A Bogoliubov Transformation to Diagonalize Tunneling Hamiltonian

Destructive interference of direct and crossed Andreev pairing in a system of two nanowires coupled via an -wave superconductor

Abstract

We consider a system of two one-dimensional nanowires coupled via an -wave superconducting strip, a geometry that is capable of supporting Kramers pairs of Majorana fermions. By performing an exact analytical diagonalization of a tunneling Hamiltonian describing the proximity effect (via a Bogoliubov transformation), we show that the excitation gap of the system varies periodically on the scale of the Fermi wavelength in the limit where the interwire separation is shorter than the superconducting coherence length. Comparing with the excitation gaps in similar geometries containing only direct pairing, where one wire is decoupled from the superconductor, or only crossed Andreev pairing, where each nanowire is considered as a spin-polarized edge of a quantum Hall state, we find that the gap is always reduced, by orders of magnitude in certain cases, when both types of pairing are present. Our analytical results are further supported by numerical calculations on a tight-binding lattice. Finally, we show that treating the proximity effect by integrating out the superconductor using the bulk Green’s function does not reproduce the results of our exact diagonalization.

pacs:
74.45.+c,71.10.Pm,73.21.Hb,74.78.Na

Introduction.

Topological superconductivity has garnered a great deal of attention in recent years Alicea (2012); Leijnse and Flensberg (2012); Beenakker (2013) both theoretically and experimentally because the localized excitations of such systems, known as Majorana fermions, obey non-Abelian statistics and can be utilized for applications in quantum computing Kitaev (2001); Nayak et al. (2008). The proposals which have received the most attention to date involve engineering Majorana states in nanowires with Rashba spin-orbit coupling in the presence of a Zeeman field Lutchyn et al. (2010); Oreg et al. (2010); Klinovaja et al. (2012); Mourik et al. (2012); Deng et al. (2012); Das et al. (2012); Rokhinson et al. (2012); Churchill et al. (2013); Finck et al. (2013); Chang et al. (2015); Albrecht et al. (2016); Zhang et al. (2017) or in ferromagnetic atomic chains Nadj-Perge et al. (2014); Pawlak et al. (2016); Klinovaja et al. (2013); Vazifeh and Franz (2013); Braunecker and Simon (2013); Nadj-Perge et al. (2013); Pientka et al. (2013). In the absence of any Zeeman splitting, it is possible to generate an even more exotic time-reversal invariant topological superconducting phase which supports Kramers pairs of Majorana fermions Wong and Law (2012); Nakosai et al. (2013); Zhang et al. (2013); Keselman et al. (2013); Schrade et al. (2015); Haim et al. (2014); Klinovaja and Loss (2014); Gaidamauskas et al. (2014); Klinovaja et al. (2014); Danon and Flensberg (2015); Haim et al. (2016). One such proposal involves coupling two Rashba nanowires via an -wave superconductor Klinovaja and Loss (2014); Gaidamauskas et al. (2014). In this system, superconductivity is induced in the nanowires via direct Cooper pair tunneling, where both electrons of a Cooper pair tunnel into the same wire, and crossed Andreev tunneling, where one electron from a Cooper pair tunnels into each wire Byers and Flatté (1995); Choi et al. (2000); Deutscher and Feinberg (2000); Lesovik et al. (2001); Recher et al. (2001); Yeyati et al. (2007). The topological phase can be realized if the strength of crossed Andreev pairing exceeds that of direct pairing. However, to date, the direct and crossed Andreev pairing strengths have been treated as theoretical parameters Klinovaja and Loss (2014); Gaidamauskas et al. (2014); Klinovaja et al. (2014); Danon and Flensberg (2015) and no rigorous treatment of the proximity effect in this system has been carried out.

In this paper, we study the interplay between direct and crossed Andreev pairing in a noninteracting double-nanowire system by calculating the proximity-induced excitation gap as a function of the interwire separation (). We show that the two pairing types always interfere destructively. When the tunneling strengths into each nanowire are equal, the excitation gap in the presence of both types of pairing is simply the difference between the gap in the presence of only direct pairing and the gap in the presence of only crossed Andreev pairing, with the direct gap always being larger than the crossed Andreev gap. When the interwire separation is shorter than the superconducting coherence length (), this destructive interference can lead to an order of magnitude reduction in the size of the excitation gap of the system.

Our results are based on an exact analytical diagonalization of the tunneling Hamiltonian via a Bogoliubov transformation. We derive a set of effective Bogoliubov-de Gennes (BdG) equations that we then solve to determine the excitation gap as a function of . Additionally, we show that integrating out the superconducting degrees of freedom using the bulk superconducting Green’s function, a common method for treating the proximity effect in low-dimensional systems Alicea (2012); Sau et al. (2010); Potter and Lee (2011); Stanescu et al. (2011); Kopnin and Melnikov (2011); Zyuzin et al. (2013); Takane and Ando (2014); van Heck et al. (2016); Reeg and Maslov (2017), yields incorrect and qualitatively different results in our finite geometry.

Model.

The system we consider is displayed in Fig. 1. Two one-dimensional nanowires are coupled to a superconducting strip of finite width , taken to occupy . The system is taken to be infinite in the -direction, allowing us to define a conserved momentum . We consider a Hamiltonian of the form

(1)

The nanowire Hamiltonian can be expressed generally as

(2)

where [] creates (annihilates) an electron of spin and momentum in nanowire and the Hamiltonian density of each wire is left unspecified. We describe the superconductor by a BCS Hamiltonian,

(3)

where , [] creates (annihilates) an electron of spin and momentum at position inside the superconductor, ( is the effective mass and is the Fermi energy of the superconductor), is the superconducting pairing potential, and are Pauli matrices acting in Nambu space. We also allow for electrons to tunnel between superconductor and wire, assuming that this process preserves both spin and momentum. Tunneling is described by

(4)

where is a wire-dependent tunneling amplitude and denotes the position of wire .

Figure 1: Geometry of considered model. A 2D conventional -wave superconductor of width () separates two 1D nanowires. The system is infinite in the -direction.

Bogoliubov-de Gennes Equations.

To solve the model under consideration, we perform an exact diagonalization of Hamiltonian (1) by introducing a transformation of the form

(5a)
(5b)

where describes the new quasiparticles of the system and are electron (hole) wave functions. It is straightforward to show sup () that transformation (5) diagonalizes Hamiltonian (1) provided that the wave functions obey a set of BdG equations given by

(6a)
(6b)
(6c)
(6d)

In Eqs. (6), we introduce the spinor electron (hole) wave function for and denote the Pauli matrix acting in spin space by .

While Eqs. (6) were derived without making any assumptions about the nanowire Hamiltonian, for the remainder of the paper we focus on the simple case where each nanowire is a normal conductor that can be described by , with ( and are the effective mass and Fermi energy of the nanowires). With this simple choice for the nanowire Hamiltonian, we are able to eliminate the trivial spin sector from the BdG equations; essentially, we can reduce Eqs. (6) from matrix equations to scalar equations. Equations (6a) and (6b) form an independent algebraic system that yields the solutions

(7)

Substituting Eq. (7), we can decouple Eqs. (6c) and (6d) to obtain a system of differential equations describing the wave functions in the superconductor,

(8)

The solution to Eq. (8) within the left (), middle () and right () regions of the superconductor is

(9)

where is a spinor wave function, , , and . To determine the eight unknown coefficients , we must impose appropriate boundary conditions at and (note that a vanishing boundary condition at and has already been imposed). In addition to continuity of the wave function, the boundary conditions on the derivatives of the wave functions are determined by the delta-function terms of Eqs. (8) and are obtained by direct integration:

(10a)
(10b)

In Eqs. (10) we introduce an energy scale associated with tunneling which is proportional to the square of the tunneling amplitude, , where is the Fermi velocity of the superconductor. Assuming that the Fermi momentum of the superconductor greatly exceeds that of the nanowires () allows us to approximate (because is conserved, typical values take ; we also expand in the limit ). However, even by making these simplifications the solvability condition of Eqs. (10) cannot be solved besides numerically for an arbitrary parameter set sup ().

In order to proceed analytically, we assume that the superconductor is only weakly coupled to each nanowire, so that . In this limit, the relevant pairing energies in the nanowires are small and we can focus our attention on energies . We also assume that the nanowires are (symmetrically) located near the ends of the superconductor, such that and with . The solvability condition in this limit can be expressed as , with the complicated expressions for the coefficients given in the Supplementary Material sup (). This equation can be solved exactly for the energy spectrum , which is plotted for several values of in Fig. 2. When [Fig. 2(a)], the spectrum consists of two parabolic bands and has a gap given by ; this represents the decoupling of two nanowires with a large spatial separation. When the wires are brought closer together [Fig. 2(b-c)], crossed Andreev pairing reduces the size of the gap while single-particle couplings induced by tunneling effectively shift the chemical potentials of each band Klinovaja et al. (2014); sup ().

Excitation gap in the weak-coupling limit.

Our goal is to calculate the excitation gap (the global minimum of the spectrum) as a function of . Although we are able to solve for the spectrum exactly, it is still quite cumbersome to determine the excitation gap for all when tunneling is asymmetric ().

Figure 2: Excitation spectra for (a) , (b) , and (c) . For all plots, we choose and .

Let us consider the symmetric-tunneling case , leaving the asymmetric case for the supplemental material sup (). Under the assumption of symmetric tunneling, it is quite straightforward to solve for the gap for any value of sup (). Assuming that , the gap is

(11)

[The gap in principle also depends on the wire position through an additional factor ; because this is a rather arbitrary quantity, we simply replace it by its mean value throughout.] When the superconductor is very wide (), the gap approaches . When the superconductor is very narrow (), the gap oscillates on the scale of between its maximum value , attained for (), and its minimum value , attained for .

Note that because we chose , we are unable to distinguish between direct and crossed Andreev pairing in our result for the gap [Eq. (11)]. We again must stress that we are not solving an effective model, so the direct and crossed Andreev pairing functions are not parameters of our theory as in Refs. Gaidamauskas et al. (2014); Klinovaja and Loss (2014); Klinovaja et al. (2014); Danon and Flensberg (2015). Instead, we identify direct terms as being proportional to () and crossed Andreev terms as being proportional or (). In an attempt to differentiate between the two contributions, we compare the gap in the presence of both pairing types to that of similar systems containing only one type of pairing.

First, we isolate direct pairing in our system by decoupling one of the wires from the superconductor. Setting in Eq. (8), we find a direct gap of the form sup ()

(12)

If the superconductor is very wide, the gap approaches as before. If the superconductor is very narrow, the gap is . The gap is sharply peaked near and has a maximum value . The gap is minimized near and takes the value .

Figure 3: Proximity-induced gaps plotted as a function of superconductor width. (a) Analytical results for . Black curve corresponds to symmetric tunneling () [Eq. (11)], green curve corresponds to case of a single wire () [Eq. (12)], and blue curve corresponds to quantum Hall regime ( and ) [Eq. (13)] (b) Numerical results for , , , and , where is the hopping parameter in the superconductor. Inset: plot of , showing very good quantitative agreement with Eq. (14).

To isolate crossed Andreev pairing in our system, we consider a situation where both nanowires are spin-polarized and have opposite spin; i.e., they are edge states of two quantum Hall systems with opposite chirality. In this case, we introduce a spin dependence to the tunneling amplitudes, . Assuming for example that while , we set in the equation for the electron wave function (which has spin-up) and in the equation for the hole wave function (which has spin-down) to find a crossed Andreev gap given by sup ()

(13)

If the superconductor is very wide, the gap oscillates on the scale and decays on the scale , . If the superconductor is very narrow, we expand to find . Similarly to the direct pairing case, the gap is sharply peaked near , having a maximum value . The crossed Andreev gap is minimized near , where it vanishes. The vanishing of the gap indicates a change in sign of the crossed Andreev pairing function (see also sup ()). Therefore, it should be possible to form a -junction by coupling two systems of different . Such phase shifts are crucial for engineering Majorana fermions in similar setups Keselman et al. (2013); Schrade et al. (2015).

The three gaps that we have calculated are plotted in Fig. 3(a). The gaps are related through the remarkably simple expression

(14)

indicating that direct and crossed Andreev pairing interfere with one another destructively. This effect is maximized when the superconductor is very narrow, as crossed Andreev reflection is not significantly suppressed by the interwire separation. Quite interestingly, because the direct and crossed Andreev gaps attain their maxima at the same thickness (), the gap is minimized when pairing is maximized. Furthermore, destructive interference between the two pairing processes at these points leads to an order of magnitude reduction of the gap [specifically, a reduction by a factor of order ].

We also support our analytical results with a standard tight-binding calculation in the geometry of Fig. 1 sup (). Results of the tight-binding calculation are plotted in Fig. 3(b), showing very good qualitative agreement with Fig. 3(a). We also plot the quantity in the inset of Fig. 3(b), showing very good quantitative agreement with Eq. (14).

Integrating out superconductor.

Finally, we show that integrating out the superconducting degrees of freedom from Eq. (1) using the bulk superconducting Green’s function does not reproduce the results of our exact diagonalization in a finite geometry. Assuming that tunneling is weak and symmetric () and that the superconductor is very narrow (), we integrate out the superconductor to yield an effective Hamiltonian describing superconductivity induced in the nanowires sup (). This effective Hamiltonian yields a low-energy spectrum , where differentiate one-wire (two-wire) tunneling processes, , , and . In obtaining the low-energy spectrum, we expanded the effective Hamiltonian to order . The minimum excitation gap of the spectrum is . Therefore, if the superconductor is integrated out using the bulk Green’s function, one would incorrectly find that crossed Andreev pairing always dominates over direct pairing in the limit [note that direct pairing shows up in the effective Hamiltonian only at order ].

Physically, this procedure gives a false result because it fails to properly account for the boundary conditions that must be imposed when evaluating the Gaussian path integral. When the width of the superconductor is small compared to the coherence length, these boundary effects cannot be neglected. We find that integrating out the superconductor using the bulk Green’s function reproduces the correct spectrum only in the limit , when the boundary effects can be safely neglected sup ().We also discuss in the supplemental material how one can properly account for the boundary effects when integrating out sup ().

Conclusions.

We have shown that direct and crossed Andreev pairing interfere destructively in a system of two nanowires coupled via an -wave superconducting strip. When the interwire separation is shorter than the coherence length , this destructive interference can lead to an order of magnitude reduction in the size of the excitation gap when compared to similar systems containing only a single type of pairing. Our analytical solution is based on an exact treatment of the proximity effect through the diagonalization of the tunneling Hamiltonian (via a Bogoliubov transformation) and is supported by numerical tight-binding calculations. Additionally, we have explicitly shown that integrating out the superconductor using the bulk Green’s function does not reproduce the results of our exact diagonalization.

The interference effects discussed in this paper, which are manifested through oscillations of the excitation gap on the scale of the Fermi wavelength , can most easily be observed when the interwire separation is smaller than the coherence length . If the superconductor is metallic, observing these oscillations is not feasible. However, proximity-inducing superconductivity in a low-density semiconducting two-dimensional electron gas (2DEG) such as InGaAs/InAs (as in Kjaergaard et al. (2016)), has several advantages. Inducing superconductivity by the proximity effect will make both and larger and will allow the density of the 2DEG to be tuned with a gate voltage (so that can be varied using a single sample). Due to our assumption of translational invariance along the -direction, the interface between superconductor and nanowire must be made smooth (on the scale of ) and must be made uniform.

Finally, we note that crossed Andreev pairing is always weaker than direct pairing in the absence of interactions. Therefore, intrawire repulsive electron-electron interactions are needed to stabilize the time-reversal-invariant topological phase in the double-nanowire system, as they can significantly reduce direct pairing while leaving crossed Andreev pairing unaffected Recher and Loss (2002); Bena et al. (2002). In this case, the nanowires support Kramers pairs of Majorana fermions and parafermions Klinovaja and Loss (2014). However, the destructive interference between direct and crossed Andreev pairing in the double-wire setup allows for a conventional topological superconducting phase to form at significantly reduced magnetic field strengths compared to the case of a single wire with only direct pairing Schrade et al. (2017).

Acknowledgments.

We thank C. Schrade and Y. Volpez for useful discussions. This work was supported by the Swiss National Science Foundation and the NCCR QSIT.

Supplemental Material

Appendix A Bogoliubov Transformation to Diagonalize Tunneling Hamiltonian

We consider a tunneling Hamiltonian which couples an -wave superconductor to two nanowires. Such a Hamiltonian can be written generally as

(1)

For now, we leave the nanowire Hamiltonian unspecified,

(2)

where denotes the coordinate in wire and is the Hamiltonian of wire . We describe the superconductor using a conventional BCS Hamiltonian given by

(3)

where denotes the coordinate in the superconductor. We define the (constant in space) pairing potential matrix by and ( and are the effective mass and Fermi energy of the superconductor, respectively). Finally, we have allowed for tunneling between the superconductor and each wire; such a tunneling Hamiltonian can be expressed generally as

(4)

where is a tunneling matrix element coupling a state in the superconductor of spin at position to a state in wire of spin at position . The matrix element also satisfies the identity . To diagonalize Hamiltonian (1), we introduce a Bogoliubov transformation of the form

(5)

To ensure appropriate fermionic commutation relations for the quasiparticle operators , the wave functions in the nanowires must obey the following relations:

(6)

Similarly, the wave functions in the superconductor must obey

(7)

Additionally, the product of nanowire and superconducting wave functions must obey

(8)

Inverting Eqs. (5), the quasiparticle operators can be expressed in terms of the nanowire and superconductor fermion operators as

(9)

Inverting also the relations in Eqs. (6)-(8), we obtain two additional constraints on the wave functions:

(10)

We now substitute transformation (5) into Eq. (1) and define the Bogoliubov-de Gennes (BdG) equations by

(11)

After defining the BdG equations, the Hamiltonian can be expressed as

(12)

Making use of Eqs. (10) to evaluate , we find

(13)

Hence, our Bogoliubov transformation indeed diagonalizes the Hamiltonian.

We will now specify an explicit geometry and model for the tunneling Hamiltonian to coincide with what is considered in the main text. We assume that the system is infinite in the -direction; owing to the translational invariance, the momentum is then a good quantum number. We assume that tunneling preserves the spin and is local in space,

(14)

where denotes the position of wire . Such a tunneling term also preserves the momentum . Fourier transforming the BdG equations to momentum space and substituting Eq. (14), we arrive at the form quoted in the main text:

(15)

where we introduce the electron and hole spinors and (and similarly for and ) and suppress explicit reference to quantum number .

Appendix B Solving BdG Equations to Determine Excitation Gap

To determine the excitation spectrum of Hamiltonian (1), we must solve Eqs. (15) for . We can solve algebraically for the wave functions of the wires to yield

(16)

Substituting Eqs. (16) into Eqs. (15), we obtain a set of coupled differential equations describing the superconducting wave functions,

(17)

The nanowires induce delta-function terms in the BdG equations that describe the superconducting wave functions, meaning that the presence of the wires simply introduce boundary conditions to the BdG equations. These boundary conditions can be found by directly integrating Eqs. (17):