaAs: E crit ≈ × V / m Dynamically assisted Sauter–Schwinger effect: ω crit ≈
80 keV ↔ GaAs: ω crit ≈ .
12 eV
TABLE I: Comparison between the scales in the DiracHamiltonian and the analog quantities in the semicon-ductor model.laser with a wavelength of . µ m . The correspondingphoton energy, .
117 eV , measures less than 8% of theband gap, so pair creation via multiphoton processes isstrongly suppressed. The background field strength andthe laser frequency together yield the combined Keldyshparameter γ ω = 1 . . While this value is fixed, wecan easily vary the laser amplitude. The critical am-plitude (43) is then given by E critwave /E stat = 0 . inthis example, which corresponds to a laser-beam intensityof I crit = ( E critwave ) / / cm . References [39–41](which consider only pulsed radiation though) suggestthat a GaAs sample of sufficient quality will probablynot be destroyed by this amount of incident power—thedamage threshold for CO -laser pulses with a halfwidthof
100 ns = 10 − s given in Ref. [41] is of the order of
10 MW / cm , for example. We will also show later (inSec. III D) that the threshold intensity is reduced signif-icantly in a space-dependent static background field offinite spatial extent.The analog quantities given in this subsection includ-ing the values for GaAs are summarized in Table I. III. SPACETIME-DEPENDENT CASE E = E ( t, x ) In this section, we generalize the semiconductor modelpresented in the previous section to spacetime-dependentelectric fields and compare it to the corresponding DiracHamiltonian again.
A. Hamiltonians
Here, we choose a different gauge, E ( t, x ) = ∂ x Φ( t, x ) ,with a vanishing vector potential A ; that is, the field isdescribed by the spacetime-dependent scalar potential Φ ,which enters the position-space Hamiltonians ˆ H D (3) and ˆ H full s (6) as an additional potential term − q Φ . The mo-mentum-space form of the Dirac Hamiltonian thus con-tains the convolution of the spatial Fourier transform ofthe scalar potential, ˜Φ( t, k ) , and the momentum-spacefield operator ˆ (cid:101) Ψ [see Eq. (4) for the conventions we use]: ˆ H D ( t ) = ∞ (cid:90) −∞ ˆ (cid:101) Ψ † ( t, k ) (cid:18) m kk − m (cid:19) ˆ (cid:101) Ψ( t, k ) d k − q √ π ∞ (cid:90) −∞ ˆ (cid:101) Ψ † ( t, k ) ∞ (cid:90) −∞ ˜Φ( t, k − k (cid:48) ) ˆ (cid:101) Ψ( t, k (cid:48) ) d k (cid:48) d k . (44)As in the time-dependent case, we want to bring thisHamiltonian into a form in which the matrix in the upperline is diagonal. This is accomplished by inserting thesame transformed field operator ˆΥ (17) as in the previoussection (but with A = 0 of course). However, the verysame transformation gives rise to a matrix M in the lower( ˜Φ ) part of the Hamiltonian. This matrix reads as M ( k, k (cid:48) ) = 1 (cid:112) d ( k ) 1 (cid:112) d ( k (cid:48) ) × (cid:18) d ( k ) − d ( k ) 1 (cid:19) · (cid:18) − d ( k (cid:48) ) d ( k (cid:48) ) 1 (cid:19) (45)with the auxiliary function d defined in Eq. (18) but with-out any time dependence here ( A = 0 ). For the trans-formed Dirac Hamiltonian, we thus get ˆ H D ( t ) = ∞ (cid:90) −∞ ˆΥ † ( t, k ) (cid:18) Ω( k ) 00 − Ω( k ) (cid:19) ˆΥ( t, k ) d k − q √ π ∞ (cid:90) −∞ ˆΥ † ( t, k ) ∞ (cid:90) −∞ ˜Φ( t, k − k (cid:48) ) M ( k, k (cid:48) ) ˆΥ( t, k (cid:48) ) d k (cid:48) d k ,(46)again with the same (but time-independent) eigenvalues ± Ω from Eq. (19).Let us now derive the semiconductor Hamiltonian inthe spacetime-dependent field. We start with the fullHamiltonian (6) again but with A = 0 and the additionalpotential term − q Φ . After the insertion of the two-bandapproximation (8), our semiconductor Hamiltonian readsas ˆ H s ( t ) = π/(cid:96) (cid:90) − π/(cid:96) ˆ a † ( t, K ) (cid:18) E + ( K ) 00 E − ( K ) (cid:19) ˆ a ( t, K ) d K − q π/(cid:96) (cid:90) − π/(cid:96) ˆ a † ( t, K ) π/(cid:96) (cid:90) − π/(cid:96) M ( t, K, K (cid:48) )ˆ a ( t, K (cid:48) ) d K (cid:48) d K (47)with the matrix M ( t, K, K (cid:48) ) = (cid:18) (cid:104) + , K | Φ | + , K (cid:48) (cid:105) (cid:104) + , K | Φ |− , K (cid:48) (cid:105)(cid:104)− , K | Φ | + , K (cid:48) (cid:105) (cid:104)− , K | Φ |− , K (cid:48) (cid:105) (cid:19) .(48) Further transformations of the operators ˆ a ± are not nec-essary in this case since the matrix in the upper line of ˆ H s is already diagonal for the present gauge.It is important to notice here that the diagonal ele-ments E ± in ˆ H s are generally not symmetric (for all K )as in the Dirac case ( ± Ω ) in Eq. (46). In the purely time-dependent field, we could make these diagonal elementsin ˆ H s symmetric via a suitable gauge transformation (seeSec. II B). However, the same approach is not valid in aspacetime-dependent field since the Φ part of the Hamil-tonian couples particles with different values of K witheach other, so K is not a conserved quantity anymore,and thus ˆ a † ( t, K )ˆ a ( t, K ) = 1 is not valid in general herefor each K . As we will see in the next subsection, thisfact requires us to make an additional assumption con-cerning the effective masses in the semiconductor in orderto draw the quantitative analogy to the Dirac Hamilto-nian. B. Analogy between the Φ -independent parts ofthe Hamiltonians At this point, we can start to compare the upper linesof the Hamiltonians ˆ H D (46) and ˆ H s (47), which do notdepend on the potential Φ . We focus on the vicinities ofthe band gaps at k = K = 0 again.Up to the lowest nonvanishing order of the small quan-tity k/m near the gap, the diagonal elements in the Diraccase are ± Ω( k ) = ± mc ± k m + O (cid:34)(cid:18) kmc (cid:19) (cid:35) (49)with c written explicitly. According to our notion, theanalogy to ˆ H s is valid if ± Ω coincides with E ± [fromEq. (32)] up to the quadratic order in k or K after sub-stituting the physical scales m and c with effective con-stants. (As in Secs. II C and II D in the case of a purelytime-dependent E field, the physical roles of k and K areequivalent close to the gaps.)We find that the analogy works with the same effec-tive constants, c (cid:63) (28) and m (cid:63) (34), as in the A ( t ) case,but we have to assume in addition that the effective elec-tron mass in the conduction band, m (cid:63),e , equals the effec-tive hole mass m (cid:63),h (in which case m (cid:63) = m (cid:63),e = m (cid:63),h ).Graphically, that means that the parabolic curvaturesof the energy curves E + and E − in Fig. 1(b) must beidentical at the gap. From a practical point of view,this is an important constraint regarding the simula-tion of nonperturbative vacuum pair production in space-time-dependent fields in semiconductors, which can onlybe met approximately. The effective masses in GaAs, m GaAs (cid:63),e = 0 . m and m GaAs (cid:63),h = 0 . m (light holes), dif-fer by about 20%, for example—compared to other com-mon semiconductors with a direct band gap, this is aquite good agreement (values taken from Ref. [36]).We will assume m (cid:63) = m (cid:63),e = m (cid:63),h in the remainder ofthis section.0 C. Analogy between the Φ parts for spatiallyslowly varying potentials We still have to show that the analogy is also truefor the Φ parts [lower lines in Eqs. (46) and (47)] of theHamiltonians in the vicinity of the band gap. We thushave to compare the matrix ˜Φ( t, k − k (cid:48) ) M ( k, k (cid:48) ) / √ π inthe Dirac case with M ( t, K, K (cid:48) ) in the semiconductorcase since the other terms in the Φ parts are equiva-lent. These matrices cannot be the same for arbitrary(quasi)momenta and potentials Φ( t, x ) , so we have tomake reasonable assumptions about these quantities andthen compare the matrices (approximately).Let us start with the Dirac case. As we can see inthe Hamiltonian (46), the Fourier components of the po-tential Φ couple particle states which differ by k − k (cid:48) intheir wave vectors. Since we want to concentrate on theparabolic vicinity of the band gap (the range | k | (cid:28) m )and electron transitions therein, we assume that the po-tential only has nonvanishing Fourier components ˜Φ( t, k ) for small wave vectors which satisfy | k/m | (cid:28) . That is,the potential and thus the electric field is slowly varyingin space compared to the Compton wavelength of an elec-tron, and therefore an electron close to the gap cannotbe excited (directly) to a point far beyond the gap in k space.This assumption is also consistent with the fact that weare interested in nonperturbative pair creation: for thisreason, the electric field should only incorporate photonenergies far below the mass gap m , which correspond towave numbers | k | (cid:28) m —leading basically to the sameassumption as above.For a Dirac-sea electron with a k near the gap( | k/m | (cid:28) ), which may be excited into another statewith the small wave vector k (cid:48) ( | k (cid:48) /m | (cid:28) ) due to thepotential, we may therefore Taylor expand the matrix M ( k, k (cid:48) ) and neglect terms of second order in these smallwave vectors. We get M ( k, k (cid:48) ) = (cid:18) (cid:19) + k − k (cid:48) mc (cid:18) − (cid:19) + O (cid:20) k m c (cid:21) + O (cid:20) ( k (cid:48) ) m c (cid:21) + O (cid:20) kk (cid:48) m c (cid:21) (50)with the speed of light written explicitly.In the semiconductor case, we have to approximatethe matrix M for slowly varying potentials. These arepotentials which only include wavelengths much greaterthan the lattice constant (cid:96) . We therefore assume thatits spatial Fourier transform, ˜Φ( t, K ) , vanishes exceptfor | K | (cid:28) π/(cid:96) . In analogy to the Dirac case, this K -space region coincides with the parabolic vicinity of thesemiconductor band gap; cf. Fig. 1(b).We think that this long-wavelength assumption is prac-tically always satisfied in the context of nonperturbativeelectron–hole pair creation, which requires the photonenergies in the electric field to be much smaller thanthe band gap: ω (cid:28) E g . Let us do a simple estimate to show this: Writing ω as π/ ( nλ ) , where n is the re-fractive index in our semiconductor (for the frequencyunder consideration), the condition ω (cid:28) E g becomes λ (cid:29) π/ ( n E g ) . It is generally justified to assume that E g is (much) smaller than the Fermi energy E F = π / (2 m(cid:96) ) in the empty lattice. Inserting this relation into theabove inequality lets us conclude that λ (cid:29) π/ ( n E F ) ,which can also be written as λ (cid:29) (8 /n )( (cid:96)/λ C ) (cid:96) , where λ C ≈ − m is the Compton wavelength of the elec-tron. For typical semiconductors, (cid:96)/λ C is much greaterthan , while /n is of order . Hence, λ (cid:29) (cid:96) should bereasonable to assume provided ω (cid:28) E g for all photons inthe external field.Since we are especially interested in GaAs here, letus consider this case in particular: The assumption ω (cid:28) E GaAs g = 1 . corresponds to vacuum wave-lengths much greater than
816 nm . The refractive indexof GaAs around the band gap measures about . (seeRef. [42] and cf., e.g., Ref. [36]), so the wavelengths withinthe medium must be much greater than approximately
220 nm —a length scale which is very large compared tothe lattice constant .
565 nm of GaAs. The assumptionof a slowly varying potential in the semiconductor caseis thus not problematic in the context of nonperturba-tive pair creation in GaAs, and, as argued above, thisstatement presumably also holds in most other semicon-ductors.This assumption together with the fact that we con-sider quasimomenta obeying | K | (cid:28) π/(cid:96) and | K (cid:48) | (cid:28) π/(cid:96) lets us derive the (still exact) expression M ( t, K, K (cid:48) ) = 1 √ π ˜Φ( t, K − K (cid:48) ) × (cid:18) (cid:104) + , K | + , K (cid:48) (cid:105) u (cid:104) + , K |− , K (cid:48) (cid:105) u (cid:104)− , K | + , K (cid:48) (cid:105) u (cid:104)− , K |− , K (cid:48) (cid:105) u (cid:19) (51)for the matrix in Eq. (48); see Appendix D for the calcu-lation.Since we are close to the band gap, we may expandthe Bloch factors which appear in the (cid:104) . . . (cid:105) u products[defined in Eq. (11)] around K = 0 up to the first orderin K or K (cid:48) using k · p perturbation theory and the two-band approximation again (cf. Appendix C). Insertingthese expansions from Eq. (C2) and also using the Bloch-factor orthonormality relation (11) yields M ( t, K, K (cid:48) ) = ˜Φ( t, K − K (cid:48) ) √ π (cid:26)(cid:18) (cid:19) + κ ( K − K (cid:48) ) m E g × (cid:18) − (cid:19) + O [ K ] + O [( K (cid:48) ) ] + O [ KK (cid:48) ] (cid:27) . (52)Let us now identify the correct effective scales: Weconsider the expression m (cid:63) c (cid:63) . According to Eq. (28),this quantity is equal to (cid:112) m (cid:63) E g , which in turn becomes1 m E g /κ by means of Eq. (27). We can thus write M as M ( t, K, K (cid:48) ) = ˜Φ( t, K − K (cid:48) ) √ π (cid:26)(cid:18) (cid:19) + K − K (cid:48) m (cid:63) c (cid:63) × (cid:18) − (cid:19) + O [ K ] + O [( K (cid:48) ) ] + O [ KK (cid:48) ] (cid:27) . (53)Comparing this equation to Eq. (50) shows that the Φ parts of the Hamiltonians are equivalent close to the bandgaps as well, with the same scale substitutions as before.Hence, we have derived the analogy between ˆ H D and ˆ H s also in the spacetime-dependent case. D. Dynamically assisted Sauter–Schwinger effect inthe spacetime-dependent case
We close this section by considering an experimentallyoriented setup, which is a spacetime-dependent versionof the dynamically assisted Sauter–Schwinger effect in asemiconductor analog.
1. Assisting temporal Sauter pulse
A spacetime-dependent QED scenario has been stud-ied analytically in Ref. [11] via the worldline instantonmethod. In this reference, the superposition of a spatialand a temporal Sauter pulse is considered: E ( t, x ) = E cosh ( kx ) + E cosh ( ωt ) . (54)If the spatial pulse is very broad [quasi-homogeneous, cf.the linear, middle part in the band diagram in Fig. 2(a)],the “ordinary” dynamically assisted Sauter–Schwinger ef-fect [6] known from the purely time-dependent case inSec. II F is recovered. This effect starts at γ crit ω = mω crit qE = π . (55)The spatial turning points (between which tunneling hap-pens) read as x (cid:63) ± = ± m/ ( qE ) in this case.This situation changes when we narrow the spatialpulse E / cosh ( kx ) by increasing its k , while E iskept constant and subcritical here ( E (cid:28) E QEDcrit ). Thetotal electrostatic energy the pulse provides reads as q ∆Φ = 2 qE /k . When this energy approaches the massgap from above, q ∆Φ (cid:38) m , due to an increasing k , weget a band diagram like in Fig. 2(b). The spatial turningpoints grow according to x (cid:63) ± ∼ ±| ln( q ∆Φ − m ) | in thislimit, so the tunneling rate due to the spatial pulse aloneis low then. These turning points are also the positionsin Euclidean spacetime where the corresponding instan-ton trajectory (we are just referring to the spatial Sauterpulse at the moment) crosses the spatial axis ( x ). The positions ± τ where this instanton trajectory intersectsthe τ axis (imaginary time, τ = i t ) are given by τ = mqE arcsin γ k γ k (cid:112) − γ k (56)according to Ref. [43], where γ k = mkqE (57)is the Keldysh parameter of the spatial Sauter pulse.Hence, the positions ± τ diverge like / √ q ∆Φ − m inthe limit q ∆Φ (cid:38) m , which is equivalent to γ k (cid:37) ; seeRefs. [11].The effect of the additional temporal Sauter pulse ∝ cosh − ( ωt ) is, in the instanton picture, that it givesrise to “walls” at ± τ sing = ± π/ (2 ω ) , which “reflect” theinstanton trajectory when touched. The value of ω forwhich the unperturbed (no temporal pulse) instanton tra-jectory just touches these “walls” is precisely ω crit , theonset frequency scale for dynamical assistance. The in-stanton-trajectory scalings explained above let us con-clude that ω crit ∼ √ q ∆Φ − m in the limit q ∆Φ (cid:38) m ;cf. Ref. [11]. Hence, if q ∆Φ is only slightly larger than m [Fig. 2(b)], even low-frequency pulses should lead toan exponential enhancement of nonperturbative (tunnel-ing) pair creation via the dynamically assisted Sauter–Schwinger effect. Since the space dependence of suchpulses is slow, their purely time-dependent treatmentshould be valid.Now let us transfer this situation to the semiconductoranalog. A localized, time-independent E field within asemiconductor gives rise to the same schematic band dia-grams depicted in Fig. 2. For example, the band bendingmay be due to a suitable doping profile plus an additionalexternal bias if required. The exact form of the bandswill not be that of a hyperbolic tangent in general as inFig. 2, which corresponds to a spatial Sauter-pulse E field ∝ cosh ( kx ) . However, we assume that the spatial E fieldwithin the semiconductor does also decay exponentiallyfor large | x | , just like a spatial Sauter pulse does—butwe do not prescribe an exact pulse shape near the fieldmaximum (the region around x = 0 in Fig. 2). Note thatthis assumption is not compatible with the conventionaldepletion approximation (see, e.g., Ref. [36]) accordingto which the density of ionized dopants is piecewise con-stant, which leads to parabolic potential curves withinthese ionized regions and constant potential values out-side. But, the idea of sharp transitions between ionizedand unionized regions is generally considered unrealis-tic, and one expects “smeared” transitions instead (seeRef. [36]). We think that it is physically reasonable toassume exponential “tails” at the edges of such transi-tions, which, together with the Boltzmann statistics ofthe free charge carriers, should lead to a built-in fieldapproaching zero exponentially (far away from x = 0 ).Even if thermal effects are negligible (low temperatures),we nevertheless still expect the built-in field to decay ex-ponentially due to quantum effects: if we think of the2 x - ★ x + ★ x ℰ χ q ΔΦ (a) Large potential step q ∆Φ (cid:29) χ . x - ★ x + ★ x ℰ χ q ΔΦ (b) Slightly above threshold q ∆Φ = χ . x ℰ χ q ΔΦ (c) No tunneling for q ∆Φ < χ . FIG. 2: Two energy bands separated by a gap χ are bentby a localized, space-dependent electric field centered at x = 0 (schematically; tanh profiles). The solid curves arethe lower edge of the upper energy band and the upperedge of the lower band, respectively. We have χ = 2 m inQED and χ = E g in the semiconductor analog. (a) Forfield profiles which give rise to a large potential differ-ence q ∆Φ (cid:29) χ , there are many different states betweenwhich tunneling is possible (e.g., along the dashed line).(b) For q ∆Φ (cid:38) χ , the number of possible tunneling tran-sitions approaches zero and the spatial turning points x (cid:63) ± diverge. In this ∆Φ range, the tunneling rate can signif-icantly be increased via additional electric low-frequencypulses according to Ref. [11]. (c) If the energy step q ∆Φ is smaller than the gap χ , the bands are separated ener-getically as indicated by the dashed constant-energy line,so tunneling becomes impossible. ionized, spatially fixed dopants on either side of the junc-tion as creating an effective, finite potential well for therespective majority carriers, the wave functions of thesecarriers will leak into the forbidden region (which beginssomewhere on the other side of the junction)—an effectwhich is in accordance with the exponential decay of thebuilt-in field.Assuming that the time-independent (built-in) fieldwithin the semiconductor decays exponentially, we con-clude that the spatial turning points x (cid:63) ± scale like ln( q ∆Φ − E g ) in the critical limit q ∆Φ (cid:38) E g [Fig. 2(b)],just like in the QED case above. Now, let us imaginethe unperturbed (no additional temporal Sauter pulse)instanton trajectory in this limit: x (cid:63) ± will be large, sothe instanton trajectory will be a huge closed loop overthe x range [ x (cid:63) − , x (cid:63) + ] . Except near x = 0 , where we donot know the exact pulse shape of the spatial field in thesemiconductor, the instanton trajectory is the same asthat of a spatial Sauter pulse (QED case above) becausethe E fields in both cases decay exponentially; this func-tional form is sufficient to fix the shape of the instantontrajectory. The imaginary-time ( τ ) positions where theinstanton trajectory crosses the τ axis will thus also di-verge like / (cid:112) q ∆Φ − E g in the limit q ∆Φ (cid:38) E g . Thisis because the exponential tails of the field let the in-stanton trajectory grow so large in this limit that thedetails close to the maximum field (around x = 0 ) arenot important for the scaling anymore. Consequently,we expect the same scaling, ω crit ∼ (cid:112) q ∆Φ − E g , as inthe QED case [11] to be exhibited by an analog of thedynamically assisted Sauter–Schwinger effect in a semi-conductor with a localized, time-independent inner fieldin the limit q ∆Φ (cid:38) E g as well.Note that this scaling law solely depends on the waythe electric field approaches zero asymptotically (here:exponentially). See Refs. [44, 45] for more informationon universal pair-creation phenomena in the no-tunnel-ing limit.
2. Assisting harmonic oscillation
Another way to assist tunneling dynamically in thisspacetime-dependent scenario is via a harmonic oscilla-tion instead of a temporal Sauter pulse. This profile, E ( t, x ) = E cosh ( kx ) + E cos( ωt ) , (58)is more appropriate to describe experiments in whichpair creation is assisted via laser beams, for example.The purely time-dependent version of this profile (ho-mogeneous background field instead of a spatial Sauterpulse) has been studied in Ref. [10], also via the worldlineinstanton method. In contrast to the temporal Sauterpulse, the oscillation does not give rise to “walls” (sin-gularities) parallel to the x axis in Euclidean spacetimebecause cos( ωt ) = cosh( ωτ ) is well behaved for all imag-inary times τ = i t . Hence, the onset of dynamical as-3sistance by the oscillation is not as sharply defined as inthe Sauter-pulse case. We have formulated the thresh-old condition (43) for the oscillation amplitude E in thecase of a homogeneous background field ( k = 0 limit)in Sec. II F 3 (with E stat → E and E wave → E here).Let us now estimate how this critical oscillation ampli-tude E crit2 changes when the background field becomes aspatial Sauter pulse ( k > ), while the maximum back-ground field strength E and the oscillation frequency ω remain fixed.We consider the instanton trajectory of the spatialSauter pulse again (see Ref. [43]). This closed loop in Eu-clidean spacetime has its largest extent [from − τ to + τ with τ from Eq. (56)] in the imaginary-time direction onthe τ axis ( x = 0 ), where the field strength of the spatialSauter pulse measures E —and that of the oscillationwould be E cosh( ωτ ) . We assume that this instantontrajectory will be noticeably deformed (dynamical assis-tance) by the additional oscillation if the amplitude E is large enough such that the term E cosh( ωτ ) has amagnitude comparable to E . Equation (43) can be un-derstood as defining a certain “threshold ratio” betweenthese two terms for the special case of a homogeneousbackground field [ k = 0 , in which case τ = m/ ( qE ) ]: E crit2 ( k = 0) E cosh[ ωτ ( k = 0) (cid:124) (cid:123)(cid:122) (cid:125) γ ω ] ! = const. (59)When we now increase k (i.e., decrease the pulse width), τ ( k ) grows according to Eq. (56). As a simple esti-mate, we determine the critical amplitude E crit2 ( k ) forthis nonzero k by demanding that the constant on theright-hand side of the above equation remains invariant.Hence, E crit2 ( k > must be smaller than E crit2 ( k = 0) tocompensate the increase of cosh[ ωτ ( k > . By consid-ering the ratio between both critical amplitudes, we caneliminate the constant and find E crit2 ( k ) E crit2 ( k = 0) = cosh γ ω cosh[ ωτ ( k )] . (60)Note that this way to derive E crit2 ( k ) is not guaranteedto preserve the property that we have originally imposedto find the critical amplitude in the homogeneous-fieldcase (the oscillation enhances the pair-creation yield by50%; see Sec. II F 3)—rather, we have presented a simpleway to estimate how E crit2 ( k ) changes when increasing k from zero, and our main intention here is to show thatthe critical amplitude decreases when the spatial extentof the static background field gets smaller.By squaring Eq. (60) (and inserting τ ), we finally findan expression for the critical (laser-beam) intensity as afunction of the inverse Sauter-pulse length scale k : I crit ( k ) I crit ( k = 0) = cosh γ ω cosh (cid:104) γ ω arcsin( γ k ) / (cid:16) γ k (cid:112) − γ k (cid:17)(cid:105) ,(61) L nm01020304050 I crit ( L ) kW / cm I crit ( L → ∞ ) L FIG. 3: Threshold CO -laser-beam intensity (61) fordynamical assistance of tunneling as a function of thewidth L = 2 π/k of the static Sauter pulse E / cosh ( kx ) in GaAs. The parameter values in this plot are E = E GaAscrit / ≈
60 MV / m , ω = 0 .
117 eV (so γ ω = 1 . ),and I crit ( L → ∞ ) = I crit ( k = 0) = 47 kW / cm (seeSec. II F 3). Tunneling vanishes in the limit γ k (cid:37) [cf.Fig. 2(b)], which corresponds to L (cid:38) L = 76 nm here.where the threshold for a constant background field, I crit ( k = 0) , can be calculated via Eq. (43). Note that I crit ( k ) decreases for increasing k until the critical ampli-tude becomes zero at a certain k value with γ k = 1 . Thisis precisely the k value at which tunneling due to thespatial Sauter pulse alone vanishes [cf. Fig. 2(b)], so theconcept of assisted tunneling breaks down there. Hence,by decreasing the width of the static background fieldappropriately, we can make the threshold intensity fordynamical assistance via the oscillation arbitrarily smallin principle—however, in order to really verify this ef-fect under controlled conditions in the laboratory, thetunneling currents (assisted and non-assisted) should notbecome too tiny, so that they remain measurable. Thisrequirement poses a practical limit on how narrow thespatial Sauter pulse (built-in field) may become.Let us exemplify the result of Eq. (61) for a semicon-ductor analog by reconsidering the experimental setupfrom Sec. II F 3 (time-dependent case; i.e., homogeneousfields only): we said there that tunneling pair creationin GaAs induced by a constant background field E = E GaAscrit / ≈
60 MV / m will significantly be assisted bya CO -laser wave E cos( ωt ) (with ω = 0 .
117 eV fixed,so γ ω = 1 . ) if the beam intensity is about I crit ( k =0) = 47 kW / cm . If we replace the constant backgroundfield with a spatial Sauter pulse E / cosh ( kx ) with anassociated length scale L = 2 π/k , Eq. (61) gives us the L -dependent critical laser intensity plotted in Fig. 3.We emphasize that the dynamical assistance mecha-nisms from Refs. [6, 10, 11] considered here are fullynonperturbative effects, which are based on a classi-cal-field description of the external fields. So, eventhough we assume the assisting temporal Sauter pulse4and the time-dependent oscillation to be weak in ampli-tude ( E (cid:28) E (cid:28) E QEDcrit ), they still must incorporatea large number of photons (high intensity) as to allowfor the classical field picture. The dynamically assistedSauter–Schwinger effect in semiconductors should thusnot be confused with the Franz–Keldysh effect [46, 47](see also Refs. [48]), which is related to a shift in thephoton-absorption edge. The QED analog of this effectwas considered in Refs. [49–51].
IV. GENERALIZATION TOELECTROMAGNETIC FIELDS IN 2+1DIMENSIONS
In this section, we briefly discuss the feasibility to gen-eralize the analogy between Bloch electrons and holes insemiconductors and Dirac’s theory to 2+1 spacetime di-mensions, including known results.The step from one to two spatial dimensions is inter-esting because it also allows for external magnetic fields,not just electric fields as in the one-dimensional case. TheDirac field operator ˆΨ still has two components in twodimensions since there is a third Pauli matrix ( σ x ) forthe additional required gamma matrix γ . This absenceof spin simplifies the calculations and is typically irrele-vant in the context of tunneling pair creation [43, 48].In two-dimensional space, the magnetic field is scalarand acts like the B z component for charge carriers con-fined to the ( x, y ) plane in three dimensions. It is givenby the components of the vector potential (cid:126)A ( t, x, y ) via B = − ∂ x A y + ∂ y A x .Graphene (see Refs. [52, 53]) is a well-known exam-ple for a two-dimensional system which mimics relativis-tic electron motion near the points where the conduc-tion band touches the valence band in the Brillouin zone(Dirac cones); see also Ref. [31]. However, the associ-ated effective electron rest mass is zero, so the analog ofthe Schwinger limit E QEDcrit ∝ m vanishes in graphene,and thus there is no characteristic exponential suppres-sion of the Sauter–Schwinger effect; see Refs. [15, 16].But by generating an offset (symmetry breaking) betweenthe two triangular carbon lattices, which in combinationmake up the honeycomb structure of graphene, it is pos-sible to separate both energy bands by a finite energygap. The Dirac cones of this so-called semiconductinggraphene become shaped like paraboloids near the gaps,which corresponds to a nonvanishing effective rest mass.Semiconducting graphene has already been produced suc-cessfully in the laboratory via epitaxial growth as re-ported in Ref. [54], and it has been studied in Ref. [17] asan analog for electron–positron pair creation in constantand oscillating (in time) electric fields.One possible problem with analogs of Dirac’s theory inmultiple space dimensions is that the vacuum is isotropic,so m and c are scalar quantities, while material proper-ties of semiconductors, for example, can depend on direc-tion (effective mass tensor, direction-dependent effective speed of light, etc.). Since these anisotropies have nocounterpart in Dirac theory, we focus on materials whichbehave isotropically around the band gap (scalar effectivequantities) or at least whose anisotropies do not interferefor the electromagnetic field profile under consideration.A simple profile which is interesting to study in 2+1 di-mensions consists of perpendicular electric ( x direction)and magnetic fields, both constant. In Dirac theory, themagnetic field decreases the pair-creation rate inducedby the E field because we can always Lorentz-transformto a frame according to which the magnetic field is zeroand the pair-creating electric field measures E − B ;see, e.g., Ref. [4, 55]. That means that Sauter–Schwingerpair creation vanishes completely for strong enough mag-netic fields ( B = E/c or higher in SI units). In Ref. [29],the authors state that for the same reason the equiv-alent effect also happens in a two-band semiconductor,but again with the effective scales m → m (cid:63) and c → c (cid:63) [Eqs. (28) and (34)]. (As in Sec. III B, we have to assume m (cid:63),e = m (cid:63),h = m (cid:63) here.) Their reasoning is that theelectrons in the semiconductor obey an effective Diracequation (near the band gap) since this type of equa-tion models a simple two-band system. The validity ofa Dirac-type equation implies the existence of an ana-log Lorentz transformation (with c → c (cid:63) ), which is thenused to show that tunneling vanishes for B = E/c (cid:63) in thesemiconductor. More detailed explanations of this Dirac-type two-band model are given in Refs. [28, 48], whichalso study the crossed-field profile, and in Refs. [30, 31].We can understand the reduction of Landau–Zenertunneling in a semiconductor due to a perpendicular B field as well by starting with the same approach as in theprevious sections, which deal with the QED–semiconduc-tor analogy in 1+1 dimensions. That is, we begin withthe Schrödinger Hamiltonian (6) again but for 2+1 di-mensions and with the vector potential (cid:126)A ( x ) = − Bx(cid:126)e y and the additional scalar potential Φ( x ) = Ex (crossedconstant fields). We then insert the 2+1-dimensional ver-sion of the two-band approximation (8). The resultingtwo-band Hamiltonian contains first- and second-orderderivatives with respect to the component K x of the crys-tal momentum, which arise from the Bloch-basis repre-sentations of x and x (see, e.g., Refs. [32, 56, 57] forthe calculation of these matrix elements). As a simple,semiclassical approach, we then consider just the centerof the Brillouin zone (cid:126)K = 0 (where we, again, assumethe direct band gap to be located) and derive the corre-sponding x -dependent band energies from the Hamilto-nian ( i ∂ K x → x ). What we find is an expression whichlooks similar to the relativistic counterpart E ± ( x ) = − qEx ± (cid:112) m c + ( cqBx ) (62)(for the same crossed-field profile and (cid:126)k = 0 ) but withthe known effective constants c → c (cid:63) and m → m (cid:63) ,plus additional terms under the square root. However,these additional terms can be neglected for typical values m (cid:63) /m ≈ − – − , c (cid:63) /c ≈ − – − (see the data for5GaAs in Table I, for example), a not too strong tunneling-inducing electric field E ≈ − E crit ≈ V / m , a per-pendicular magnetic field in the range B (cid:46) E/c (cid:63) ≈
10 T ,and x values of the order of the unperturbed (by the B field) tunneling length E g / ( qE ) . The E ± ( x ) graphs inthe semiconductor thus look like the relativistic version,which was also found in Ref. [48].We emphasize that the reduction of the tunneling cur-rent in perpendicular B fields has just been explainedby referring to the local dispersion relations of the Diracequation and the two-band semiconductor model, respec-tively. So, although the same effect happens in both sys-tems, this does not necessarily imply the analogy betweenthe full underlying Hamiltonians/equations of motion. V. CONCLUSIONS
We studied the quantitative analogy between theSauter–Schwinger effect and interband tunneling in suit-able semiconductors with special emphasis on fieldswhich depend on space and time. To this end, we com-pared the Dirac Hamiltonian [Eqs. (5) and (44)] in 1+1dimensions with the effective two-band Hamiltonian ofa semiconductor [Eqs. (15) and (47)]. In the case ofpurely time-dependent electric fields E ( t ) , one may de-rive a quantitative analogy for every k mode after a spa-tial Fourier transform. In this case, the analog of theSchwinger critical field (40) is determined by materialconstants such as the band gap E g and the interbandcoupling κ , which is related to the effective mass m (cid:63) viaEq. (27). For GaAs, for example, we obtain a value ofapproximately E GaAscrit ≈ . × V / m , which is far be-low the QED critical field E QEDcrit ≈ . × V / m andabout one order of magnitude above the typical break-down field strength of a few ( – ) V / m in GaAs ac-cording to Ref. [36]. This is a very natural result becausethe analog of the QED critical field yields the ultimatequantum limit until which the semiconductor can retainits insulating behavior: no matter how perfect and free ofdefects the sample is and how low the temperature, tun-neling will become strong at that field strength (unless itis suppressed, e.g., by a magnetic field; see below).This scenario of purely time-dependent electric fields E ( t ) would already allow us to study the analog of thedynamically assisted Sauter–Schwinger effect [6] with anadditional Sauter pulse, for example, where the thresh-old frequency (for E stat = E GaAscrit / ) lies around .
12 eV (instead of
80 keV as in real QED), which is favorablefor an experimental verification. For the experimentallyprobably more relevant case of an additional sinusoidalfield (instead of a Sauter pulse), we get the additional re-quirement that the field strength of this additional fieldshould be large enough to assist tunneling. This indicatesan important difference to the well-known Franz–Keldysheffect [46, 47] corresponding to tunneling assisted by asingle photon (which can be treated perturbatively). Asingle photon with an energy of .
12 eV would not have a significant impact because its energy is far below theband gap. However, a field oscillating at this frequencywith sufficient intensity can assist tunneling, which showsthat it is necessary to treat this field beyond (first-order)perturbation theory; see also Refs. [10, 58].For electric fields depending on space and time, E ( t, x ) ,more approximations are necessary to obtain a quan-titative analogy. For example, because electrons andpositrons in real QED are limited by the same speed oflight, one has to neglect the difference in the velocities ofparticles and holes (more precisely, the curvature of theirbands at the gap) in the semiconductor and to approxi-mate both by the same effective mass of around 7% of theelectron mass. This scenario E ( t, x ) includes additionalinteresting cases. For example, if the strong and staticfield is inhomogeneous and close to the edge of the tun-neling regime, the frequency and/or field strength of theadditional weaker time-dependent field required for dy-namical assistance is reduced; see Sec. III D and Ref. [11].Finally, we discussed the generalization to 2+1 dimen-sions. Apart from facilitating the distinction betweentransverse and longitudinal fields (see also Ref. [10]),this case also allows us to introduce a magnetic field.For the Sauter–Schwinger effect in real QED, it is wellknown that an additional magnetic field can suppressthe tunneling probability. Here, we find an analogoussuppression for the tunneling in semiconductors; see alsoRefs. [28, 29, 48]. For example, in GaAs with an electricfield of 1% of the critical field, E stat = E GaAscrit / (i.e.,roughly one order of magnitude below the breakdownfield strength), a magnetic field of can alreadysuppress tunneling significantly ( E stat /c GaAs (cid:63) ≈ . willstop it completely).In summary, our findings suggest that the analog ofthe Sauter–Schwinger effect and its dependence on thespatial and temporal field profile (e.g., dynamical assis-tance) should be observable with present-day technologyin suitable high-quality semiconductors at low tempera-tures, where competing mechanisms (due to defects etc.)are suppressed sufficiently. VI. OUTLOOK: INTERACTIONS
In all of our previous considerations, we neglected theCoulomb interaction between the electrons. This approx-imation is well motivated experimentally since the pic-ture of non-interacting electrons (e.g., band structure,Drude model) describes the experiments in bulk semi-conductors typically very well. Note that the situation isdifferent in quantum dots, for example, where the spatialconfinement enhances Coulomb interaction effects.The same approximation is typically used in real QED,where most of the calculations regarding the Sauter–Schwinger effect neglect the interaction between the cre-ated electrons and positrons. While this interaction isexpected to be small, it is probably fair to say that it isnot fully understood yet.6In order to obtain a rough estimate, let us comparethe Coulomb force F Coulomb of the electron–positron pairseparated by the tunneling distance to the force F ext = qE induced by the external electric field: F Coulomb F ext = 14 α QED EE QEDcrit . (63)Thus, even for a very strong field of E = E QEDcrit / , wefind a suppression of ≈ × − , which indicates thatneglecting these interactions is a good approximation.If we now perform the same estimate for the semicon-ductor case, we find F Coulomb F ext = 14 α QED EE crit cc (cid:63) . (64)As a result, due to c/c (cid:63) ≈ for GaAs (cf. Table I), theimpact of the Coulomb interactions is stronger in thissituation. Intuitively speaking, the electrons are slowerand thus have more time to interact. This enhancementis even more pronounced for graphene [52] where c/c (cid:63) ≈ . Nevertheless, even with the very strong field E = E GaAscrit / , the Coulomb force is only a 4% correction tothe external force, such that neglecting it should still bea good approximation.Turning the argument around, high-precision experi-ments in semiconductors could (at least qualitatively) il-luminate the impact of interactions, while the analogousexperiments in real QED are far more difficult. ACKNOWLEDGMENTS
The authors acknowledge financial support by theDeutsche Forschungsgemeinschaft (Grant No. SFB 1242,Projects A01, B03, and B07).
Appendix A: Absorption of the A term in thesemiconductor Hamiltonian In the time-dependent case (Sec. II), the electric poten-tial is specified in temporal gauge; that is, E ( t ) = ˙ A ( t ) and the scalar potential, Φ , is set to zero. However, in-troducing also the scalar potential Φ explicitly for themoment, the electric field becomes E = ˙ A + ∂ x Φ , so apurely time-dependent scalar potential Φ = Φ( t ) doesnot have any physical significance. The scalar potentialcouples to time derivatives ( ∂ t → ∂ t − i q Φ ) and thereforeleads to the additional term − q Φ in the Hamiltonian (6): ˆ H full s ( t ) = ∞ (cid:90) −∞ ˆ ψ † (cid:26) [ − i ∂ x + qA ( t )] m + V ( x ) − q Φ (cid:27) ˆ ψ d x .(A1)We may thus absorb the quadratic A term in this equa-tion by setting Φ( t ) = qA ( t ) / (2 m ) and obtain the sim-plified Hamiltonian (7). Appendix B: Bloch-wave momentum matrixelements1. Underlying formula
Let us first derive a general equation for a type ofintegrals which appears regularly in calculations in theBloch wave basis. Assume that g ( x ) is an (cid:96) -periodicfunction—i.e., g ( x + (cid:96) ) = g ( x ) —and we want to calculatethe integral (cid:82) ∞−∞ exp(i kx ) g ( x ) d x with a real k satisfying | k | < π/(cid:96) .We start by writing the (cid:96) -periodic g as a Fourier series, g ( x ) = ∞ (cid:88) j = −∞ ˜ g j e π i jx/(cid:96) , (B1)with complex Fourier coefficients ˜ g j . Insertion into theabove integral yields ∞ (cid:90) −∞ e i kx g ( x ) d x = ∞ (cid:88) j = −∞ ˜ g j ∞ (cid:90) −∞ e i( k +2 πj/(cid:96) ) x d x = 2 π ∞ (cid:88) j = −∞ ˜ g j δ (cid:18) k + 2 π(cid:96) j (cid:19) . (B2)Since | k | < π/(cid:96) , the delta distribution vanishes exceptfor the case j = 0 ; cf., e.g., Ref. [59]. The correspondingFourier coefficient, ˜ g , coincides with the average of g over a unit cell, so we get the result ∞ (cid:90) −∞ e i kx g ( x ) d x = 2 π(cid:96) (cid:96) (cid:90) g ( x ) d x δ ( k ) . (B3)
2. Momentum matrix elements
We start to calculate the matrix elements by insertingthe general Bloch wave form (10): (cid:104) n, K | − i ∂ x | n (cid:48) , K (cid:48) (cid:105) = ∞ (cid:90) −∞ f ∗ n ( K, x )( − i ∂ x ) f n (cid:48) ( K (cid:48) , x ) d x = ∞ (cid:90) −∞ e i( K (cid:48) − K ) x u ∗ n ( K, x ) × (cid:20) K (cid:48) u n (cid:48) ( K (cid:48) , x ) − i ∂u n (cid:48) ( K (cid:48) , x ) ∂x (cid:21) d x . (B4)Since the Bloch factors are (cid:96) periodic with respect to x and | K (cid:48) − K | < π/(cid:96) (because K and K (cid:48) are restrictedto the first Brillouin zone), we may apply Eq. (B3) and7find (cid:104) n, K | − i ∂ x | n (cid:48) , K (cid:48) (cid:105) = (cid:104) K (cid:104) n, K | n (cid:48) , K (cid:105) u (cid:124) (cid:123)(cid:122) (cid:125) δ nn (cid:48) + (cid:104) n, K | − i ∂ x | n (cid:48) , K (cid:105) u (cid:105) δ ( K (cid:48) − K ) ,(B5)cf., e.g., Ref. [32]. Note that we used the unit-cell scalarproduct defined in Eq. (11) to write the remaining single-cell integrals. Furthermore, the first product just gives aKronecker delta due to the Bloch-factor orthonormaliza-tion (11). Appendix C: Taylor expansion of κ ( K ) around K = 0 We are interested in the first-order K dependence of κ [Eq. (13)], so we need to evaluate the first K derivative of κ at K = 0 . Together with the definition of the single-cellproduct in Eq. (11), we get ( K derivatives are denotedas superscript numbers in parentheses) κ (1) (0) = (cid:68) u (1) − (0 , x ) (cid:12)(cid:12)(cid:12) − i ∂ x (cid:12)(cid:12)(cid:12) u + (0 , x ) (cid:69) u + (cid:68) u − (0 , x ) (cid:12)(cid:12)(cid:12) − i ∂ x (cid:12)(cid:12)(cid:12) u (1)+ (0 , x ) (cid:69) u . (C1)The K derivatives of the Bloch factors at K = 0 canbe calculated by expanding u ± ( K, x ) in powers of K via k · p perturbation theory. Again, we apply the two-bandapproximation, so we only take into account correctionsfrom the valence band and the conduction band. Theresulting expansions, u ± ( K, x ) = u ± (0 , x ) ± κ Km E g u ∓ (0 , x ) + O ( K ) (C2)(cf. Ref. [34]), inserted above immediately give κ (1) (0) = (cid:28) − κ m E g u + (0 , x ) (cid:12)(cid:12)(cid:12)(cid:12) − i ∂ x (cid:12)(cid:12)(cid:12)(cid:12) u + (0 , x ) (cid:29) u + (cid:28) u − (0 , x ) (cid:12)(cid:12)(cid:12)(cid:12) − i ∂ x (cid:12)(cid:12)(cid:12)(cid:12) κ m E g u − (0 , x ) (cid:29) u = − κ E g ∆ v (0) = 0 (C3)since the group velocities v ± ( K ) = (cid:104)± , K | − i ∂ x |± , K (cid:105) u /m vanish at the direct bandgap at K = 0 in both energy bands.The Taylor series of κ around K = 0 thus does not in-clude a linear term (according to k · p perturbation theoryand the two-band model); κ ( K ) = κ + O ( K ) . Appendix D: Matrix elements of M ( t, K, K (cid:48) ) forspatially slowly varying potentials The elements of the matrix M ( t, K, K (cid:48) ) in Eq. (48)are expressions of the form (cid:104) n, K | Φ | n (cid:48) , K (cid:48) (cid:105) . For slowlyvarying potentials, this general scalar product can be cal-culated. We start by inserting the Bloch-wave form (10)and the spatial Fourier transform [cf. Eq. (4)] of the po-tential. After changing the order of integration, we get (cid:104) n, K | Φ | n (cid:48) , K (cid:48) (cid:105) = 1 √ π ∞ (cid:90) −∞ ˜Φ( t, k ) × ∞ (cid:90) −∞ e i( k + K (cid:48) − K ) x u ∗ n ( K, x ) u n (cid:48) ( K (cid:48) , x ) d x d k . (D1)Let us now reconsider our assumptions: The slowlyvarying potential satisfies ˜Φ( t, k ) = 0 unless | k | (cid:28) π/(cid:96) ,so we only need to calculate the x integral (correctly) forsmall values of k . Furthermore, we are interested in thequasimomentum region near the band gap to draw theanalogy to Dirac theory; that is, we evaluate the matrixelements between values of K and K (cid:48) near the Brillouinzone center and thus | K (cid:48) − K | is significantly smallerthan π/(cid:96) , the total zone width. Altogether, we mayassume | k + K (cid:48) − K | < π/(cid:96) and thus apply the formulain Eq. (B3) again: (cid:104) n, K | Φ | n (cid:48) , K (cid:48) (cid:105) = 1 √ π ∞ (cid:90) −∞ ˜Φ( t, k ) δ ( k + K (cid:48) − K ) × π(cid:96) (cid:96) (cid:90) u ∗ n ( K, x ) u n (cid:48) ( K (cid:48) , x ) d x d k . (D2)Now, the k integral can easily be calculated and thesingle-cell x integral is expressed via the Bloch-factorscalar product introduced in Eq. (11). That yields ourend result (cid:104) n, K | Φ | n (cid:48) , K (cid:48) (cid:105) = ˜Φ( t, K − K (cid:48) ) √ π (cid:104) n, K | n (cid:48) , K (cid:48) (cid:105) u . (D3)Note that this equation is exact as long as the conditionmentioned above is true. [1] F. Sauter, “Über das Verhalten eines Elektrons im homo-genen elektrischen Feld nach der relativistischen Theorie Diracs,” Z. Phys. , 742–764 (1931). [2] F. Sauter, “Zum ‘Kleinschen Paradoxon’,” Z. Phys. ,547–552 (1932).[3] W. Heisenberg and H. 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