## Main

The interplay of electron–electron interactions, disorder and spin correlation in solids is the origin of many competing ground states and phase transitions between them1,2,3, which are typically observed in strongly correlated complex materials ranging from heavy-fermion compounds4 to high-temperature superconductors5,6. An intriguing question in complex interacting systems is how the interactions of microscopic particles lead to macroscopic phenomena such as superconductivity, charge and spin density waves and heavy fermions. The Kondo effect very often plays a central role in understanding the correlated ground states of electron systems and electrical transport (Fig. 1). Doping is a versatile tool with which to address this question because of its ability to control the interaction strength and the way particles interact with each other and with external perturbations such as magnetic fields and pressure7,8.

Here, we report observations of the Kondo interaction and an exotic Bardeen–Cooper–Schrieffer (BCS)-type pseudogap in highly P-doped degenerate silicon (Si:P) with doping concentrations of n ≈ 2−5 × 1019 cm−3 at very low temperatures (Methods). The existence and formation of magnetic moments in metallic Si:P are well understood in terms of Anderson’s picture and well explained in the pioneering works of Bhatt and Lee and others9,10,11,12. Because of the high doping concentration (higher than ~2 × 1019 cm3), the Fermi energy (EF) in Si:P lies in the conduction band13, as it does in a metal (Supplementary Fig. 4a). In these circumstances, it is very likely that the local moments are entangled with the conduction electrons to form micrometre-sized Kondo clouds14,15 that overlap with each other (Fig. 1b), leading to a correlated ground state in the Si:P metal. In this work, tunnelling density-of-states (DOS) spectroscopy and bulk electrical transport measurements allow the electronic and magnetic nature of competing ground states in the Si:P metal to be explicitly identified and the relevant phase diagram in the temperature–magnetic field (TB) plane to be constructed.

We first measured the differential resistance Rd of a bulk Si:P metal as a function of T and B. At B = 0 G, Rd exhibited strong temperature dependence at high temperatures (Fig. 2a). As T decreased below approximately 2 K, a clear anomaly appeared as a broadened step-like increase in Rd with decreasing T, and then it decreased slightly below ~1 K. As only a limited temperature range was available, the observed step-like increase is difficult to describe analytically but the ln(T) behaviour (blue line) may be relevant. The temperature dependence of the resistance is reminiscent of that of a Kondo lattice compound with disorder16. The increase in ln(T) and the subsequent decrease in the resistivity of heavy-fermion compounds is attributed to Kondo scattering and the crossover to a coherent state of conduction and localized electrons. A modest magnetic field switches the non-Fermi liquid metal state to a conventional disordered Fermi liquid (DFL), confirming that the observed resistivity anomaly below 2 K is associated with magnetic interactions. On the other hand, the B-dependent Rd at 10 and 77 mK exhibited a small dip in the low-magnetic-field region (Fig. 2b), supporting the presence of the coupling of the local magnetic moments and conducting electrons. A similar magnetoresistance feature is observed in Ce-based heavy-fermion compounds and is attributed to magnetic quantum criticality17.

In fact, the interplay between the competing interactions may switch the ground state of the Si:P metal. The competition between the ground states can be elucidated by observing the evolution of tunnelling DOS spectra near EF of the Si metal at various T and B. For this purpose, we fabricated tunnel junction devices consisting of the Si:P metal with silver (Ag) electrodes separated by a thin SiO2 tunnel barrier for the tunnelling DOS spectroscopy measurements (Fig. 2c and Methods). We also conducted experiments with Al–SiO2–Si:P tunnel junctions that revealed the expected Al superconducting energy gap and proved that tunnelling was the major transport mechanism in such structures. Because Ag behaves similarly to a Fermi liquid with a flat DOS region in the vicinity of EF, the measured differential tunnelling conductance G is directly proportional to the DOS of Si:P: G(V)  DOS(E), where E = EF + eV (V is the voltage across the tunnel barrier and e is the elementary charge). Some of the basic results measured below 200 mK are presented in Fig. 2d,e, which displays the measured GV characteristics at various temperatures at B = 0 G and an intensity plot of G in the TV plane, respectively. At T = 18 mK, a partial depletion of the tunnelling DOS near EF, which may be referred to as a pseudogap, is seen in the GV characteristics near V = 0 V, and anomalous peaks appear outside the pseudogap. As T increases, the depleted DOS within the pseudogap is restored, and the peaks become closer in a nonlinear manner. Above ~160 mK, the U-shaped pseudogap with the side peaks changes into a V1/2-type Altshuler–Aronov gap (also called a zero-bias anomaly (ZBA))18, which is anticipated for metallic bulk Si with disorder and Coulomb correlations in the paramagnetic DFL phase.

Next, we present results that show how the anomalous DOS spectrum at EF varies with B at T = 18 mK. Figure 3a,b presents examples of the measured GV characteristics at various B below 3,000 G and an intensity plot of G in the BV plane. As B increases, the peaks become closer together, and the depth and width of the U-shaped pseudogap decrease. Most noticeably, the pseudogap and peaks smoothly change into the Altshuler–Aronov ZBA of the paramagnetic DFL phase. Afterwards, the |V|1/2-dependent GV curve remains unchanged and independent of B. Interestingly, in the intermediate B region between approximately 1,000 and 2,000 G, G(V) increases linearly as a function of V up to the position of the considerably smaller peaks and then remains constant. Thus, this |V|1-dependent GV curve has a V-shaped groove, which is distinct from the ZBA. The inset in Fig. 3a shows a comparison of this V-shaped pseudogap and the ZBA. The depth and width of the V-shaped pseudogap continue to decrease with increasing B and T (Supplementary Fig. 5); however, its dependence on B is much weaker than that of the U-shaped pseudogap in the low-B region. We have confirmed that the observed anomalous DOS spectra are not hysteretic in B by sweeping B in both directions. Thus, the observed smooth B-driven phase transition, together with the electrical properties of a bulk Si:P metal, reveal that an exotic magnetically correlated state (which is not generally anticipated in a simple elemental semiconductor) seems to exist in the degenerately doped Si:P metal. T- and B-dependent characteristics of the pseudogap and anomalous peaks are presented in the Supplementary Figs. 6 and 7.

Nonlinear behaviour is observed in the B-dependent restoration of the DOS at EF (G at V = 0 V) at various T from 18 mK to 160 mK. The smooth magnetic phase transition from the magnetic pseudogap phase to the paramagnetic DFL phase is demonstrated in the intensity plot for the zero-bias differential conductance, G(V,B), at 18 mK for each value of B (Supplementary Fig. 8). In the low-T region below approximately 100 mK, the DOS minimum (pseudogap) at EF is reinstated superlinearly with B and then sublinearly up to the value of B (Supplementary Fig. 9a) at which the non-magnetic |V|1/2-dependent ZBA starts to appear in the DFL phase. There exists a clear inflection point (marked with the vertical arrows in the inset of Supplementary Fig. 9a) in the derivative of the G(V = 0 V)–B curve, and we regard this point as the boundary between the superlinear (strongly B-dependent) and sublinear (weakly B-dependent) regions. As T increases, the inflection B point shifts to lower B values, making a boundary in the TB plane (black diamonds in Supplementary Fig. 9b). The measured tunnelling conductance of Al–SiO2–Si:P devices confirms a strong modification in the DOS near EF of the Si:P metal (Supplementary Fig. 10).

The TB phase diagram of the Si:P metal shown in Fig. 4 summarizes our main experimental results. Two different phases are clearly visible in the intensity map of the DOS at EF in the TB plane with an intervening region. The blue region corresponds to the magnetic metal phase where the U-shaped pseudogap and large anomalous DOS peaks outside the gap appear. The black diamonds show the inflection point in the derivative of the G(V = 0 V)–B curve as a function of T. In this phase, the DOS spectrum at EF (G at V = 0 V) changes strongly as B varies. The red stars represent the temperature at which the non-magnetic |V|1/2-type ZBA is detected first, T*, at different B, creating the most important characteristic boundary between the magnetically correlated metal phase and the non-magnetic DFL phase. The magnetic metal phase transforms smoothly into the paramagnetic DFL phase (yellow region) through the intervening weakly magnetic metal phase (green region), where the V-shaped DOS spectrum appears with considerably weaker DOS peaks. In this intervening phase, the depleted DOS at EF is reinstated relatively weakly with increasing B. The boundary consisting of the green circles indicates the temperature at which the anomalous DOS peaks are first detected (TΔ; Supplementary Fig. 6).

Observations of the abnormal electronic DOS spectra in the Si:P metal and their B-driven tuning are very surprising because neither the host Si nor the dopant P show a magnetic order in the crystal structure, and degenerate Si:P is regarded as a metal (EF is located within the conduction band). We ruled out several possible mechanisms that could be responsible for our exotic DOS spectra. First, we ruled out a Coulomb gap scenario because it cannot explain the magnetic behaviour. Second, we ruled out spin glass-type magnetic gaps19 because no noticeable differences were seen between the field-cooled and zero-field-cooled DOS spectra. Neither of these mechanisms can explain the existence of DOS peaks outside the gap or the observed non-Fermi-liquid behaviours in the resistance of bulk Si:P. We also eliminated the possibility of the Kondo lattice gap because the local moments in our sample were randomly distributed. Indeed, our DOS was highly symmetric, while the DOS of the Kondo lattice system has a characteristic asymmetry20. In fact, Si:P is not a heavy-fermion system. Third, the ground state with RKKY interaction-induced magnetic order, for example, the random singlet state (Fig. 1d), is not relevant because it does not have any gap at EF (ref. 21). In addition, the spin density wave caused by the RKKY interaction may produce a BCS-like gap at EF but the preliminary condition for such a mechanism is the lattice translation symmetry, which is absent in our Si metal due to random impurities. Finally, unlike boron (B)-doped degenerate Si (ref. 22), superconductivity has not been observed in P-doped Si and a clear signature of superconductivity (a collapse of the bulk resistance) was not also detected in our Si metal down to 10 mK. However, we do not entirely exclude the occurrence of incipient or hidden superconductivity.

All of the T- and B-dependent bulk transport, the DOS spectroscopy measurements and the existence of magnetic moments suggest that Kondo physics plays a key role in the formation of a correlated electron ground state in the Si:P metal. The underlying physics of the correlation behaviour is proposed here. That is, for a Si:P metal with a doping density of ~3 × 1019 cm3, more than a 10−5 fraction of the total impurities induce residual, unscreened localized moments9, and the mean distance between the moments is less than 1 μm, which may be comparable to (or even smaller than) the size of a Kondo cloud. Our proposed ground state is the condensation of overlapping Kondo clouds (Fig. 1b), and as a consequence of Kondo condensation, a fraction of itinerant electrons entangled with magnetic impurities are correlated to form a many-body ground state. This Kondo condensation model is analogous to a Bose–Einstein condensate and, similar to BCS Cooper pairs, a singlet ground state with a small BCS-like gap is formed in the DOS at EF. In fact, the shape and behaviour of the observed pseudogap in the Si:P metal is very similar to a BCS-like gap. In the lower doping regime where the metal–insulator transition occurs, this discovered ground state has not been reported. This is presumably because, although the density of magnetic impurities may be higher in this regime, the density of the itinerant electrons captured by the magnetic impurities to form Kondo clouds is not sufficiently high. This is the reason why the observed Kondo phenomenon occurs in the degenerate metal regime, rather than the impurity band regime. To sum up, based on the tunnelling DOS and bulk transport measurements, as well as the fact that magnetic moments are present in the Si:P metal, the proposed microscopic picture responsible for the BCS-like DOS spectrum is Kondo condensation (that is, the overlap of randomly distributed Kondo clouds) and the mechanism for the zero-bias anomaly at elevated temperatures and magnetic fields is electron–electron interaction effects. Thus, the relevant balance between these competing ground states, which leads to a magnetic phase boundary, is controlled by the temperature and magnetic field—as demonstrated by the DOS in the TB domain.

Understanding many-impurity Kondo physics at a microscopic level remains challenging, especially for disordered solids. In particular, the Kondo effect is a many-body interaction, so it would be inappropriate to apply a single-electron picture. The usual way to tackle the disorder effect is to consider the distribution of the Kondo temperature, which means that the system is described as a sum of independent subsystems with some probability of a given coupling. This is analogous to treating the system as a mixed state, which is an approximation that is only valid when the individual local components of the system (that is, individual Kondo clouds) do not correlate with each other. In such cases, the system can be treated as an ensemble of subsystems with random couplings, thus treating the disorder by summing over the random couplings9. However, the result of such approach is the scaling law in T, which seems to contradict our experimental finding: the observation of the BCS-like pseudogap that is convincing evidence for macroscopic coherence.

The Kondo condensation scenario proposed here can be used to reproduce the observed DOS spectra by considering the correlation effect on the spectrum of itinerant fermions. For this purpose, we introduce the charge neutral scalar field $$\varphi = < c_ \uparrow f_ \downarrow ^ + - c_ \downarrow \,f_ \uparrow ^ + >$$, where c is an itinerant electron and f+ is an ion impurity with a net spin of 1/2. Kondo condensation is the configuration in which φ is non-vanishing. If we consider φ as a constant part of field φ(x), its coupling to the fermion can describe the effects of Kondo condensation on the spectrum of the fermion23. When T or B increases sufficiently, the condensation is destroyed by thermal fluctuations or by Zeeman flips; thus, the pseudogap will disappear, and the system will undergo a transition from a gapped scalar ordered state with Kondo condensation to a paramagnetic DFL state.

The calculation for the DOS spectra of this system encounters two sources of difficulty in many-body theories; that is, randomness and strong correlations. A mean-field description of the system that works even for a strongly interacting system would be helpful. One such theory is the holographic theory24,25,26, the validity of which relies on the universality of the system near the quantum critical point. The principle of holography works even after the scale symmetry is slightly broken27, similar to our situation where the scale symmetry is broken due to the presence of Kondo condensation (Supplementary Fig. 9). Figure 5a reveals the modelled pseudogap in the DOS and two DOS peaks at opposite energies, calculated at B = 0 V. A gap is clearly visible in the DOS at T = 18 mK, whereas for a larger value of T > 160 mK, the pseudogap is closed, which is in qualitative agreement with the experimental GV curve at B = 0 G and T = 18 mK (Fig. 2d). Because there is a non-zero DOS value at EF (V = 0 V), the corresponding state is a metal. For larger magnetic fields, no strong pseudogap is present in the computed DOS (Fig. 5b). The calculated magnetic-field-dependent DOS spectra are qualitatively similar to the experimental GV curves (Fig. 3a). In addition to our holographic model, it will be worth checking the reproducibility of our results using other theoretical approaches (Supplementary Information 10).

In conclusion, similarly shaped DOS spectra at EF have been observed in cuprate, pnictide and heavy-fermion superconductors5,6,28,29. In fact, the TB phase diagram of our Si:P metal in Fig. 4 is also similar to that of the quantum materials listed above. This result is not a surprise because the observed pseudogap behaviours in these materials can be commonly explained by coherent electronic states with a correlation. Although a microscopic understanding of how the order parameter depends on external perturbations such as T, B, pressure, and doping is still challenging, the observation of Kondo condensation and its phase transition in degenerately doped elemental semiconductor Si will contribute to understanding other quantum materials such as Kondo lattices, spin glasses and high-TC superconductors. It would also be interesting to experimentally probe the Hall coefficient and shot noise near the quantum critical point, which are expected to display non-Fermi liquid physics30,31,32.

## Methods

### Sample preparation and characterization

The structural, electrical and magnetic properties of bulk Si:P metal were studied using transmission electron microscopy, secondary ion mass spectrometry, magnetoresistance and superconducting quantum interference device measurements (Supplementary Figs. 13). Tunnel junction devices were fabricated on a [100]-oriented Si substrate with a P concentration of ~2.6 × 1019 cm3. The SiO2 tunnel barrier was formed by thermal oxidation of the Si:P layer. The tunnel junction device was then completed by depositing a 200-nm-thick top Ag or Al electrode.