Manipulating and Imaging the Shape of an Electronic Wave Function by Magnetotunneling Spectroscopy A. Patane `, 1, * N. Mori, 2 O. Makarovsky, 1 L. Eaves, 1 M. L. Zambrano, 1,† J. C. Arce, 3 L. Dickinson, 1 and D. K. Maude 4 1 School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom 2 Department of Electronic Engineering, University of Osaka, 2-1 Yamada-Oka, Osaka 5650871, Japan 3 Departamento de Quı ´mica, Universidad del Valle, A.A. 25360, Cali, Colombia 4 Laboratoire National des Champs Magne ´tiques Intenses, CNRS-UJF-UPS-INSA, 38042 Grenoble, France (Received 2 August 2010; published 1 December 2010) We measure the current due to electrons tunneling through the ground state of hydrogenic Si donors placed in a GaAs quantum well in the presence of a magnetic field tilted at an angle to the plane of the well. The component of ~ B parallel to the direction of current compresses the donor wave function. By measuring the current as a function of the perpendicular component of ~ B, we probe how the magneto- compression affects the spatial form of the wave function and observe directly the transition from Coulombic to magnetic confinement at high fields. DOI: 10.1103/PhysRevLett.105.236804 PACS numbers: 73.21.b, 73.40.Gk, 74.50.+r, 74.55.+v The possibility of fine-tuning and imaging the electronic wave function in a quantum system is of fundamental interest and has potential for applications in quantum information processing and other advanced technologies [1,2]. This field is still in its infancy and requires the development of sensitive methods to probe and understand the nature of the quantum states. In many cases, our theoretical understanding is based on the solution of the Schro ¨dinger equation, which gives the eigenvalues and corresponding eigenfunctions [3]. Spectroscopic studies, on the other hand, focus traditionally on measuring tran- sition energies and intensities and their change in response to external perturbations. The spatial form of the wave functions is generally more difficult to probe. Although there have been successful examples of wave function imaging by scanning tunneling microscopy [4–8], magnetotunneling spectroscopy (MTS) [9,10], and magnetocapacitance-voltage spectroscopy [11–13], the ef- fect of an external force on the shape of a wave function has received less attention. In this Letter, we use MTS to probe in situ the spatial compression of a quantum state induced by an applied magnetic field. Our study focuses on the electronic wave function of a donor atom in a semiconductor. In GaAs, the donor eigen- states are well described by the hydrogenic effective mass approximation with an effective Bohr radius a B ¼ 10:1 nm. Hence the Hamiltonian for a donor and for a hydrogen atom in a magnetic field, ~ B, reduces to the same problem by using an effective magnetic field strength given by  ¼ B=B 0 , where B 0 ¼ B H ða 0 =a B Þ 2 , B H ¼ 4:7  10 5 T, and a 0 ¼ 0:053 nm is the hydrogenic Bohr radius [14]. In our MTS experiment, we use magnetic fields of up to B ¼ 22 T, which correspond to strengths of up to 10 6 T when scaled to the hydrogen problem. We study the current due to electron tunneling through the ground state of Si donors within a GaAs quantum well (QW) in a magnetic field tilted at an angle to the QW plane. The component of ~ B parallel to the direction of current provides us with a means of increasing the donor binding energy [15,16] and hence of compressing the donor wave function in the QW plane. By measuring the current as a function of the perpendicular component of ~ B, we probe how the magnetocompression affects the spatial form of the donor wave function. Our resonant tunneling diode structures were grown by molecular beam epitaxy on a n þ -GaAs substrate. Each sample consists of a 9-nm-wide GaAs QW embedded between two 5.7-nm Al 0:4 Ga 0:6 As tunnel barriers. The central plane of the QW is -doped with Si donors at 4  10 9 cm 2 corresponding to an average donor separation of 0:16 m, which is much larger than a B . Undoped GaAs spacer layers, each of width 21 nm, separate the Al 0:4 Ga 0:6 As barriers from an n-doped (2  10 16 cm 3 ) GaAs layer of width 51 nm, an n-doped (2  10 17 cm 3 ) GaAs layer of width 80.6 nm, and an n-doped (2  10 18 cm 3 ) GaAs layer. The samples were processed into circular mesa diodes of diameter between 10 and 200 m, with Ohmic contacts alloyed to the top and bottom doped GaAs layers. In the following, we define positive bias with the cap layer biased positive. Similar results were obtained in both bias directions. Figure 1(a) shows the current-voltage, IðV Þ, character- istics at T ¼ 4:2K. At low biases (< 0:05 V), the IðV Þ curve shows weak resonant features, whose bias position and intensity are sample-dependent. These are due to resonant tunneling of electrons through the states of Si- donor pairs, which exist in the QW due to the random positioning of the donors [17]. At higher bias, tunneling of electrons into the ground state of the QW-confined isolated Si donors generates a stronger resonant peak D [18]. At still higher bias voltages ( > 0:1V), electrons tunnel into the lowest energy subband E1 of the QW, thus leading to a PRL 105, 236804 (2010) PHYSICAL REVIEW LETTERS week ending 3 DECEMBER 2010 0031-9007= 10=105(23)=236804(4) 236804-1 The American Physical Society