Although the Cu doped Bi 2 Se 3 topological insulator was discovered and intensively studied for almost a decade, its electrical and magnetic properties in normal state, and the mechanism of 'high-T c ' superconductivity regarding the relatively low-carrier density are still not addressed yet. In this work, we report a systematic investigation of magnetic susceptibility, critical fields, and electrical transport on the nominal Cu 0.20 Bi 2 Se 3 single crystals with T onset c = 4.18 K, the highest so far. The composition analysis yields the Cu stoichiometry of x = 0.09(1). The magnetic susceptibility shows considerable anisotropy and an obvious kink at around 96 K was observed in the magnetic susceptibility for H c, which indicates a charge density anomaly. The electrical transport measurements indicate the two-dimensional (2D) Fermi liquid behavior at low temperatures with a high Kadowaki-Woods ratio, A/γ 2 = 30.3a 0 . The lower critical field at 0 K limit was extracted to be 6.0 Oe for H ab. In the clean limit, the ratio of energy gap to T c was determined to be Δ 0 /k B T c = 2.029 ± 0.124 exceeding the standard BCS value 1.764, suggesting Cu 0.09 Bi 2 Se 3 is a strong-coupling superconductor. The in-plane penetration depth at 0 K was calculated to be 1541.57 nm, resulting in an unprecedented high ratio of T c /λ −2 (0) ∼ = 9.86. Moreover, the ratio of T c to Fermi temperature is estimated to be T c /T 2D F = 0.034. Both ratios fall into the region of unconventional superconductivity according to Uemura's regime, supporting the unconventional superconducting mechanism in Cu x Bi 2 Se 3 . Finally, the enhanced T c value higher than 4 K is proposed to arise from the increased density of states at Fermi energy and strong electron-phonon interaction induced by the charge density instability. 1 K, such as SrTiO 3 (T c ∼ 0.05-0.29 K, n e = 0.65 to 42.4 × 10 20 cm −3 ) [3,4], GeTe (T c ∼ 0.07-0.3 K, n e = 8.6 to 15.2 × 10 20 cm −3 ) [5], and SnTe (T c ∼ 0.034-0.214 K, n p = 7.5 to 20.0 × 10 20 cm −3 ) [6]. To compare with these systems, the T c value of Cu x Bi 2 Se 3 is relatively 'high' given it is a highly doped narrow semiconductor with lower carrier density. However, the mechanism of such anomalous enhanced T c phenomenon remains unclear despite nearly a decade of extensive research. © 2020 The Author(s). Published by IOP Publishing Ltd on behalf of the Institute of Physics and Deutsche Physikalische Gesellschaft New J. Phys. 22 (2020) 053026 Y Fang et alThe emergence and enhancement of superconductivity may occur in various cases by tailoring the ground states of quantum materials. For example, suppression of charge density wave (CDW) by chemical doping or pressure is observed in many systems [7-10], including low dimensional layered Cu x TiSe 2 and three dimensional (3D) alloys Lu(Pt 1−x Pd x ) 2 In and (Sr, Ca) 3 Ir 4 Sn 13 , 3D oxides Ba 1−x K x BiO 3 . Similarly, the magnetic ordering or spin density wave (SDW) have also been suppressed in strongly correlated heavy fermions materials, high-T c cuprates and i...