A method for determining valley-splitting in a semiconductor device

Abstract

A method of determining a valley splitting of a semiconductor device comprises setting a field strength of an external magnetic field B, for splitting entangled spin states associated with a double quantum dot generated in the semiconductor device; evolving the entangled spin states for an evolution period; measuring a state of the entangled spin states in a basis of the entangled spin states, the basis comprising at least two basis states; repeating the evolving and the measuring to determine a probability of the state of the entangled spin states being at least one of the at least two basis states; and assessing, based on the determined probability, the valley splitting.

Description

DescriptionTitle: A method for determining valley-splitting in a semiconductor device

[0001] The present invention relates to a method for determining valley-splitting in a semiconductor heterostructure.Background of the invention

[0002] Silicon is semiconductor used extensively in current micro- and nanoelectronics. Silicon has enabled the semiconductor industry to meet Moore’s law during the last few decades. Nowadays, billions of nanometre-scale transistors per chip are being manufactured.

[0003] Silicon is also a promising candidate for quantum computing based on qubits encoded in spins (spin states) of electrons confined in quantum dots. Recent advances show that a two-dimensional architecture based on a silicon heterostructure and structured electrode gates for quantum computing with several qubits is in reach (see Matthias Kiinne et al. arXiv: 2306.16348 (2023)).

[0004] In order to make such silicon-based quantum computing possible, however, a degeneracy involving six conduction band minima (valleys) in a silicon structure, including a silicon heterostructure, must be overcome. One way of lifting the degeneracy is by straining the crystal lattice of the silicon structure (e.g., by depositing a layer of silicon onto a silicon germanium (SiGe) substrate) or by exposing the silicon structure to an electrical field. The strain or the exposure to the electrical field can reduce or eliminate the valley degeneracy by bringing the degenerate valleys to different energy levels. The energy difference between a ground state of one valley and the next higher valley is often referred to as valley splitting (Evs).

[0005] One disadvantage of straining, and / or exposing to electrical fields of the silicon structure is that electron mass and electron mobility are affected, which impacts on the functioning of the silicon structure with dimensions in the nano-range.

[0006] However, valley splitting that is sufficiently large enables the generation of well- defined (i.e., unlikely to transition to other quantum states) and long-lived (i.e., with sufficiently long decoherence times to enable qubit operations) spin-qubits in the silicon structure.

[0007] Moreover, for valley-qubits in silicon based on two of the six valleys, the valley splitting has to be known. For hybrid qubits, e.g., hybrid of a spin-qubit and a charge-qubit, the valley splitting has to be both known and adjustable.

[0008] For quantum processor architectures based on the shuttling of spin-qubits (see Matthias Kiinne et al. arXiv: 2306.16348 (2023)), the valley splitting has to be sufficiently large in order to enable coherent shuttling of the spin-qubits. As the valley splitting depends on local properties (local strain, local electrical fields, atomic details) of the silicon structure, a two-dimensional determination of the valley splitting across the silicon structure is desirable.Summary of the disclosure

[0009] The present disclosure relates to method for determining valley-splitting of a semiconductor heterostructure.

[0010] A method of determining a valley splitting of a semiconductor device comprises setting a field strength of an external magnetic field B, for splitting entangled spin states associated with a double quantum dot generated in the semiconductor device; evolving the entangled spin states for an evolution period; measuring a state of the entangled spin states in a basis of the entangled spin states, the basis comprising at least two basis states; repeating the evolving and the measuring to determine a probability of the state of the entangled spin states being at least one of the at least two basis states; and assessing, based on the determined probability, the valley splitting.

[0011] The entangled spin states may be associated with a first electron or hole and a second electron or hole, arranged in the double quantum dot.

[0012] The method may further comprise arranging the double quantum dot at a lateral position of the semiconductor device.

[0013] The method may further comprise moving at least one moveable quantum dot of the double quantum dot by a distance d along at least one shuttling path of the semiconductor device.

[0014] The moving may be performed before the evolving of the entangled spin states.

[0015] The moving may be performed after the evolving of the entangled spin states.

[0016] The repeating may comprise maintaining the shuttling distance d or altering the shuttling distance d.

[0017] The repeating may comprise maintaining the field strength of the external magnetic field B or altering the field strength of the external magnetic field B.

[0018] The method may further comprise initialising a first quantum state of the first electron or hole, and initialising a second quantum state of the second electron or hole.

[0019] The method may further comprise measuring the occupancy state of the loadable quantum dot.

[0020] The method may further comprise loading the first electron or hole and the second electron or hole in the double quantum dot.

[0021] The method may further comprise identifying a valley splitting by detecting anomalous behaviour the oscillations of the entangled spin states between the two basis states.

[0022] In the method according to the disclosure, the spin states are used as local probes of the valley splitting in the semiconductor device.Brief description of the drawings

[0023] FIG. 1 shows a top view of a semiconductor device.

[0024] FIG. 2 shows two energy landscapes of quantum states with an anticrossing.

[0025] FIG. 3 shows oscillations of entangled spins between two quantum states for different external magnetic fields.

[0026] FIG: 4 shows a singlet probability with reference to its mean value for different external magnetic fields and different shuttling distances.

[0027] FIG: 5 shows the singlet probability of FIG. 4, further indicating the field strengths of the magnetic field, at which anomalous behaviour occurs.

[0028] FIG. 6 shows the singlet probability, like in FIG. 4, for a different lateral position at the semiconductor device 10.

[0029] FIG. 7 shows a further semiconductor device.Detailed description

[0030] The present disclosure relates to a method of determining a valley splitting of a semiconductor device 10 shown in FIG. 1. The valley splitting may be determined along one or two directions with respect to a surface 14 of the semiconductor device 10. In one aspectof the disclosure, the valley splitting may be determined in a direction along at least one path 45 along the semiconductor device 10 and / or in a direction perpendicular to the at least one path 45. The at least one path 45 extends along an x-direction and a -direction (see FIG. 1)

[0031] FIG. 1 shows an example of a semiconductor device 10. The semiconductor device 10 may comprise a bulk semiconductor material. An example of the bulk semiconductor material is silicon. The semiconductor device 10 may comprise a semiconductor heterostructure. An example of the semiconductor heterostructure is a silicon heterostructure, such as Si / SiGe, a germanium semiconductor heterostructure, such as GaAs / AlGaAs or Ge / SiGe, or a silicon MOS heterostructure, such as Si / SiO2. Using semiconductor materials to form the semiconductor heterostructure facilitates manufacturing due to easy handling and low costs of the materials, such as in the case of silicon. There are established technologies for using silicon in computing hardware. A two-dimensional electron gas (2DEG) or a two-dimensional hole gas (2DHG) is confinable within the semiconductor heterostructure, formed from the semiconductor materials, in a quantum well (not shown). The 2DEG or the 2DHG may further be confined based on electrical potentials. The electric potentials may be static electric potentials or non-static electric potentials. The electrical potentials may form at least one quantum dot QD1, QDm. At least one electron or hole of the 2DEG or 2DHG is trappable or confinable in the at least one quantum dot QD1, QDm. A plurality of quantum states may be associated with the at least one electron or hole. Examples of the plurality of quantum states are a plurality of spin states, a plurality of orbital states, a plurality of valley states, and any combination thereof. The plurality of quantum states may serve to implement at least one qubit. For example, the spin of the trapped (confined) at least one electron or hole is usable to implement at least one spin-qubit.

[0032] Moving the electrical potentials results in moving the at least one moveable quantum dot QDm (indicated in FIG. 1 by the dashed circles at two positions at the at least one path 45). The moving of the at least one moveable quantum dot QDm enables moving the trapped (confined) at least one electron / hole. The moving of the at least one moveable quantum dot QDm further enables moving the plurality of quantum states associated with the trapped (confined) at least one electron / hole. Altering a strength of the electrical potentials alters the degree of confinement of the trapped (confined) at least one electron or hole.

[0033] The semiconductor heterostructure comprises several layers of differing material composition. As explained above, the semiconductor heterostructure may be a Si / SiGe heterostructure, a GaAs / AlGaAs heterostructure, a Si-MOS heterostructure, or a Ge / SiGe heterostructure. Generally, the semiconductor heterostructure may be made of one or more materials in which a two-dimensional electron gas (2DEG) or a two-dimensional hole gas (2DHG) can be generated.

[0034] In one aspect, the semiconductor heterostructure may further comprise a silicon cap (not shown). In a further aspect, the semiconductor heterostructure may further comprise at least one layer of strained silicon. In yet a further aspect, the semiconductor heterostructure may further comprise at least one layer of silicon dioxide.

[0035] The semiconductor device 10 shown in FIG. 1 is a substantially two-dimensional device, as defined by two dimensions of at least one surface 14. A thickness of the semiconductor device 10 is defined by a third dimension of the semiconductor device 10 in a direction perpendicular to the plane of FIG. 1.

[0036] On the semiconductor heterostructure or on the silicon cap, one or more dielectric or insulating layers may be arranged. A plurality of gate electrodes LP, LB1, LB2, Gl, G2, G3, SI, S2, S3, S4, Fl, F2 may be arranged on and / or in between the one or more dielectric or insulating layers.

[0037] The plurality of gate electrodes LP, LB1, LB2, Gl, G2, G3, SI, S2, S3, S4, Fl, F2 may further be arranged to move (shuttle) the at least one electron or hole along the at least one path 45. Likewise, the plurality of gate electrodes LP, LB1, LB2, Gl, G2, G3, SI, S2, S3, S4, Fl, F2 may further be arranged to move (shuttle) the plurality of quantum states along the at least one path 45. The movement (shuttling) may occur in either one of the two directions (back and forth) along the at least one path 45.

[0038] The plurality of gate electrodes LP, LB1, LB2, Gl, G2, G3, SI, S2, S3, S4, Fl, F2 may be made of metal. The plurality of gate electrodes LP, LB1, LB2, Gl, G2, G3, SI, S2, S3, S4, Fl, F2 may be individually provided with individual voltages by one or more voltage sources. The plurality of gate electrodes LP, LB1, LB2, Gl, G2, G3, SI, S2, S3, S4, Fl, F2 may be superconducting. The individual voltages may serve to define the at least one path 45 and / or move (shuttle) the at least one electron or hole. The provided individual voltages may comprise DC (direct current) voltages and AC (alternating current) voltages. Theindividual voltages may comprise one or more stationary voltages and one or more non- stationary voltages. The individual voltages may be applied by means of DC lines, AC lines, and / or bias tees. The individual voltages may be adjustable.

[0039] An external magnetic field splits a plurality of quantum states, e.g., ones of the plurality of quantum states associated with the at least one qubit (e.g., spin states such as the spin-up and the spin-down), into separated energy levels (Zeeman splitting). The external magnetic field may be provided by an external magnet, e.g., an electromagnet (not shown), that is placed in the vicinity of the semiconductor device 10. In case the semiconductor device 10 is comprised in, or connected to, a quantum processor, a computational basis for the at least one qubit may be selected from the plurality of quantum states.

[0040] The plurality of gate electrodes LP, LB1, LB2, Gl, G2, G3, SI, S2, S3, S4, Fl, F2 may comprise one or more laterally positioning gate electrodes (also termed "screening gates”) Fl, F2 (indicated by two pairs of dotted lines in FIG. 1) arranged to define and / or modify a lateral position of a trajectory in the quantum well at the at least one path 45 for moving (shuttling) the at least one electron or hole. The trajectory may be a trajectory of one or more potential wells (further described below), in which the at least one electron or hole is arrangeable. The one or more potential wells may thus be one or more travelling potential wells. The trajectory of the one or more potential wells may thus correspond to a trajectory of the at least one electron or hole. Thus, the lateral position of the trajectory may correspond to a lateral position of the one or more potential wells and / or of the at least one electron or hole. The trajectory at the at least one path 45 may be directed in the x-direction and / or the y-di recti on (see FIG. 1). The trajectory may overlap in the x-direction and the j'-di recti on with the at least one path 45.

[0041] The plurality of gate electrodes LP, LB1, LB2, Gl, G2, G3, SI, S2, S3, S4, Fl, F2 may further comprise one or more shuttling gate electrodes (also termed “conveyor gates” or “finger gates” or “clavier gates”) SI, S2, S3, S4 (see 1) arranged to move (shuttle) the at least one electron or hole along the at least one path 45. The moving (shuttling) of the at least one electron or hole enables bringing a first electron (or hole) of the at least one electron (or hole) into proximity of a further electron (or hole) of the at least one electron (or hole), for performing an operation involving the first electron (or hole) and the further electron (or hole). In case the semiconductor device 10 is comprised in, or connected to, a quantumprocessor, the moving (shuttling) of the at least one electron or hole enables bringing a first qubit of the at least one qubit, associated with the at least one electron or hole, into proximity of a further qubit of the at least one qubit for performing a two-qubit operation. The two- qubit operation enables quantum computing. For performing the moving (shuttling) of the at least one electron or hole, the one or more shuttling gate electrodes SI, S2, S3, S4 are provided with a time-varying (i.e., non-stationary) ones of the voltages. For example, the time-varying or non-stationary ones of the voltages may be substantially sinusoidally shaped, with time-varying phases 2Tift + A(pLand phase differences A(pLbetween the shuttling gate electrodes SI, S2, S3,= 7T, to generate travelling waves for generating the one or more travelling potential wells. In one example, the amplitudes of the time-varying voltages may range between 150 and 200 mV depending on an individual distance of the shuttling gate electrodes SI, S2, S3, S4 to the quantum well (the individual distance resulting from the one or more dielectric or insulating layers being arranged between the shuttling gate electrodes SI, S2, S3, S4 to electrically insulate the shuttling gates electrodes SI, S2, S3, S4 from each other). In a further example, the frequency may be 10 MHz. In another example, the time-varying voltages applied to the shuttling gate electrodes SI, S2, S3, S4 may include stationary offsets ranging between approximately 0.5 V and 1.2 V.

[0042] The plurality of gate electrodes LP, LB1, LB2, Gl, G2, G3, SI, S2, S3, S4, Fl, F2 may further comprise electron-handling gate electrodes (or hole-handling gate electrodes) Gl, G2, G3 arranged for performing at least one action on the at least one electron or hole. The electron-handling electrodes Gl, G2, G3 may be used as plunger gates and / or barrier gates. The electron-handling electrodes Gl, G2, G3 may be used to control an occupancy of a loadable quantum dot QD1. The loadable quantum dot QD1 may be a stationary quantum dot. The electron-handling electrodes Gl, G2, G3 may further be used to control a detuning and / or a tunnel barrier between the loadable quantum dot QD1 and the at least one moveable quantum dot QDm, when the at least one moveable quantum dot QDm is adjacent to the loadable quantum dot QD1 (thus forming a double quantum dot QD1, QDm), e.g., for performing an exchange of the at least one electron (or hole) of the at least one electron (or hole) and the further electron (or hole) of the at least one electron (or hole). For performing these functions of the electron-handling electrodes Gl, G2, G3, the electron-handling electrodes Gl, G2, G3 and at least some of the shuttling gate electrodes SI, S2, S3, S4 maybe provided with the voltages in a coordinated manner. For example, the stationary offsets applied to the shuttling gate electrodes SI, S2, S3, S4 may be set in a manner to provide the double quantum dot QD1, QDm which has a pre-defined shape of the electrical potential.

[0043] The double quantum dot QD1, QDm is understood to be a pair of the loadable quantum dot QD1 and the moveable quantum dot QDm that are arranged in the semiconductor device 10 such that an exchange of the at least one electron (or hole) of the at least one electron (or hole) and the further electron (or hole) of the at least one electron (or hole) may be achieved. In one aspect of the disclosure, the loadable quantum dot QD1 and the moveable quantum dot QDm of the double quantum dot QD1, QDm are arranged in proximity of the electron-handling electrodes Gl, G2, G3.

[0044] The semiconductor device 10 may further comprise at least one pitch-enhancing gate electrode (also termed “top gate”; not shown) arranged to enable enhancing a conveyor gate pitch or spacing z of the conveyor gates SI, S2, S3, S4.

[0045] The semiconductor device 10 may further comprise at least one vertically positioning gate electrode (also termed “back gate”; not shown) arranged to define and / or modify a vertical position of the trajectory in the quantum well and / or at the at least one path 45 for moving (shuttling) the at least one electron or hole within the unit cell or beyond the unit cell. The vertical position of the trajectory may correspond to a vertical position of the at least one electron or hole.

[0046] The semiconductor device 10 serves to move (shuttle) the at least one moveable quantum dot QDm in the semiconductor heterostructure for moving (shuttling) the at least one electron or hole within the unit cell or beyond the unit cell. The semiconductor device 10 may thus form a “shuttling lane”. Aspects of the shuttling lane are disclosed in international patent applications no. WO 2021 / 052531 Al, the disclosure of which is incorporated herein by reference in its entirety.

[0047] The plurality of gate electrodes LP, LB1, LB2, Gl, G2, G3, SI, S2, S3, S4, Fl, F2 further comprise gates LP, LB 1 , LB2, which together with a current path LTop form a singleelectron transistor (SET). The current path LTop may also serve as an electron reservoir.

[0048] The SET may be used to load at least one electron into the loadable quantum dot QD1. The semiconductor device 10 may serve to initialize the at least one electron or hole, which is loaded in the loadable quantum dot QD1, to be in a chosen one of the plurality ofquantum states. For example, the semiconductor device 10 may serve to initialize the at least one qubit associated with the at least one electron or hole, which is loaded into the loadable quantum dot QD1. For example, waiting for at least a relaxation time lets a current one of the plurality of quantum states of the at least one electron or hole transition (or relax) to (or remain at) a lowest-energy quantum state of the plurality of quantum states, e.g., based on interactions with a lattice of the semiconductor heterostructure. When the at least one electron or hole has been initialized, the current one of the plurality of quantum states, associated with the at least one electron or hole, (e.g., the spin-up or the spin-down) has been selected.

[0049] In one aspect of the disclosure, the at least one electron (or hole) comprises two electrons (or holes). The two electrons (or holes) may be loaded in the loadable quantum dot QD1. The two electrons (or holes) may be initialised to a singlet state S. The waiting may, for instance, result in the two electrons (or holes) being found in the spin-singlet state S after the waiting (e.g., when two or more electrons or holes are loaded into the loadable quantum dot QD1).

[0050] In another aspect, the at least one electron (or hole) comprises four electrons (or holes). The four electrons (or holes) may be loaded in the loadable quantum dot QD1. The four electrons or holes in the loadable quantum dot QD1 may occupy two of the valleys. For example, two valleys comprise the lowest-energy valley and the next-higher energy valley. The loading of the four electrons or holes in the loadable quantum dot QD1 enables enhancing an energy gap (or energy splitting) between the singlet state S and one or more triplet states, e.g., a triplet state To. The singlet state S and the triplet state To may be formed by two of the four electrons (or holes) in the next-higher energy valley. The enhancing of the energy gap may increase a range of voltages applied to the gate electrodes Gl, G2, G3 for performing Pauli spin blockade (see below).

[0051] The SET may be further used as a charge sensor sensitive to the occupancy of the loadable quantum dot QD1. The semiconductor device 10 may thus serve to read out the current one of the plurality of quantum states of the at least one electron or hole.

[0052] The semiconductor device 10 enables the construction of a quantum processor in which the first one electron (or hole) of the at least one electron (or hole), and along-side the current one of the plurality of quantum states, is moved or shuttled to the further electron (orhole) of the at least one electron (or hole), for an operation involving the first electron (or hole) and the further electron (or hole). For example, the semiconductor device 10 enables a two-qubit operation on a first qubit associated with the first electron (or hole) and a further qubit associated with the further electron (or hole). The two-qubit operation may be used to perform a quantum algorithm. In order to achieve such two-qubit operation, the at least one electron or hole, along with the current one of the plurality of quantum states, is shuttled along the at least one path 45 in a coherent manner. This enables coherently shuttling the at least one qubit, e.g., the at least one spin-qubit.

[0053] One way of determining a coherence of the shuttling of the current one of the plurality of quantum states (and of the at least one qubit, e.g., the spin-qubit) is detecting coherent oscillations between two quantum states of a plurality of quantum states.

[0054] For example, the plurality of quantum states may comprise a first quantum state, associated with the first electron (or hole), and a further quantum state, associated with the further electron (or hole). The first electron (or hole) may be arranged in the at least one moveable quantum dot QDm. The further electron (or hole) may be arranged in the loadable quantum dot QD1. In case the semiconductor device 10 is comprised in, or connected to, a quantum processor, the first quantum state may be associated with a first spin-qubit, and the further quantum state may be associated with a further spin-qubit.

[0055] The method according to the disclosure may comprise a step 100 of setting the field strength of the external magnetic field B. The field strength of the magnetic field determines a Zeeman splitting Ezbetween the spin states spin-up and spin-down (see FIG. 2). As explained above, if the field strength of the external magnetic field B equals Bvs, the anomalous behaviour may occur (see below). The field strength of the external magnetic field B may have, but is not limited to, values ranging from 0 T to 0.6 T.

[0056] The method according to the disclosure may further comprise a step 120 of setting a shuttling distance d for the moveable quantum dot QDm to be shuttled.

[0057] The method may further comprise a step 150 of forming the double quantum dot QD1, QDm at the gate electrodes G2 and SI (i.e., the finger of SI closest to the gate electrode G2). The step 150 of forming the double quantum dot QD1, QDm may comprise a step 242 of applying the individual voltages to the gate electrodes G2 and SI, to form the potential wells to generate the double quantum dot QDL, QDm.

[0058] The method according to the disclosure may further comprise a step 160 of laterally defining a lateral position (along the y-direction in FIG. 1) the double quantum dot QD1, QDm. The defined lateral position may be a starting position of a trajectory in the quantum well at the at least one path 45 for moving (shuttling) the at least one electron or hole.

[0059] The method may further comprise a step 180 of loading the at least one electron or hole, e.g., one or more electrons, into the double quantum dot QD1, QDm. The loading 180 may comprise loading the at least one electron or hole, e.g., the one or more electrons, into the loadable quantum dot QD1, for instance, two electrons may be loaded in the loadable quantum dot QD1. In this case, the quantum double dot QD1, QDm will be found in an occupancy state (n, m) = (2, 0), where n and m are the electron occupancy numbers of the loadable quantum dot QD1 and the at least one moveable quantum dot QDm, respectively. The two electrons (or holes) may provide the first spin-qubit and the second spin-qubit. The loading step 180 may include a step 182 of lowering the voltage applied to gate electrode Gl. The loading step 180 may further include a step 184 of raising the voltage applied to gate electrode Gl subsequent to the step 182 of lowering the voltage. Thereby, the reservoir LTop and the loadable quantum dot QD1 are first electrically coupled to let the two electrons access the loadable quantum dot QD1 and electrically subsequently decoupled.

[0060] The method according to the present disclosure may further comprise a step 200 of initialising the at least one electron or hole, e.g., the one or more electrons, loaded in the loadable quantum dot QD1. In the previous example, the two electrons loaded into the loadable quantum dot QD1 may be initialised to the spin singlet state S (see above), by waiting in an occupancy state equal to (n, m) = (2, 0) for approximately 1 ms.

[0061] In another aspect of the disclosure, the loading 180 may result in the occupancy state of the double quantum dot QD1, QDm being equal (n,ni) = (4, 0), i.e., in which four electrons are loaded in the loadable quantum QD1. In this aspect, the initialising 200 may result in the two of the four electrons (or holes) in the next-higher energy valley to initialise to the singlet state S.

[0062] The method according to the present disclosure may further comprise the step 240 of changing an occupancy state of the double quantum dot QD1, QDm. The step 240 of changing the occupancy state the double quantum dot QD1, QDm may further comprise applying the individual voltage to the gate electrode G3 to form an adjustable tunnellingbarrier (inter-dot tunnel coupling) between the loadable quantum dot QD1 and the at least one moveable quantum dot QDm.

[0063] For example, the method may comprise the step 242 of lowering the tunnel barrier by means of the gate electrode G3. The step 240 of changing the occupancy state may comprise the step 244 of adiabatically pulsing from the (2,0) occupancy state to the (1,1) occupancy state. The step 240 of changing the occupancy state may comprise the step 246 of raising the tunnel barrier by means of the gate electrode G3.

[0064] The method according to the present disclosure includes a step 260 of entangling the first quantum state, e.g., a first spin state, of the first electron (or hole) and further quantum state, e.g., a further spin state, of the further electron (or hole). For example, the first spinqubit, associated with the first electron (or hole), and the second spin-qubit, associated with the first electron (or hole), may be entangled. The two spin-qubits in the at least one moveable quantum dot QDm and the electron in the loadable quantum dot QD1 may thus form a spin singlet state S.

[0065] The first electron (or hole) and the further electron (or hole) may oscillate between the first quantum state and the further quantum state, e.g., the first spin state and the further spin state. The first spin state may be the spin singlet state S. The further spin state may be the spin triplet state, e.g., the spin triplet state To. The first electron (or hole) and the further electron (or hole) may thus oscillate between the spin singlet state S and the spin triplet state To (which oscillations are referred to as ST0oscillations). The ST0oscillations may result from a difference of the Zeeman energy at the loadable quantum dot QD1 and the at least one moveable quantum dot QDm. The difference may result from different electron ^-factors (i.e., a Ag) and Overhauser-energies due to hyperfine contact interaction (i.e., a AF]^). The electron ^-factor may depend on a valley state of the loadable quantum dot QD1 and / or a valley state of the at least one moveable quantum dot QDm. The electron ^-factor may depend on a confinement at the loadable quantum dot QD1 and / or a confinement at the at least one moveable quantum dot QDm. A further source of ^-factor variability may be spinorbit-valley coupling (i.e., a coupling of spin-orbit coupling and orbit-valley coupling), e.g., at an interface of the semiconductor heterostructure of the semiconductor device 10. An example of the interface may be, but is not limited to, a Si / SiGe interface. The electron g- factor may further vary along the at least one path 45 (in the x-direction in FIG. 1). The g-factor may further vary in a direction lateral to the shuttling direction (in the j'-di recti on in FIG. 1). This spatially varying ^-factor results in a spatially varying difference Ag(x, y) of the ^-factor between the loadable quantum dot QD1 and the at least one moveable quantum dot QDm.

[0066] The spatially varying difference Eg(x, y) leads to the Zeeman energy difference EZof the entangled spins depending on a shuttling position x along the at least one path 45 of the at least one moveable quantum dot QDm. Likewise, a frequency vST()of the ST0oscillations depend on the shuttling position x along the at least one path 45 of the at least one moveable quantum dot QDm. During a shuttling of the at least one quantum dot, the frequency vST()may change depending on the shuttling position x and a shuttling distance d. The frequency vST()may be calculated by an averaging along the shuttling distance d:1 rdVST0(d) = — J0dxAEz (x) . Furthermore, the spatially varying difference Eg (x, y) of the g- factor, leads to the Zeeman energy difference EE2depending on the position y (i.e., the lateral shift between the screening gates Fl, F2).

[0067] In the event that the external magnetic field B equals a field strength Bvssuch that the Zeeman energy ggBBvsresulting from the field strength Bvsequals the valley splitting Evsbetween the spin states spin-up and spin-down, i.e., Evs= ggBBvs, the frequency vSTf)of the ST0oscillations may feature anomalous behaviour due to a spin-valley-coupling.

[0068] The two energy diagrams shown in FIG. 2 illustrate the spin-valley coupling. The two energy diagrams show an effect of the spin-valley coupling on eigenenergies of low- energy states of the plurality of quantum states. Each of the two energy diagrams shows four lines. In both energy diagrams, the two upper lines correspond to the two spin states spin-up and spin-down in a higher-energy valley state (higher-energy spin-valley states). These two upper lines illustrate the dependency of the energies of the higher-energy spin-valley states on the field strength of the external magnetic field B. The two upper lines originate from the same energy level at a value of the field strength of the external magnetic field B equal to zero, which reflects the degeneracy of the higher-energy spin-valley states when the external magnetic field is absent.

[0069] Furthermore, in both energy diagrams, the two lower lines correspond to the two spin states spin-up and spin-down in a lower-energy valley state (lower-energy spin-valleystates). These two lower lines illustrate the dependency of the energies of the lower-energy spin-valley states on the field strength of the external magnetic field B. The two lower lines originate from the same energy level at a value of the field strength of the external magnetic field B equal to zero, which reflects the degeneracy of the lower-energy spin-valley states when the external magnetic field is absent.

[0070] The spin-valley coupling occurs at an anticrossing (avoided crossing) shown in either of the two energy diagrams of FIG. 2. The anticrossing is also termed hot spot. The anomalous behaviour, which results from the spatially varying difference g(x, y) of the g- factor, is due to the anti crossing. At the anti crossing, the spin-up lower-energy spin-valley state and the spin-down higher-energy spin-valley state have approximately the same energy levels. The closeness of the energy levels facilitates mixing of the spin-valley states. The mixing may result in transitions between the spin-valley states. In these transitions, both the valley state and the spin state of the corresponding at least one electron (or hole) may change. A likelihood of relaxation of the quantum state of the at least one electron (or hole) is increased.

[0071] The anomalous behaviour comprises breakdown of the entanglement of the first quantum state, e.g., the first spin state, and the second quantum state, e.g., the second spin state. This breakdown may result from a dephasing of the first spin state or second spin state, depending on which of the associated first electron (or hole) and second electron (or hole) is located at the hot spot.

[0072] The insets of FIG. 2 illustrate two examples of a valley energy landscape, to which either the loadable quantum dot QD1 or the at least one moveable quantum dot QDm is exposed, when the field strength of the external magnetic field B is equal to, or in the vicinity of, Bvs. The two diagrams represent to different situations of the valley energy landscape, in which the valley splitting has a different value (see value of Evs at B = 0, as indicated by the double-headed arrow). In the upper diagram of FIG. 2, e.g., showing the situation for the moveable quantum dot QDm, the field strength Bvshas a lower value compared with the lower diagram, e.g., showing the situation for the loadable quantum dot QD1.

[0073] In a step 300 of the method, the at least one movable quantum dot QDm is moved by the shuttling distance d along the at least one path 45 to a position Xd. The moving (shuttling) 300 of the at least one movable quantum dot QDm is performed by means of the above-described travelling potential wells. The shuttling begins at an initial position xo of the at least one moveable quantum dot QDm, in which the loadable quantum dot QD1 and the at least one moveable quantum dot QDm form the double quantum dot QD1, QDm.

[0074] Based on the shuttling distance d, the one or more shuttling gate electrodes SI, S2, S3, S4 are provided with the time-varying (i.e., non-stationary) ones of the voltages. A momentary phase <p(t) = 2nf(t — t0) allows to estimate a current position x(t) of moveable quantum dot QDm based on the formula x(t) = f(t — t0), where is the conveyor gate pitch or spacing of the conveyor gates SI, S2, S3, S4, and f is the frequency of the applied time-varying ones of the voltages. The momentary phase <p(t) thus allows to determine a time tdwhen the at least one moveable quantum dot QDm has been moved (shuttled) to a position x(td) corresponding to the distance d. A shuttling speed v = f may be larger than 1 m / s, e.g., 2.8 m / s or 10 m / s, but are not limited thereto.

[0075] The method may further comprise a step 400 of evolving the first quantum state and the second quantum state, which are entangled. For example, in case the semiconductor device 10 is comprised in, or connected to, a quantum processor, the entangled first qubit and second qubit may evolve. The quantum state and second quantum state may evolve for an evolution period Tev. The evolution period Tevmay be, for example, 300 ns long, but is not limited thereto. During the evolution period Tev, the first quantum state may be associated with the first electron (or hole), which may be located in the loadable quantum dot QD1. During the evolution period Tev, the second quantum state may be associated with the further electron (or hole), which may be located in the moveable quantum dot QDm.

[0076] The method may further comprise a step 500 of returning the further electron (or hole) to the double quantum dot QD1, QDm. During the step 500, the at least one moveable quantum dot QDm is moved from the position x(td), corresponding to the shuttling distance < , back to the initial position x0. The time-varying voltages applied to the one or more shuttling gate electrodes SI, S2, S3, S4 may be time-reversed ones of the time-varying voltages which are applied to the one or more shuttling gate electrodes SI, S2, S3, S4 in step 300. When the at least one moveable quantum dot QDm has returned to the initial position x0, the double quantum dot QD1, QDm has been reformed. The double quantum dot QD1, QDm will be found in the occupancy state (1,1).

[0077] The method further comprises a step 600 of reversing the occupancy state of the double quantum dot QD1, QDm from the occupancy state (1,1) to the occupancy state (2,0).

[0078] The step 600 of reversing the occupancy state the double quantum dot QD1, QDm may comprise applying the individual voltage to the gate electrode G3 to form an adjustable tunnelling barrier (inter-dot tunnel coupling) between the loadable quantum dot QD1 and the at least one moveable quantum dot QDm.

[0079] For example, the step 600 may comprise the step 602 of lowering the tunnel barrier by means of the gate electrode G3, to let the second electron or hole tunnel from the at least one moveable quantum dot QDm to the loadable quantum dot QD1.

[0080] The method may further comprise a step 700 of measuring the occupancy state of the loadable quantum dot QD1. The measuring of the occupancy state may include using Pauli spin blockade (PSB). Using the PSB may include pulsing the electron-handling electrodes Gl, G2, G3, used as plunger gates and / or barrier gates. The pulsing may result in changing of energy levels in the loadable quantum QD1 and / or the at least one moveable quantum dot QDm. As explained above, the SET, formed by the gates LP, LB1, LB2 together with the current path LTop, may be used as a charge sensor sensitive to the occupancy of the loadable quantum dot QD1. When the PSB is used, the spin states, e.g., the spin-up and the spin-down, manifest as charge states. Depending on the measured charge, it may be determined whether the spin singlet state or the spin-triplet state is found. It may furthermore be determined whether the entanglement of the two spin states has been preserved.

[0081] In a further step 800, some or all of the afore-mentioned steps of the method according to the disclosure are repeated. The repeating allows to generate a statistically valid result. In one example, the method may be repeated 50,000 times. In another example, the method may be repeated 100 times. The repeating 800 may comprise maintaining the set field strength of the external magnetic field B or altering the field strength of the external magnetic field B. The repeating 800 may comprise maintaining the shuttling distance d or altering the shuttling distance d.

[0082] In a further step 900, the method according to the disclosure may further comprise identifying the valley splitting. The identifying 900 of the valley splitting may comprise detecting the anomalous behaviour in data based on the repeating 800 of the measuring 700 of the occupancy state of the of the loadable quantum dot QD1. The identifying 900 of thevalley splitting may comprise detecting the anomalous behaviour in the STo oscillations that may be detected in the data.

[0083] An example of a measurement of the anomalous behaviour is indicated in FIG. 3 by means of the arrows. FIG. 3 shows a grey-scale-coded landscape of a singlet probability Psof measuring the entangled first quantum state, e.g., the first spin state, and the second quantum state, e.g., the second spin state, to be in the singlet state S. The landscape shows the singlet probability Psas a function of time (along the horizontal axis; measured in nanoseconds (ns)) and of the external magnetic field B (along the vertical axis; measured in Tesla (T)). The singlet probability Psis shown in FIG. 3 with reference to its mean value Ps), i.e., the grey scales correspond to values of Ps— Ps). It can be seen that for times up to between approximately 800 ns and 1200 ns (depending on the field strength of the magnetic field) the measured singlet probability exhibits a pattern (singlet probability pattern). The pattern reflects temporal oscillations of the singlet probability Psfor a given value of the field strength of the external magnetic field B (along a horizontal direction in FIG. 3). The two arrows indicate areas where the singlet probability pattern is interrupted and the oscillatory behaviour of the singlet probability Pschanges abruptly. These two areas are associated with the loadable quantum dot QD1 and the at least one moveable quantum dot QDm, respectively. Which of the two areas is associated with which of the loadable quantum dot QD1 and the at least one moveable quantum dot QDm, requires further experimental steps (i.e., the shuttling of the at least on electron or hole) described with respect to FIG. 4. The two areas indicated that the values of the valley splitting at the loadable quantum dot QD1 and the at least one moveable quantum dot QDm are different. In case one single area that interrupts the singlet probability pattern is found, the values of the valley splitting at the loadable quantum dot QD1 and the at least one moveable quantum dot QDm are substantially the same.

[0084] The measurements leading to the results shown in FIG. 3 involve the above-described steps 100, 150, 160, 180, 200, 240, 260, 400, 600, 700, and 800.

[0085] FIG. 4 shows measurements of the singlet probability Pswith reference to its mean value Ps), i.e., Ps— Ps), as a function of the field strength of the external magnetic field B (along the vertical axis; measured in Tesla (T)) and of the shuttling distance d (along the horizontal axis; measured in nanometers (nm)). In the case shown in FIG. 4, the lateralposition (in the -direction, see FIG. 1) of the double quantum dot QD1, QDm is set symmetrically, i.e., to a middle position (e.g., y = 0) between the one or more laterally positioning gate electrodes Fl, F2.

[0086] The measured singlet probability shown in FIG. 4 exhibits a pattern, which in several areas has horizontal structures. Two of the horizontal structures in FIG. 4 correspond to the two areas of FIG. 3 (indicated by the two arrows) where the singlet probability pattern is interrupted. In the case of FIGS. 3 and 4, the two horizontal structures of FIG. 4 and the two areas of FIG. 3 are associated with slightly different values of the field strength of the external magnetic field B (vertical axis) due to different DC tunings in the corresponding experiments. These horizontal structures of FIG. 4 corresponding to the two areas in FIG. 3 appear in FIG. 4 across the distances d (the horizontal axis of FIG. 4) because the method according to the disclosure that results in FIG. 4 begins with an arrangement of the first electron or hole and the second electron or hole, in which the first electron or hole and the second electron or hole are located in the double quantum dot QD1, QDm. For corresponding ones of the values of the field strength of the magnetic field , the valley splitting therefore shows up.

[0087] The two areas shown in FIG. 3 may be assigned to the loadable quantum dot QD1 and the at least one moveable quantum dot QDm, respectively, based on results shown in FIG. 4. The lower horizontal structure shows branching structures that branch off from the lower horizontal structure (these are indicated in FIG. 5 by the black dots). As the at least one moveable quantum QDm is moved (shuttled) along the at least one path 45, it is expected, based on the assumption that the semiconductor device 10 is not homogeneous, that the valley splitting changes when the at least one moveable quantum dot QDm is moved (shuttled). The changing valley splitting is represented in FIG. 4 by the branching structures (indicated in FIG. 5 by the black dots).

[0088] The upper horizontal structure of FIG. 4 shows a single sine-shaped structure that branches off in a downwards direction (towards lower values of the field strength) from the upper horizontal structure (at the distance d ~ 50 nm) and recombines with the upper horizontal structure (at the distance d ~ 225 nm). This sine-shaped structure is attributed to crosstalk of the loadable quantum dot QD1 (the stationary quantum dot) and the at least one moveable quantum dot QDm (the shuttling quantum dot).

[0089] Vertical structures shown in FIG. 4 are attributed to positions (x, y) where Eg between the loadable quantum dot QD1 and the at least one moveable quantum dot QDm is equal to zero.

[0090] The measurements leading to the results shown in FIG. 4 involve the above-described steps 100, 120, 150, 160, 180, 200, 240, 260, 300, 400, 500, 600, 700, and 800.

[0091] As explained above, FIG. 5 shows, for the case of the measurements of FIG. 4, the identified field strengths Bvsof the external magnetic field 7>, for which the Zeeman energy equals the valley splitting, i.e., Evs= g^B^vs- The identified field strengths Bvsare indicated by the black dots. In the case of FIG. 5, the identified field strengths Bvscorrespond to the positions (x,y) = (d, 0).

[0092] FIG. 6 shows similar results as FIG. 5, with the lateral position (y-position) of the double quantum dot QD1, QDm offset, i.e., y = y', with respect to the middle position (y = 0) between the one or more laterally positioning gate electrodes Fl, F2. The identified field strengths Bvsare indicated by the black dots. In the case of FIG. 6, the identified field strengths Bvscorrespond to the positions (x, y) = (d,y').

[0093] In one aspect of the disclosure, the identifying of the field strength Bvsmay be done by means of a computer.

[0094] Based on the identified field strength Bvs, the method of determining the valley splitting results in a valley splitting across the semiconductor device 10 (or across the at least one path 45). This may also be regarded as a spatial mapping of the valley splitting. Based on the determined spatial mapping of the valley splitting, a value of the field strength of the external magnetic field B may be selected that allows for high-fidelity shuttling of the at least one electron or hole along the at least one path 45. For example, in one aspect of the disclosure, the selected value of the field strength of the external magnetic field B may be chosen to be smaller than the identified field strength Bvsfor all positions (x,y), at which the method according to the disclosure was performed. In another aspect, in the case of low values of the identified field strength Bvsat one or more so-called hot spots (HS,yHS), a trajectory along the at least one path 45 may be chosen that circumvents the one or more hot spots xHS, yHS).

[0095] FIG. 7 shows a semiconductor device 11 according to an aspect of the disclosure, which comprises several ones of the semiconductor device 10. Some of the laterally positioning gate electrodes Fl, F2, F3, F4 in this aspect act on two of the at least one path 45-1, 45-2, 45-3 on either side of the corresponding one of the laterally positioning gate electrodes Fl, F2, F3, F4. For simplicity, several ones of the plurality of gate electrodes Gl, G2, G3, SI, S2, S3, S4, arranged over the path 45-1, the path 45-2, and the path 45-3, respectively, are not shown. The semiconductor device 11 may, for example, be used as an electron pump (or hole pump) for delivering one or more electrons (or holes) to an output position (not shown). The semiconductor device 11 may be analysed, based on the method according to the disclosure, across several portions of a surface semiconductor device 11 with respect to the valley splitting.

[0096] For a selected one of the position (x,y), x < dmax,ymln< y < ymax, and a selected one of the field strength of the external magnetic field B, B < Bmax, a measurement may take around 3 ps.

[0097] The measurement may be made with a resolution in the x-direction and y-direction of 10 nm. The resolution may be chosen dependent on the correlation length of the valley splitting of approximately 20 nm (which corresponds to a typical size of the loadable quantum dot QD1 or the at least one moveable quantum dot QDm).

[0098] The measurement may further be made with a resolution in the field strength of the external magnetic field B of 10 mT, which corresponds to a valley splitting of approximately 1 peV.

[0099] Therefore, covering a surface (in x-direction and y-direction) of 10 pm * 100 nm of the semiconductor device 10, and covering field strengths up to 2 T results in a measurement time of approximately 10 min.

[0100] The method according to the disclosure allows to assess a quality of the semiconductor device 10. The allows to assess the semiconductor device 10 dependent on growth conditions, e.g., a growth direction, of the semiconductor device 10. The assessment enables optimising a manufacturing of the semiconductor device 10.

[0101] Subsequently, the semiconductor device 10 may be adjusted with respect to a value of the field strength of the external magnetic field B as well as to the voltages applied to the gate electrodes LP, LB1, LB2, Gl, G2, G3, SI, S2, S3, S4, Fl, F2.

Claims

Claims1. A method of determining a valley splitting of a semiconductor device (10,11), the method comprising- setting (100) a field strength of an external magnetic field B for splitting entangled spin states associated with a double quantum dot (QDl,QDm) generated in the semiconductor device (10, 11);- evolving (400) the entangled spin states for an evolution period (TA);- measuring (700) a state of the entangled spin states in a basis of the entangled spin states, the basis comprising at least two basis states;- repeating (800) the evolving (400) and the measuring (700) to determine a probability of the state of the entangled spin states being one of the at least two basis states; and- assessing, based on the determined probability, the valley splitting.

2. The method of claim 1, wherein the entangled spin states are associated with a first electron or hole and a second electron or hole, arranged in the double quantum dot (QD1, QDm).

3. The method of claim 1 or 2, further comprising arranging (160) the double quantum dot (QD1, QDm) at a lateral position (y) of the semiconductor device (10).

4. The method of any one of claims 1 to 3, further comprising moving (300, 500) at least one moveable quantum dot (QDm) of the double quantum dot (QD1, QDm) by a distance d along at least one shuttling path (45) of the semiconductor device (10).

5. The method of claim 4, wherein the moving (300) is performed before the evolving (400) of the entangled spin states.

6. The method of claim 4 or 5, wherein the moving (500) is performed after the evolving (400) of the entangled spin states.

7. The method of any one of claims 1 to 6, wherein the repeating (800) comprises maintaining the shuttling distance d or altering the shuttling distance d.

8. The method of any one of claims 1 to 7, wherein the repeating (800) comprises maintaining the field strength of the external magnetic field B or altering the field strength of the external magnetic field B.

9. The method of any one of claims 1 to 8, further comprising initialising (200) a first quantum state of the first electron or hole, and initialising (200) a second quantum state of the second electron or hole.

10. The method of any one of claims 1 to 9, further comprising measuring (700) the occupancy state of the loadable quantum dot (QD1).

11. The method of any one of claims 1 to 10, further comprising loading (180) the first electron or hole and the second electron or hole in the double quantum dot (QD1, QDm).

12. The method of any one of claims 1 to 11, further comprising identifying (900) a valley splitting by detecting anomalous behaviour the oscillations of the entangled spin states between the two basis states.

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